%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP098^2 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n024.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:32 AM UTC 2026
% Result : Theorem 2.25s 1.28s
% Output : Refutation 2.25s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.06 % Problem : ITP098^2 : TPTP v9.2.1. Released v7.5.0.
% 0.00/0.07 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.24 % Computer : n024.cluster.edu
% 0.07/0.24 % Model : x86_64 x86_64
% 0.07/0.24 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.24 % Memory : 8042.1875MB
% 0.07/0.24 % OS : Linux 3.10.0-693.el7.x86_64
% 0.07/0.24 % CPULimit : 300
% 0.07/0.24 % WCLimit : 300
% 0.07/0.24 % DateTime : Tue Feb 24 03:26:10 EST 2026
% 0.07/0.25 % CPUTime :
% 0.07/0.25 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.07/0.31 ---- Original DTF file ---
% 0.07/0.31 thf(spec,logic,$$dhol).
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % File : ITP098^2 : TPTP v9.2.1. Released v7.5.0.
% 0.07/0.31 % Domain : Interactive Theorem Proving
% 0.07/0.31 % Problem : Sledgehammer ListInf problem prob_167__5410008_1
% 0.07/0.31 % Version : Especial.
% 0.07/0.31 % English :
% 0.07/0.31
% 0.07/0.31 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.07/0.31 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.07/0.31 % Source : [Des21]
% 0.07/0.31 % Names : ListInf/prob_167__5410008_1 [Des21]
% 0.07/0.31
% 0.07/0.31 % Status : Theorem
% 0.07/0.31 % Rating : 0.00 v8.1.0, 0.25 v7.5.0
% 0.07/0.31 % Syntax : Number of formulae : 325 ( 86 unt; 32 typ; 0 def)
% 0.07/0.31 % Number of atoms : 804 ( 270 equ; 0 cnn)
% 0.07/0.31 % Maximal formula atoms : 13 ( 2 avg)
% 0.07/0.31 % Number of connectives : 3843 ( 27 ~; 3 |; 32 &;3381 @)
% 0.07/0.31 % ( 0 <=>; 400 =>; 0 <=; 0 <~>)
% 0.07/0.31 % Maximal formula depth : 18 ( 8 avg)
% 0.07/0.31 % Number of types : 4 ( 3 usr)
% 0.07/0.31 % Number of type conns : 278 ( 278 >; 0 *; 0 +; 0 <<)
% 0.07/0.31 % Number of symbols : 30 ( 29 usr; 0 con; 1-6 aty)
% 0.07/0.31 % Number of variables : 1246 ( 65 ^;1131 !; 20 ?;1246 :)
% 0.07/0.31 % ( 30 !>; 0 ?*; 0 @-; 0 @+)
% 0.07/0.31 % SPC : TH1_THM_EQU_NAR
% 0.07/0.31
% 0.07/0.31 % Comments : This file was generated by Sledgehammer 2021-02-23 16:16:12.554
% 0.07/0.31 %------------------------------------------------------------------------------
% 0.07/0.31 % Could-be-implicit typings (4)
% 0.07/0.31 thf(ty_t_Set_Oset,type,
% 0.07/0.31 set: $tType > $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_tf_c,type,
% 0.07/0.31 c: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_tf_b,type,
% 0.07/0.31 b: $tType ).
% 0.07/0.31
% 0.07/0.31 thf(ty_tf_a,type,
% 0.07/0.31 a: $tType ).
% 0.07/0.31
% 0.07/0.31 % Explicit typings (28)
% 0.07/0.31 thf(sy_cl_Lattices_Obounded__lattice,type,
% 0.07/0.31 bounded_lattice:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_HOL_Otype,type,
% 0.07/0.31 type:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Ouminus,type,
% 0.07/0.31 uminus:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Oord,type,
% 0.07/0.31 ord:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Otop,type,
% 0.07/0.31 top:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Oorder,type,
% 0.07/0.31 order:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Lattices_Olattice,type,
% 0.07/0.31 lattice:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Olinorder,type,
% 0.07/0.31 linorder:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Opreorder,type,
% 0.07/0.31 preorder:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Orderings_Oorder__top,type,
% 0.07/0.31 order_top:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.07/0.31 boolean_algebra:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.07/0.31 semilattice_sup:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.07/0.31 ordered_ab_group_add:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 0.07/0.31 bounded_lattice_top:
% 0.07/0.31 !>[A: $tType] : $o ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Finite__Set_OFpow,type,
% 0.07/0.31 finite_Fpow:
% 0.07/0.31 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Ocomp,type,
% 0.07/0.31 comp:
% 0.07/0.31 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Oinj__on,type,
% 0.07/0.31 inj_on:
% 0.07/0.31 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Fun_Othe__inv__into,type,
% 0.07/0.31 the_inv_into:
% 0.07/0.31 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.07/0.31 uminus_uminus:
% 0.07/0.31 !>[A: $tType] : ( A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Lattices_Osup__class_Osup,type,
% 0.07/0.31 sup_sup:
% 0.07/0.31 !>[A: $tType] : ( A > A > A ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.07/0.31 ord_less_eq:
% 0.07/0.31 !>[A: $tType] : ( A > A > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.07/0.31 top_top:
% 0.07/0.31 !>[A: $tType] : A ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_OCollect,type,
% 0.07/0.31 collect:
% 0.07/0.31 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_Set_Oimage,type,
% 0.07/0.31 image:
% 0.07/0.31 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_c_member,type,
% 0.07/0.31 member:
% 0.07/0.31 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_f,type,
% 0.07/0.31 f: a > b ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_g,type,
% 0.07/0.31 g: c > a ).
% 0.07/0.31
% 0.07/0.31 thf(sy_v_h,type,
% 0.07/0.31 h: c > a ).
% 0.07/0.31
% 0.07/0.31 % Relevant facts (256)
% 0.07/0.31 thf(fact_0_o__ext,axiom,
% 0.07/0.31 ! [C: $tType,A: $tType,B: $tType,H: B > A,F: A > C,G: A > C] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ ( image @ B @ A @ H @ ( top_top @ ( set @ B ) ) ) )
% 0.07/0.31 => ( ( F @ X )
% 0.07/0.31 = ( G @ X ) ) )
% 0.07/0.31 => ( ( comp @ A @ C @ B @ F @ H )
% 0.07/0.31 = ( comp @ A @ C @ B @ G @ H ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % o_ext
% 0.07/0.31 thf(fact_1_o__cong,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,A: $tType,H: A > B,I: A > B,F: B > C,G: B > C] :
% 0.07/0.31 ( ( H = I )
% 0.07/0.31 => ( ! [X: B] :
% 0.07/0.31 ( ( member @ B @ X @ ( image @ A @ B @ I @ ( top_top @ ( set @ A ) ) ) )
% 0.07/0.31 => ( ( F @ X )
% 0.07/0.31 = ( G @ X ) ) )
% 0.07/0.31 => ( ( comp @ B @ C @ A @ F @ H )
% 0.07/0.31 = ( comp @ B @ C @ A @ F @ I ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % o_cong
% 0.07/0.31 thf(fact_2_o__inj__on,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,H: A > C] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.07/0.31 = ( comp @ C @ B @ A @ F @ H ) )
% 0.07/0.31 => ( ( inj_on @ C @ B @ F @ ( sup_sup @ ( set @ C ) @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) @ ( image @ A @ C @ H @ ( top_top @ ( set @ A ) ) ) ) )
% 0.07/0.31 => ( G = H ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % o_inj_on
% 0.07/0.31 thf(fact_3_ex__o__conv,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,G: A > C,F: B > C] :
% 0.07/0.31 ( ( ? [H2: A > B] :
% 0.07/0.31 ( G
% 0.07/0.31 = ( comp @ B @ C @ A @ F @ H2 ) ) )
% 0.07/0.31 = ( ! [X2: C] :
% 0.07/0.31 ( ( member @ C @ X2 @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) )
% 0.07/0.31 => ? [Y: B] :
% 0.07/0.31 ( X2
% 0.07/0.31 = ( F @ Y ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ex_o_conv
% 0.07/0.31 thf(fact_4_o__eq__conv,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,F: C > B,H: A > C,G: C > B] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ F @ H )
% 0.07/0.31 = ( comp @ C @ B @ A @ G @ H ) )
% 0.07/0.31 = ( ! [X2: C] :
% 0.07/0.31 ( ( member @ C @ X2 @ ( image @ A @ C @ H @ ( top_top @ ( set @ A ) ) ) )
% 0.07/0.31 => ( ( F @ X2 )
% 0.07/0.31 = ( G @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % o_eq_conv
% 0.07/0.31 thf(fact_5_sup__top__left,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( bounded_lattice_top @ A )
% 0.07/0.31 => ! [X3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ ( top_top @ A ) @ X3 )
% 0.07/0.31 = ( top_top @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_top_left
% 0.07/0.31 thf(fact_6_sup__top__right,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( bounded_lattice_top @ A )
% 0.07/0.31 => ! [X3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ ( top_top @ A ) )
% 0.07/0.31 = ( top_top @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_top_right
% 0.07/0.31 thf(fact_7_inj__on__Un__image__eq__iff,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 => ( ( ( image @ A @ B @ F @ A2 )
% 0.07/0.31 = ( image @ A @ B @ F @ B2 ) )
% 0.07/0.31 = ( A2 = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_Un_image_eq_iff
% 0.07/0.31 thf(fact_8_inj__compose,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
% 0.07/0.31 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_compose
% 0.07/0.31 thf(fact_9_fun_Oinj__map,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,F: A > B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.inj_map
% 0.07/0.31 thf(fact_10_comp__inj__on,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,G: B > C] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_inj_on
% 0.07/0.31 thf(fact_11_inj__on__imageI,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A2 )
% 0.07/0.31 => ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_imageI
% 0.07/0.31 thf(fact_12_comp__inj__on__iff,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,F2: B > C] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( inj_on @ B @ C @ F2 @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F2 @ F ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_inj_on_iff
% 0.07/0.31 thf(fact_13_range__ex1__eq,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,B3: B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 0.07/0.31 = ( ? [X2: A] :
% 0.07/0.31 ( ( B3
% 0.07/0.31 = ( F @ X2 ) )
% 0.07/0.31 & ! [Y: A] :
% 0.07/0.31 ( ( B3
% 0.07/0.31 = ( F @ Y ) )
% 0.07/0.31 => ( Y = X2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % range_ex1_eq
% 0.07/0.31 thf(fact_14_inj__image__eq__iff,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( ( image @ A @ B @ F @ A2 )
% 0.07/0.31 = ( image @ A @ B @ F @ B2 ) )
% 0.07/0.31 = ( A2 = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_image_eq_iff
% 0.07/0.31 thf(fact_15_inj__image__mem__iff,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A3: A,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 = ( member @ A @ A3 @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_image_mem_iff
% 0.07/0.31 thf(fact_16_comp__surj,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 = ( top_top @ ( set @ C ) ) )
% 0.07/0.31 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_surj
% 0.07/0.31 thf(fact_17_sup_Oright__idem,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 0.07/0.31 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.right_idem
% 0.07/0.31 thf(fact_18_sup__left__idem,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.07/0.31 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_left_idem
% 0.07/0.31 thf(fact_19_sup_Oleft__idem,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 0.07/0.31 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.left_idem
% 0.07/0.31 thf(fact_20_sup__idem,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ X3 )
% 0.07/0.31 = X3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_idem
% 0.07/0.31 thf(fact_21_sup_Oidem,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A] :
% 0.07/0.31 ( ( sup_sup @ A @ A3 @ A3 )
% 0.07/0.31 = A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.idem
% 0.07/0.31 thf(fact_22_sup__apply,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ B )
% 0.07/0.31 => ( ( sup_sup @ ( A > B ) )
% 0.07/0.31 = ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_apply
% 0.07/0.31 thf(fact_23_comp__apply,axiom,
% 0.07/0.31 ! [C: $tType,A: $tType,B: $tType] :
% 0.07/0.31 ( ( comp @ B @ A @ C )
% 0.07/0.31 = ( ^ [F3: B > A,G2: C > B,X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_apply
% 0.07/0.31 thf(fact_24_sup__left__commute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
% 0.07/0.31 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_left_commute
% 0.07/0.31 thf(fact_25_sup_Oleft__commute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,A3: A,C2: A] :
% 0.07/0.31 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 0.07/0.31 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.left_commute
% 0.07/0.31 thf(fact_26_sup__commute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( sup_sup @ A )
% 0.07/0.31 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_commute
% 0.07/0.31 thf(fact_27_sup_Ocommute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( sup_sup @ A )
% 0.07/0.31 = ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.commute
% 0.07/0.31 thf(fact_28_sup__assoc,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 0.07/0.31 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_assoc
% 0.07/0.31 thf(fact_29_sup_Oassoc,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A,C2: A] :
% 0.07/0.31 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 0.07/0.31 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.assoc
% 0.07/0.31 thf(fact_30_boolean__algebra__cancel_Osup2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B2: A,K: A,B3: A,A3: A] :
% 0.07/0.31 ( ( B2
% 0.07/0.31 = ( sup_sup @ A @ K @ B3 ) )
% 0.07/0.31 => ( ( sup_sup @ A @ A3 @ B2 )
% 0.07/0.31 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % boolean_algebra_cancel.sup2
% 0.07/0.31 thf(fact_31_boolean__algebra__cancel_Osup1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A2: A,K: A,A3: A,B3: A] :
% 0.07/0.31 ( ( A2
% 0.07/0.31 = ( sup_sup @ A @ K @ A3 ) )
% 0.07/0.31 => ( ( sup_sup @ A @ A2 @ B3 )
% 0.07/0.31 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % boolean_algebra_cancel.sup1
% 0.07/0.31 thf(fact_32_sup__fun__def,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ B )
% 0.07/0.31 => ( ( sup_sup @ ( A > B ) )
% 0.07/0.31 = ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_fun_def
% 0.07/0.31 thf(fact_33_inf__sup__aci_I5_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ( ( sup_sup @ A )
% 0.07/0.31 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_aci(5)
% 0.07/0.31 thf(fact_34_inf__sup__aci_I6_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 0.07/0.31 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_aci(6)
% 0.07/0.31 thf(fact_35_inf__sup__aci_I7_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
% 0.07/0.31 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_aci(7)
% 0.07/0.31 thf(fact_36_inf__sup__aci_I8_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.07/0.31 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_aci(8)
% 0.07/0.31 thf(fact_37_fun_Omap__comp,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.07/0.31 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.07/0.31 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.map_comp
% 0.07/0.31 thf(fact_38_comp__eq__dest__lhs,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C2: A > B,V: A] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.07/0.31 = C2 )
% 0.07/0.31 => ( ( A3 @ ( B3 @ V ) )
% 0.07/0.31 = ( C2 @ V ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_eq_dest_lhs
% 0.07/0.31 thf(fact_39_comp__eq__elim,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.07/0.31 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.07/0.31 => ! [V2: A] :
% 0.07/0.31 ( ( A3 @ ( B3 @ V2 ) )
% 0.07/0.31 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_eq_elim
% 0.07/0.31 thf(fact_40_comp__eq__dest,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D,V: A] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 0.07/0.31 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 0.07/0.31 => ( ( A3 @ ( B3 @ V ) )
% 0.07/0.31 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_eq_dest
% 0.07/0.31 thf(fact_41_comp__assoc,axiom,
% 0.07/0.31 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.07/0.31 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.07/0.31 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_assoc
% 0.07/0.31 thf(fact_42_comp__def,axiom,
% 0.07/0.31 ! [A: $tType,C: $tType,B: $tType] :
% 0.07/0.31 ( ( comp @ B @ C @ A )
% 0.07/0.31 = ( ^ [F3: B > C,G2: A > B,X2: A] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_def
% 0.07/0.31 thf(fact_43_inj__on__inverseI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ( ( G @ ( F @ X ) )
% 0.07/0.31 = X ) )
% 0.07/0.31 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_inverseI
% 0.07/0.31 thf(fact_44_inj__on__contraD,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( X3 != Y2 )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( member @ A @ Y2 @ A2 )
% 0.07/0.31 => ( ( F @ X3 )
% 0.07/0.31 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_contraD
% 0.07/0.31 thf(fact_45_mem__Collect__eq,axiom,
% 0.07/0.31 ! [A: $tType,A3: A,P: A > $o] :
% 0.07/0.31 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 0.07/0.31 = ( P @ A3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % mem_Collect_eq
% 0.07/0.31 thf(fact_46_Collect__mem__eq,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] :
% 0.07/0.31 ( ( collect @ A
% 0.07/0.31 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 0.07/0.31 = A2 ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mem_eq
% 0.07/0.31 thf(fact_47_Collect__cong,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( P @ X )
% 0.07/0.31 = ( Q @ X ) )
% 0.07/0.31 => ( ( collect @ A @ P )
% 0.07/0.31 = ( collect @ A @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_cong
% 0.07/0.31 thf(fact_48_ext,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( F @ X )
% 0.07/0.31 = ( G @ X ) )
% 0.07/0.31 => ( F = G ) ) ).
% 0.07/0.31
% 0.07/0.31 % ext
% 0.07/0.31 thf(fact_49_inj__on__eq__iff,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( member @ A @ Y2 @ A2 )
% 0.07/0.31 => ( ( ( F @ X3 )
% 0.07/0.31 = ( F @ Y2 ) )
% 0.07/0.31 = ( X3 = Y2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_eq_iff
% 0.07/0.31 thf(fact_50_inj__on__cong,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 0.07/0.31 ( ! [A5: A] :
% 0.07/0.31 ( ( member @ A @ A5 @ A2 )
% 0.07/0.31 => ( ( F @ A5 )
% 0.07/0.31 = ( G @ A5 ) ) )
% 0.07/0.31 => ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_cong
% 0.07/0.31 thf(fact_51_inj__on__def,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( inj_on @ A @ B )
% 0.07/0.31 = ( ^ [F3: A > B,A6: set @ A] :
% 0.07/0.31 ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A6 )
% 0.07/0.31 => ! [Y: A] :
% 0.07/0.31 ( ( member @ A @ Y @ A6 )
% 0.07/0.31 => ( ( ( F3 @ X2 )
% 0.07/0.31 = ( F3 @ Y ) )
% 0.07/0.31 => ( X2 = Y ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_def
% 0.07/0.31 thf(fact_52_inj__onI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 0.07/0.31 ( ! [X: A,Y3: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ( ( member @ A @ Y3 @ A2 )
% 0.07/0.31 => ( ( ( F @ X )
% 0.07/0.31 = ( F @ Y3 ) )
% 0.07/0.31 => ( X = Y3 ) ) ) )
% 0.07/0.31 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_onI
% 0.07/0.31 thf(fact_53_inj__onD,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( ( F @ X3 )
% 0.07/0.31 = ( F @ Y2 ) )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( member @ A @ Y2 @ A2 )
% 0.07/0.31 => ( X3 = Y2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_onD
% 0.07/0.31 thf(fact_54_surj__def,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: B > A] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) )
% 0.07/0.31 = ( ! [Y: A] :
% 0.07/0.31 ? [X2: B] :
% 0.07/0.31 ( Y
% 0.07/0.31 = ( F @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surj_def
% 0.07/0.31 thf(fact_55_surjI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( G @ ( F @ X ) )
% 0.07/0.31 = X )
% 0.07/0.31 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surjI
% 0.07/0.31 thf(fact_56_surjE,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,Y2: A] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ~ ! [X: B] :
% 0.07/0.31 ( Y2
% 0.07/0.31 != ( F @ X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surjE
% 0.07/0.31 thf(fact_57_surjD,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,Y2: A] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ? [X: B] :
% 0.07/0.31 ( Y2
% 0.07/0.31 = ( F @ X ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surjD
% 0.07/0.31 thf(fact_58_image__eq__imp__comp,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A2: set @ B,G: C > A,B2: set @ C,H: A > D] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ A2 )
% 0.07/0.31 = ( image @ C @ A @ G @ B2 ) )
% 0.07/0.31 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A2 )
% 0.07/0.31 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eq_imp_comp
% 0.07/0.31 thf(fact_59_image__comp,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 0.07/0.31 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
% 0.07/0.31 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_comp
% 0.07/0.31 thf(fact_60_inj__on__image__iff,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ! [Xa: A] :
% 0.07/0.31 ( ( member @ A @ Xa @ A2 )
% 0.07/0.31 => ( ( ( G @ ( F @ X ) )
% 0.07/0.31 = ( G @ ( F @ Xa ) ) )
% 0.07/0.31 = ( ( G @ X )
% 0.07/0.31 = ( G @ Xa ) ) ) ) )
% 0.07/0.31 => ( ( inj_on @ A @ A @ F @ A2 )
% 0.07/0.31 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 0.07/0.31 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_image_iff
% 0.07/0.31 thf(fact_61_inj__def,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 = ( ! [X2: A,Y: A] :
% 0.07/0.31 ( ( ( F @ X2 )
% 0.07/0.31 = ( F @ Y ) )
% 0.07/0.31 => ( X2 = Y ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_def
% 0.07/0.31 thf(fact_62_inj__eq,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,X3: A,Y2: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( ( F @ X3 )
% 0.07/0.31 = ( F @ Y2 ) )
% 0.07/0.31 = ( X3 = Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_eq
% 0.07/0.31 thf(fact_63_injI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B] :
% 0.07/0.31 ( ! [X: A,Y3: A] :
% 0.07/0.31 ( ( ( F @ X )
% 0.07/0.31 = ( F @ Y3 ) )
% 0.07/0.31 => ( X = Y3 ) )
% 0.07/0.31 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % injI
% 0.07/0.31 thf(fact_64_injD,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,X3: A,Y2: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( ( F @ X3 )
% 0.07/0.31 = ( F @ Y2 ) )
% 0.07/0.31 => ( X3 = Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % injD
% 0.07/0.31 thf(fact_65_inj__on__imageI2,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,F2: C > B,F: A > C,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F2 @ F ) @ A2 )
% 0.07/0.31 => ( inj_on @ A @ C @ F @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_imageI2
% 0.07/0.31 thf(fact_66_fun_Oinj__map__strong,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,X3: D > A,Xa2: D > A,F: A > B,Fa: A > B] :
% 0.07/0.31 ( ! [Z2: A,Za: A] :
% 0.07/0.31 ( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
% 0.07/0.31 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
% 0.07/0.31 => ( ( ( F @ Z2 )
% 0.07/0.31 = ( Fa @ Za ) )
% 0.07/0.31 => ( Z2 = Za ) ) ) )
% 0.07/0.31 => ( ( ( comp @ A @ B @ D @ F @ X3 )
% 0.07/0.31 = ( comp @ A @ B @ D @ Fa @ Xa2 ) )
% 0.07/0.31 => ( X3 = Xa2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.inj_map_strong
% 0.07/0.31 thf(fact_67_fun_Omap__cong0,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,X3: D > A,F: A > B,G: A > B] :
% 0.07/0.31 ( ! [Z2: A] :
% 0.07/0.31 ( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
% 0.07/0.31 => ( ( F @ Z2 )
% 0.07/0.31 = ( G @ Z2 ) ) )
% 0.07/0.31 => ( ( comp @ A @ B @ D @ F @ X3 )
% 0.07/0.31 = ( comp @ A @ B @ D @ G @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.map_cong0
% 0.07/0.31 thf(fact_68_fun_Omap__cong,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,X3: D > A,Ya: D > A,F: A > B,G: A > B] :
% 0.07/0.31 ( ( X3 = Ya )
% 0.07/0.31 => ( ! [Z2: A] :
% 0.07/0.31 ( ( member @ A @ Z2 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 0.07/0.31 => ( ( F @ Z2 )
% 0.07/0.31 = ( G @ Z2 ) ) )
% 0.07/0.31 => ( ( comp @ A @ B @ D @ F @ X3 )
% 0.07/0.31 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.map_cong
% 0.07/0.31 thf(fact_69_fun_Oset__map,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
% 0.07/0.31 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
% 0.07/0.31 = ( image @ A @ B @ F @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % fun.set_map
% 0.07/0.31 thf(fact_70_surj__fun__eq,axiom,
% 0.07/0.31 ! [B: $tType,C: $tType,A: $tType,F: B > A,X4: set @ B,G1: A > C,G22: A > C] :
% 0.07/0.31 ( ( ( image @ B @ A @ F @ X4 )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ! [X: B] :
% 0.07/0.31 ( ( member @ B @ X @ X4 )
% 0.07/0.31 => ( ( comp @ A @ C @ B @ G1 @ F @ X )
% 0.07/0.31 = ( comp @ A @ C @ B @ G22 @ F @ X ) ) )
% 0.07/0.31 => ( G1 = G22 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % surj_fun_eq
% 0.07/0.31 thf(fact_71_Un__iff,axiom,
% 0.07/0.31 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 = ( ( member @ A @ C2 @ A2 )
% 0.07/0.31 | ( member @ A @ C2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_iff
% 0.07/0.31 thf(fact_72_UnCI,axiom,
% 0.07/0.31 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.07/0.31 ( ( ~ ( member @ A @ C2 @ B2 )
% 0.07/0.31 => ( member @ A @ C2 @ A2 ) )
% 0.07/0.31 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UnCI
% 0.07/0.31 thf(fact_73_iso__tuple__UNIV__I,axiom,
% 0.07/0.31 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % iso_tuple_UNIV_I
% 0.07/0.31 thf(fact_74_UNIV__I,axiom,
% 0.07/0.31 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UNIV_I
% 0.07/0.31 thf(fact_75_top__apply,axiom,
% 0.07/0.31 ! [C: $tType,D: $tType] :
% 0.07/0.31 ( ( top @ C )
% 0.07/0.31 => ( ( top_top @ ( D > C ) )
% 0.07/0.31 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top_apply
% 0.07/0.31 thf(fact_76_image__eqI,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
% 0.07/0.31 ( ( B3
% 0.07/0.31 = ( F @ X3 ) )
% 0.07/0.31 => ( ( member @ B @ X3 @ A2 )
% 0.07/0.31 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_eqI
% 0.07/0.31 thf(fact_77_inj__imp__inj__on,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_imp_inj_on
% 0.07/0.31 thf(fact_78_Un__UNIV__right,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_UNIV_right
% 0.07/0.31 thf(fact_79_Un__UNIV__left,axiom,
% 0.07/0.31 ! [A: $tType,B2: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
% 0.07/0.31 = ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_UNIV_left
% 0.07/0.31 thf(fact_80_imageI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 0.07/0.31 ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % imageI
% 0.07/0.31 thf(fact_81_image__iff,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
% 0.07/0.31 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.31 = ( ? [X2: B] :
% 0.07/0.31 ( ( member @ B @ X2 @ A2 )
% 0.07/0.31 & ( Z
% 0.07/0.31 = ( F @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_iff
% 0.07/0.31 thf(fact_82_bex__imageD,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.07/0.31 ( ? [X5: A] :
% 0.07/0.31 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.31 & ( P @ X5 ) )
% 0.07/0.31 => ? [X: B] :
% 0.07/0.31 ( ( member @ B @ X @ A2 )
% 0.07/0.31 & ( P @ ( F @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bex_imageD
% 0.07/0.31 thf(fact_83_image__cong,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 0.07/0.31 ( ( M = N )
% 0.07/0.31 => ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ N )
% 0.07/0.31 => ( ( F @ X )
% 0.07/0.31 = ( G @ X ) ) )
% 0.07/0.31 => ( ( image @ A @ B @ F @ M )
% 0.07/0.31 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_cong
% 0.07/0.31 thf(fact_84_ball__imageD,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.31 => ( P @ X ) )
% 0.07/0.31 => ! [X5: B] :
% 0.07/0.31 ( ( member @ B @ X5 @ A2 )
% 0.07/0.31 => ( P @ ( F @ X5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ball_imageD
% 0.07/0.31 thf(fact_85_rev__image__eqI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
% 0.07/0.31 ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( B3
% 0.07/0.31 = ( F @ X3 ) )
% 0.07/0.31 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rev_image_eqI
% 0.07/0.31 thf(fact_86_UNIV__eq__I,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] :
% 0.07/0.31 ( ! [X: A] : ( member @ A @ X @ A2 )
% 0.07/0.31 => ( ( top_top @ ( set @ A ) )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % UNIV_eq_I
% 0.07/0.31 thf(fact_87_UNIV__witness,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UNIV_witness
% 0.07/0.31 thf(fact_88_top__set__def,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( top_top @ ( set @ A ) )
% 0.07/0.31 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top_set_def
% 0.07/0.31 thf(fact_89_UnE,axiom,
% 0.07/0.31 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 => ( ~ ( member @ A @ C2 @ A2 )
% 0.07/0.31 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UnE
% 0.07/0.31 thf(fact_90_UnI1,axiom,
% 0.07/0.31 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( member @ A @ C2 @ A2 )
% 0.07/0.31 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UnI1
% 0.07/0.31 thf(fact_91_UnI2,axiom,
% 0.07/0.31 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 0.07/0.31 ( ( member @ A @ C2 @ B2 )
% 0.07/0.31 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % UnI2
% 0.07/0.31 thf(fact_92_bex__Un,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.07/0.31 ( ( ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 & ( P @ X2 ) ) )
% 0.07/0.31 = ( ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A2 )
% 0.07/0.31 & ( P @ X2 ) )
% 0.07/0.31 | ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ B2 )
% 0.07/0.31 & ( P @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bex_Un
% 0.07/0.31 thf(fact_93_ball__Un,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 0.07/0.31 ( ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 => ( P @ X2 ) ) )
% 0.07/0.31 = ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A2 )
% 0.07/0.31 => ( P @ X2 ) )
% 0.07/0.31 & ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ B2 )
% 0.07/0.31 => ( P @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ball_Un
% 0.07/0.31 thf(fact_94_Un__assoc,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 0.07/0.31 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_assoc
% 0.07/0.31 thf(fact_95_Un__absorb,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 0.07/0.31 = A2 ) ).
% 0.07/0.31
% 0.07/0.31 % Un_absorb
% 0.07/0.31 thf(fact_96_Un__commute,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) )
% 0.07/0.31 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_commute
% 0.07/0.31 thf(fact_97_Un__left__absorb,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.31 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_left_absorb
% 0.07/0.31 thf(fact_98_Un__left__commute,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.07/0.31 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
% 0.07/0.31 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_left_commute
% 0.07/0.31 thf(fact_99_rangeI,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rangeI
% 0.07/0.31 thf(fact_100_range__eqI,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
% 0.07/0.31 ( ( B3
% 0.07/0.31 = ( F @ X3 ) )
% 0.07/0.31 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % range_eqI
% 0.07/0.31 thf(fact_101_image__Un,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ B] :
% 0.07/0.31 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
% 0.07/0.31 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_Un
% 0.07/0.31 thf(fact_102_Inf_OINF__image,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
% 0.07/0.31 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
% 0.07/0.31 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Inf.INF_image
% 0.07/0.31 thf(fact_103_Sup_OSUP__image,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
% 0.07/0.31 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
% 0.07/0.31 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Sup.SUP_image
% 0.07/0.31 thf(fact_104_the__inv__into__comp,axiom,
% 0.07/0.31 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A2: set @ C,X3: B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) )
% 0.07/0.31 => ( ( inj_on @ C @ A @ G @ A2 )
% 0.07/0.31 => ( ( member @ B @ X3 @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) ) )
% 0.07/0.31 => ( ( the_inv_into @ C @ B @ A2 @ ( comp @ A @ B @ C @ F @ G ) @ X3 )
% 0.07/0.31 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A2 @ G ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G @ A2 ) @ F ) @ X3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_into_comp
% 0.07/0.31 thf(fact_105_the__inv__into__onto,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 = A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_into_onto
% 0.07/0.31 thf(fact_106_comp__cong,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X3: C,F2: D > A,G3: E > D,X6: E] :
% 0.07/0.31 ( ( ( F @ ( G @ X3 ) )
% 0.07/0.31 = ( F2 @ ( G3 @ X6 ) ) )
% 0.07/0.31 => ( ( comp @ B @ A @ C @ F @ G @ X3 )
% 0.07/0.31 = ( comp @ D @ A @ E @ F2 @ G3 @ X6 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp_cong
% 0.07/0.31 thf(fact_107_comp2__conv,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,C: $tType] :
% 0.07/0.31 ( ( comp @ C @ B @ A )
% 0.07/0.31 = ( ^ [F1: C > B,F22: A > C,X2: A] : ( F1 @ ( F22 @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp2_conv
% 0.07/0.31 thf(fact_108_top1I,axiom,
% 0.07/0.31 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 0.07/0.31
% 0.07/0.31 % top1I
% 0.07/0.31 thf(fact_109_the__inv__into__f__eq,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( ( F @ X3 )
% 0.07/0.31 = Y2 )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( the_inv_into @ A @ B @ A2 @ F @ Y2 )
% 0.07/0.31 = X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_into_f_eq
% 0.07/0.31 thf(fact_110_the__inv__into__f__f,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X3 ) )
% 0.07/0.31 = X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_into_f_f
% 0.07/0.31 thf(fact_111_inj__on__the__inv__into,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inj_on_the_inv_into
% 0.07/0.31 thf(fact_112_f__the__inv__into__f,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y2: B] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( member @ B @ Y2 @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 => ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y2 ) )
% 0.07/0.31 = Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % f_the_inv_into_f
% 0.07/0.31 thf(fact_113_the__inv__f__f,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,X3: A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.31 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X3 ) )
% 0.07/0.31 = X3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_f_f
% 0.07/0.31 thf(fact_114_Sup_OSUP__cong,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ( ! [X: B] :
% 0.07/0.31 ( ( member @ B @ X @ B2 )
% 0.07/0.31 => ( ( C3 @ X )
% 0.07/0.31 = ( D3 @ X ) ) )
% 0.07/0.31 => ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
% 0.07/0.31 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Sup.SUP_cong
% 0.07/0.31 thf(fact_115_Inf_OINF__cong,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ( ! [X: B] :
% 0.07/0.31 ( ( member @ B @ X @ B2 )
% 0.07/0.31 => ( ( C3 @ X )
% 0.07/0.31 = ( D3 @ X ) ) )
% 0.07/0.31 => ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
% 0.07/0.31 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Inf.INF_cong
% 0.07/0.31 thf(fact_116_comp3__conv,axiom,
% 0.07/0.31 ! [C: $tType,D: $tType,B: $tType,A: $tType,F12: D > B,F23: C > D,F32: A > C] :
% 0.07/0.31 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F12 @ F23 ) @ F32 )
% 0.07/0.31 = ( ^ [X2: A] : ( F12 @ ( F23 @ ( F32 @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % comp3_conv
% 0.07/0.31 thf(fact_117_top__empty__eq,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( top_top @ ( A > $o ) )
% 0.07/0.31 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top_empty_eq
% 0.07/0.31 thf(fact_118_top__conj_I2_J,axiom,
% 0.07/0.31 ! [A: $tType,P: $o,X3: A] :
% 0.07/0.31 ( ( P
% 0.07/0.31 & ( top_top @ ( A > $o ) @ X3 ) )
% 0.07/0.31 = P ) ).
% 0.07/0.31
% 0.07/0.31 % top_conj(2)
% 0.07/0.31 thf(fact_119_top__conj_I1_J,axiom,
% 0.07/0.31 ! [A: $tType,X3: A,P: $o] :
% 0.07/0.31 ( ( ( top_top @ ( A > $o ) @ X3 )
% 0.07/0.31 & P )
% 0.07/0.31 = P ) ).
% 0.07/0.31
% 0.07/0.31 % top_conj(1)
% 0.07/0.31 thf(fact_120_the__inv__into__into,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: B,B2: set @ A] :
% 0.07/0.31 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.31 => ( ( member @ B @ X3 @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X3 ) @ B2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % the_inv_into_into
% 0.07/0.31 thf(fact_121_rewriteR__comp__comp2,axiom,
% 0.07/0.31 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 0.07/0.31 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.07/0.31 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 0.07/0.31 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.07/0.31 = L )
% 0.07/0.31 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 0.07/0.31 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % rewriteR_comp_comp2
% 0.07/0.31 thf(fact_122_order__refl,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( preorder @ A )
% 0.07/0.31 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_refl
% 0.07/0.31 thf(fact_123_subsetI,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ( member @ A @ X @ B2 ) )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetI
% 0.07/0.31 thf(fact_124_subset__antisym,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.07/0.31 => ( A2 = B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_antisym
% 0.07/0.31 thf(fact_125_le__sup__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 0.07/0.31 = ( ( ord_less_eq @ A @ X3 @ Z )
% 0.07/0.31 & ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_sup_iff
% 0.07/0.31 thf(fact_126_sup_Obounded__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,C2: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.07/0.31 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.31 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.bounded_iff
% 0.07/0.31 thf(fact_127_Un__subset__iff,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 0.07/0.31 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Un_subset_iff
% 0.07/0.31 thf(fact_128_bex__cong2,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
% 0.07/0.31 => ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ I2 )
% 0.07/0.31 => ( ( F @ X )
% 0.07/0.31 = ( G @ X ) ) )
% 0.07/0.31 => ( ( ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ I2 )
% 0.07/0.31 & ( P @ ( F @ X2 ) ) ) )
% 0.07/0.31 = ( ? [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ I2 )
% 0.07/0.31 & ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bex_cong2
% 0.07/0.31 thf(fact_129_ball__cong2,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
% 0.07/0.31 => ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ( ( F @ X )
% 0.07/0.31 = ( G @ X ) ) )
% 0.07/0.31 => ( ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ I2 )
% 0.07/0.31 => ( P @ ( F @ X2 ) ) ) )
% 0.07/0.31 = ( ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ I2 )
% 0.07/0.31 => ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ball_cong2
% 0.07/0.31 thf(fact_130_bex__subset__imp__bex,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,P: A > $o,B2: set @ A] :
% 0.07/0.31 ( ? [X5: A] :
% 0.07/0.31 ( ( member @ A @ X5 @ A2 )
% 0.07/0.31 & ( P @ X5 ) )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ? [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ B2 )
% 0.07/0.31 & ( P @ X ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % bex_subset_imp_bex
% 0.07/0.31 thf(fact_131_ball__subset__imp__ball,axiom,
% 0.07/0.31 ! [A: $tType,B2: set @ A,P: A > $o,A2: set @ A] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ B2 )
% 0.07/0.31 => ( P @ X ) )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ! [X5: A] :
% 0.07/0.31 ( ( member @ A @ X5 @ A2 )
% 0.07/0.31 => ( P @ X5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ball_subset_imp_ball
% 0.07/0.31 thf(fact_132_top_Oextremum__uniqueI,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order_top @ A )
% 0.07/0.31 => ! [A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.07/0.31 => ( A3
% 0.07/0.31 = ( top_top @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top.extremum_uniqueI
% 0.07/0.31 thf(fact_133_top_Oextremum__unique,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order_top @ A )
% 0.07/0.31 => ! [A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 0.07/0.31 = ( A3
% 0.07/0.31 = ( top_top @ A ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top.extremum_unique
% 0.07/0.31 thf(fact_134_top__greatest,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order_top @ A )
% 0.07/0.31 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % top_greatest
% 0.07/0.31 thf(fact_135_subset__image__iff,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.31 = ( ? [AA: set @ B] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 0.07/0.31 & ( B2
% 0.07/0.31 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_image_iff
% 0.07/0.31 thf(fact_136_image__subset__iff,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 0.07/0.31 = ( ! [X2: B] :
% 0.07/0.31 ( ( member @ B @ X2 @ A2 )
% 0.07/0.31 => ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_subset_iff
% 0.07/0.31 thf(fact_137_subset__imageE,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.31 => ~ ! [C4: set @ B] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
% 0.07/0.31 => ( B2
% 0.07/0.31 != ( image @ B @ A @ F @ C4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_imageE
% 0.07/0.31 thf(fact_138_image__subsetI,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( member @ A @ X @ A2 )
% 0.07/0.31 => ( member @ B @ ( F @ X ) @ B2 ) )
% 0.07/0.31 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_subsetI
% 0.07/0.31 thf(fact_139_image__mono,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % image_mono
% 0.07/0.31 thf(fact_140_subset__UNIV,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_UNIV
% 0.07/0.31 thf(fact_141_sup_OcoboundedI2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [C2: A,B3: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.07/0.31 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.coboundedI2
% 0.07/0.31 thf(fact_142_sup_OcoboundedI1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [C2: A,A3: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.07/0.31 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.coboundedI1
% 0.07/0.31 thf(fact_143_sup_Oabsorb__iff2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( ord_less_eq @ A )
% 0.07/0.31 = ( ^ [A4: A,B4: A] :
% 0.07/0.31 ( ( sup_sup @ A @ A4 @ B4 )
% 0.07/0.31 = B4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.absorb_iff2
% 0.07/0.31 thf(fact_144_sup_Oabsorb__iff1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( ord_less_eq @ A )
% 0.07/0.31 = ( ^ [B4: A,A4: A] :
% 0.07/0.31 ( ( sup_sup @ A @ A4 @ B4 )
% 0.07/0.31 = A4 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.absorb_iff1
% 0.07/0.31 thf(fact_145_sup_Ocobounded2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.cobounded2
% 0.07/0.31 thf(fact_146_sup_Ocobounded1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.cobounded1
% 0.07/0.31 thf(fact_147_sup_Oorder__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( ord_less_eq @ A )
% 0.07/0.31 = ( ^ [B4: A,A4: A] :
% 0.07/0.31 ( A4
% 0.07/0.31 = ( sup_sup @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.order_iff
% 0.07/0.31 thf(fact_148_sup_OboundedI,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,A3: A,C2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.boundedI
% 0.07/0.31 thf(fact_149_sup_OboundedE,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,C2: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 0.07/0.31 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.boundedE
% 0.07/0.31 thf(fact_150_sup__absorb2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 => ( ( sup_sup @ A @ X3 @ Y2 )
% 0.07/0.31 = Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_absorb2
% 0.07/0.31 thf(fact_151_sup__absorb1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [Y2: A,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.07/0.31 => ( ( sup_sup @ A @ X3 @ Y2 )
% 0.07/0.31 = X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_absorb1
% 0.07/0.31 thf(fact_152_sup_Oabsorb2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.07/0.31 = B3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.absorb2
% 0.07/0.31 thf(fact_153_sup_Oabsorb1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.31 => ( ( sup_sup @ A @ A3 @ B3 )
% 0.07/0.31 = A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.absorb1
% 0.07/0.31 thf(fact_154_sup__unique,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [F: A > A > A,X3: A,Y2: A] :
% 0.07/0.31 ( ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( F @ X @ Y3 ) )
% 0.07/0.31 => ( ! [X: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X @ Y3 ) )
% 0.07/0.31 => ( ! [X: A,Y3: A,Z2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ Y3 @ X )
% 0.07/0.31 => ( ( ord_less_eq @ A @ Z2 @ X )
% 0.07/0.31 => ( ord_less_eq @ A @ ( F @ Y3 @ Z2 ) @ X ) ) )
% 0.07/0.31 => ( ( sup_sup @ A @ X3 @ Y2 )
% 0.07/0.31 = ( F @ X3 @ Y2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_unique
% 0.07/0.31 thf(fact_155_sup_OorderI,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( sup_sup @ A @ A3 @ B3 ) )
% 0.07/0.31 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.orderI
% 0.07/0.31 thf(fact_156_sup_OorderE,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [B3: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.31 => ( A3
% 0.07/0.31 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.orderE
% 0.07/0.31 thf(fact_157_le__iff__sup,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ( ( ord_less_eq @ A )
% 0.07/0.31 = ( ^ [X2: A,Y: A] :
% 0.07/0.31 ( ( sup_sup @ A @ X2 @ Y )
% 0.07/0.31 = Y ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_iff_sup
% 0.07/0.31 thf(fact_158_sup__least,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [Y2: A,X3: A,Z: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ Z @ X3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z ) @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_least
% 0.07/0.31 thf(fact_159_sup__mono,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,C2: A,B3: A,D2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ C2 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ B3 @ D2 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_mono
% 0.07/0.31 thf(fact_160_sup_Omono,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [C2: A,A3: A,D2: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ C2 @ A3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ D2 @ B3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup.mono
% 0.07/0.31 thf(fact_161_le__supI2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,B3: A,A3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X3 @ B3 )
% 0.07/0.31 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_supI2
% 0.07/0.31 thf(fact_162_le__supI1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,A3: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X3 @ A3 )
% 0.07/0.31 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_supI1
% 0.07/0.31 thf(fact_163_sup__ge2,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_ge2
% 0.07/0.31 thf(fact_164_sup__ge1,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % sup_ge1
% 0.07/0.31 thf(fact_165_le__supI,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,X3: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ B3 @ X3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_supI
% 0.07/0.31 thf(fact_166_le__supE,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( semilattice_sup @ A )
% 0.07/0.31 => ! [A3: A,B3: A,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 )
% 0.07/0.31 => ~ ( ( ord_less_eq @ A @ A3 @ X3 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ B3 @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_supE
% 0.07/0.31 thf(fact_167_inf__sup__ord_I3_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_ord(3)
% 0.07/0.31 thf(fact_168_inf__sup__ord_I4_J,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( lattice @ A )
% 0.07/0.31 => ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % inf_sup_ord(4)
% 0.07/0.31 thf(fact_169_in__mono,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( ( member @ A @ X3 @ A2 )
% 0.07/0.31 => ( member @ A @ X3 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % in_mono
% 0.07/0.31 thf(fact_170_subsetD,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( ( member @ A @ C2 @ A2 )
% 0.07/0.31 => ( member @ A @ C2 @ B2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subsetD
% 0.07/0.31 thf(fact_171_equalityE,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityE
% 0.07/0.31 thf(fact_172_subset__eq,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) )
% 0.07/0.31 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.07/0.31 ! [X2: A] :
% 0.07/0.31 ( ( member @ A @ X2 @ A6 )
% 0.07/0.31 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_eq
% 0.07/0.31 thf(fact_173_equalityD1,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD1
% 0.07/0.31 thf(fact_174_equalityD2,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.31 ( ( A2 = B2 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.07/0.31
% 0.07/0.31 % equalityD2
% 0.07/0.31 thf(fact_175_subset__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) )
% 0.07/0.31 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.07/0.31 ! [T: A] :
% 0.07/0.31 ( ( member @ A @ T @ A6 )
% 0.07/0.31 => ( member @ A @ T @ B5 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_iff
% 0.07/0.31 thf(fact_176_subset__refl,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 0.07/0.31
% 0.07/0.31 % subset_refl
% 0.07/0.31 thf(fact_177_Collect__mono,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ! [X: A] :
% 0.07/0.31 ( ( P @ X )
% 0.07/0.31 => ( Q @ X ) )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono
% 0.07/0.31 thf(fact_178_subset__trans,axiom,
% 0.07/0.31 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.31 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.07/0.31 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % subset_trans
% 0.07/0.31 thf(fact_179_le__funD,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( ord @ B )
% 0.07/0.31 => ! [F: A > B,G: A > B,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_funD
% 0.07/0.31 thf(fact_180_le__funE,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( ord @ B )
% 0.07/0.31 => ! [F: A > B,G: A > B,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_funE
% 0.07/0.31 thf(fact_181_le__funI,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( ord @ B )
% 0.07/0.31 => ! [F: A > B,G: A > B] :
% 0.07/0.31 ( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
% 0.07/0.31 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_funI
% 0.07/0.31 thf(fact_182_set__eq__subset,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 0.07/0.31 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 0.07/0.31 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % set_eq_subset
% 0.07/0.31 thf(fact_183_le__fun__def,axiom,
% 0.07/0.31 ! [B: $tType,A: $tType] :
% 0.07/0.31 ( ( ord @ B )
% 0.07/0.31 => ( ( ord_less_eq @ ( A > B ) )
% 0.07/0.31 = ( ^ [F3: A > B,G2: A > B] :
% 0.07/0.31 ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_fun_def
% 0.07/0.31 thf(fact_184_Collect__mono__iff,axiom,
% 0.07/0.31 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.07/0.31 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.07/0.31 = ( ! [X2: A] :
% 0.07/0.31 ( ( P @ X2 )
% 0.07/0.31 => ( Q @ X2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % Collect_mono_iff
% 0.07/0.31 thf(fact_185_order__subst1,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType] :
% 0.07/0.31 ( ( ( order @ B )
% 0.07/0.31 & ( order @ A ) )
% 0.07/0.31 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 0.07/0.31 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.07/0.31 => ( ! [X: B,Y3: B] :
% 0.07/0.31 ( ( ord_less_eq @ B @ X @ Y3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_subst1
% 0.07/0.31 thf(fact_186_order__subst2,axiom,
% 0.07/0.31 ! [A: $tType,C: $tType] :
% 0.07/0.31 ( ( ( order @ C )
% 0.07/0.31 & ( order @ A ) )
% 0.07/0.31 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 0.07/0.31 => ( ! [X: A,Y3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.07/0.31 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_subst2
% 0.07/0.31 thf(fact_187_ord__eq__le__subst,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType] :
% 0.07/0.31 ( ( ( ord @ B )
% 0.07/0.31 & ( ord @ A ) )
% 0.07/0.31 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 0.07/0.31 ( ( A3
% 0.07/0.31 = ( F @ B3 ) )
% 0.07/0.31 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 0.07/0.31 => ( ! [X: B,Y3: B] :
% 0.07/0.31 ( ( ord_less_eq @ B @ X @ Y3 )
% 0.07/0.31 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_eq_le_subst
% 0.07/0.31 thf(fact_188_ord__le__eq__subst,axiom,
% 0.07/0.31 ! [A: $tType,B: $tType] :
% 0.07/0.31 ( ( ( ord @ B )
% 0.07/0.31 & ( ord @ A ) )
% 0.07/0.31 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( ( F @ B3 )
% 0.07/0.31 = C2 )
% 0.07/0.31 => ( ! [X: A,Y3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X @ Y3 )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) )
% 0.07/0.31 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_le_eq_subst
% 0.07/0.31 thf(fact_189_eq__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.07/0.31 = ( ^ [X2: A,Y: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X2 @ Y )
% 0.07/0.31 & ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_iff
% 0.07/0.31 thf(fact_190_antisym,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.07/0.31 => ( X3 = Y2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % antisym
% 0.07/0.31 thf(fact_191_linear,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linorder @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % linear
% 0.07/0.31 thf(fact_192_eq__refl,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( preorder @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ( X3 = Y2 )
% 0.07/0.31 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % eq_refl
% 0.07/0.31 thf(fact_193_le__cases,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linorder @ A )
% 0.07/0.31 => ! [X3: A,Y2: A] :
% 0.07/0.31 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_cases
% 0.07/0.31 thf(fact_194_order_Otrans,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ! [A3: A,B3: A,C2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.07/0.31 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order.trans
% 0.07/0.31 thf(fact_195_le__cases3,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( linorder @ A )
% 0.07/0.31 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.31 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 0.07/0.31 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ X3 @ Z ) )
% 0.07/0.31 => ( ( ( ord_less_eq @ A @ X3 @ Z )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 0.07/0.31 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 0.07/0.31 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ Z @ X3 ) )
% 0.07/0.31 => ~ ( ( ord_less_eq @ A @ Z @ X3 )
% 0.07/0.31 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % le_cases3
% 0.07/0.31 thf(fact_196_antisym__conv,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ! [Y2: A,X3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.31 = ( X3 = Y2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % antisym_conv
% 0.07/0.31 thf(fact_197_order__class_Oorder_Oeq__iff,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.07/0.31 = ( ^ [A4: A,B4: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A4 @ B4 )
% 0.07/0.31 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % order_class.order.eq_iff
% 0.07/0.31 thf(fact_198_ord__eq__le__trans,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord @ A )
% 0.07/0.31 => ! [A3: A,B3: A,C2: A] :
% 0.07/0.31 ( ( A3 = B3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 0.07/0.31 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_eq_le_trans
% 0.07/0.31 thf(fact_199_ord__le__eq__trans,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( ord @ A )
% 0.07/0.31 => ! [A3: A,B3: A,C2: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( B3 = C2 )
% 0.07/0.31 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 0.07/0.31
% 0.07/0.31 % ord_le_eq_trans
% 0.07/0.31 thf(fact_200_order__class_Oorder_Oantisym,axiom,
% 0.07/0.31 ! [A: $tType] :
% 0.07/0.31 ( ( order @ A )
% 0.07/0.31 => ! [A3: A,B3: A] :
% 0.07/0.31 ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.31 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.32 => ( A3 = B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_class.order.antisym
% 0.07/0.32 thf(fact_201_order__trans,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( preorder @ A )
% 0.07/0.32 => ! [X3: A,Y2: A,Z: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 0.07/0.32 => ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % order_trans
% 0.07/0.32 thf(fact_202_dual__order_Orefl,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( order @ A )
% 0.07/0.32 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.refl
% 0.07/0.32 thf(fact_203_linorder__wlog,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( linorder @ A )
% 0.07/0.32 => ! [P: A > A > $o,A3: A,B3: A] :
% 0.07/0.32 ( ! [A5: A,B6: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ A5 @ B6 )
% 0.07/0.32 => ( P @ A5 @ B6 ) )
% 0.07/0.32 => ( ! [A5: A,B6: A] :
% 0.07/0.32 ( ( P @ B6 @ A5 )
% 0.07/0.32 => ( P @ A5 @ B6 ) )
% 0.07/0.32 => ( P @ A3 @ B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % linorder_wlog
% 0.07/0.32 thf(fact_204_dual__order_Otrans,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( order @ A )
% 0.07/0.32 => ! [B3: A,A3: A,C2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 0.07/0.32 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.trans
% 0.07/0.32 thf(fact_205_dual__order_Oeq__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( order @ A )
% 0.07/0.32 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 0.07/0.32 = ( ^ [A4: A,B4: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ B4 @ A4 )
% 0.07/0.32 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.eq_iff
% 0.07/0.32 thf(fact_206_dual__order_Oantisym,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( order @ A )
% 0.07/0.32 => ! [B3: A,A3: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ B3 @ A3 )
% 0.07/0.32 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 0.07/0.32 => ( A3 = B3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % dual_order.antisym
% 0.07/0.32 thf(fact_207_Un__mono,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A,D3: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 0.07/0.32 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_mono
% 0.07/0.32 thf(fact_208_Un__least,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.07/0.32 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_least
% 0.07/0.32 thf(fact_209_Un__upper1,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_upper1
% 0.07/0.32 thf(fact_210_Un__upper2,axiom,
% 0.07/0.32 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_upper2
% 0.07/0.32 thf(fact_211_Un__absorb1,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 = B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_absorb1
% 0.07/0.32 thf(fact_212_Un__absorb2,axiom,
% 0.07/0.32 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.07/0.32 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 = A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Un_absorb2
% 0.07/0.32 thf(fact_213_subset__UnE,axiom,
% 0.07/0.32 ! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 0.07/0.32 => ~ ! [A7: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 0.07/0.32 => ! [B7: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
% 0.07/0.32 => ( C3
% 0.07/0.32 != ( sup_sup @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_UnE
% 0.07/0.32 thf(fact_214_subset__Un__eq,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) )
% 0.07/0.32 = ( ^ [A6: set @ A,B5: set @ A] :
% 0.07/0.32 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 0.07/0.32 = B5 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_Un_eq
% 0.07/0.32 thf(fact_215_inj__on__subset,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.07/0.32 => ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_subset
% 0.07/0.32 thf(fact_216_subset__inj__on,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ B2 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_inj_on
% 0.07/0.32 thf(fact_217_SetInterval2_Oinj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A,A3: A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 0.07/0.32 => ( ( member @ A @ A3 @ A2 )
% 0.07/0.32 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ B2 ) )
% 0.07/0.32 = ( member @ A @ A3 @ B2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % SetInterval2.inj_on_image_mem_iff
% 0.07/0.32 thf(fact_218_inj__on__image__eq__iff,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ C3 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 0.07/0.32 => ( ( ( image @ A @ B @ F @ A2 )
% 0.07/0.32 = ( image @ A @ B @ F @ B2 ) )
% 0.07/0.32 = ( A2 = B2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_eq_iff
% 0.07/0.32 thf(fact_219_Fun_Oinj__on__image__mem__iff,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A3: A,A2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ B2 )
% 0.07/0.32 => ( ( member @ A @ A3 @ B2 )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 0.07/0.32 = ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Fun.inj_on_image_mem_iff
% 0.07/0.32 thf(fact_220_inj__image__subset__iff,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.32 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
% 0.07/0.32 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_image_subset_iff
% 0.07/0.32 thf(fact_221_rewriteL__comp__comp,axiom,
% 0.07/0.32 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 0.07/0.32 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.07/0.32 = L )
% 0.07/0.32 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 0.07/0.32 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % rewriteL_comp_comp
% 0.07/0.32 thf(fact_222_rewriteR__comp__comp,axiom,
% 0.07/0.32 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
% 0.07/0.32 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.07/0.32 = R )
% 0.07/0.32 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 0.07/0.32 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % rewriteR_comp_comp
% 0.07/0.32 thf(fact_223_rewriteL__comp__comp2,axiom,
% 0.07/0.32 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 0.07/0.32 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.07/0.32 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.07/0.32 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.07/0.32 = R )
% 0.07/0.32 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 0.07/0.32 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % rewriteL_comp_comp2
% 0.07/0.32 thf(fact_224_subset__image__inj,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,S: set @ A,F: B > A,T2: set @ B] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T2 ) )
% 0.07/0.32 = ( ? [U: set @ B] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
% 0.07/0.32 & ( inj_on @ B @ A @ F @ U )
% 0.07/0.32 & ( S
% 0.07/0.32 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % subset_image_inj
% 0.07/0.32 thf(fact_225_ex__subset__image__inj,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
% 0.07/0.32 ( ( ? [T3: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
% 0.07/0.32 & ( P @ T3 ) ) )
% 0.07/0.32 = ( ? [T3: set @ B] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
% 0.07/0.32 & ( inj_on @ B @ A @ F @ T3 )
% 0.07/0.32 & ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ex_subset_image_inj
% 0.07/0.32 thf(fact_226_all__subset__image__inj,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
% 0.07/0.32 ( ( ! [T3: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
% 0.07/0.32 => ( P @ T3 ) ) )
% 0.07/0.32 = ( ! [T3: set @ B] :
% 0.07/0.32 ( ( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
% 0.07/0.32 & ( inj_on @ B @ A @ F @ T3 ) )
% 0.07/0.32 => ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % all_subset_image_inj
% 0.07/0.32 thf(fact_227_all__subset__image,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: ( set @ A ) > $o] :
% 0.07/0.32 ( ( ! [B5: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F @ A2 ) )
% 0.07/0.32 => ( P @ B5 ) ) )
% 0.07/0.32 = ( ! [B5: set @ B] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ B ) @ B5 @ A2 )
% 0.07/0.32 => ( P @ ( image @ B @ A @ F @ B5 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % all_subset_image
% 0.07/0.32 thf(fact_228_inj__image__Compl__subset,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 0.07/0.32 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_image_Compl_subset
% 0.07/0.32 thf(fact_229_inj__on__image__Fpow,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 0.07/0.32 ( ( inj_on @ A @ B @ F @ A2 )
% 0.07/0.32 => ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F ) @ ( finite_Fpow @ A @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_on_image_Fpow
% 0.07/0.32 thf(fact_230_uminus__apply,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType] :
% 0.07/0.32 ( ( uminus @ B )
% 0.07/0.32 => ( ( uminus_uminus @ ( A > B ) )
% 0.07/0.32 = ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % uminus_apply
% 0.07/0.32 thf(fact_231_double__compl,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A] :
% 0.07/0.32 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 0.07/0.32 = X3 ) ) ).
% 0.07/0.32
% 0.07/0.32 % double_compl
% 0.07/0.32 thf(fact_232_compl__eq__compl__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,Y2: A] :
% 0.07/0.32 ( ( ( uminus_uminus @ A @ X3 )
% 0.07/0.32 = ( uminus_uminus @ A @ Y2 ) )
% 0.07/0.32 = ( X3 = Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_eq_compl_iff
% 0.07/0.32 thf(fact_233_ComplI,axiom,
% 0.07/0.32 ! [A: $tType,C2: A,A2: set @ A] :
% 0.07/0.32 ( ~ ( member @ A @ C2 @ A2 )
% 0.07/0.32 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % ComplI
% 0.07/0.32 thf(fact_234_Compl__iff,axiom,
% 0.07/0.32 ! [A: $tType,C2: A,A2: set @ A] :
% 0.07/0.32 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.07/0.32 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_iff
% 0.07/0.32 thf(fact_235_Compl__eq__Compl__iff,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 0.07/0.32 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.07/0.32 = ( A2 = B2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_eq_Compl_iff
% 0.07/0.32 thf(fact_236_compl__le__compl__iff,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 0.07/0.32 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_le_compl_iff
% 0.07/0.32 thf(fact_237_inj__uminus,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( ordered_ab_group_add @ A )
% 0.07/0.32 => ! [A2: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % inj_uminus
% 0.07/0.32 thf(fact_238_Compl__subset__Compl__iff,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 0.07/0.32 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_subset_Compl_iff
% 0.07/0.32 thf(fact_239_Compl__anti__mono,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_anti_mono
% 0.07/0.32 thf(fact_240_sup__compl__top__left2,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,Y2: A] :
% 0.07/0.32 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % sup_compl_top_left2
% 0.07/0.32 thf(fact_241_sup__compl__top__left1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,Y2: A] :
% 0.07/0.32 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ ( sup_sup @ A @ X3 @ Y2 ) )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % sup_compl_top_left1
% 0.07/0.32 thf(fact_242_sup__compl__top,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A] :
% 0.07/0.32 ( ( sup_sup @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % sup_compl_top
% 0.07/0.32 thf(fact_243_compl__sup__top,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A] :
% 0.07/0.32 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_sup_top
% 0.07/0.32 thf(fact_244_compl__mono,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,Y2: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_mono
% 0.07/0.32 thf(fact_245_compl__le__swap1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [Y2: A,X3: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 0.07/0.32 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_le_swap1
% 0.07/0.32 thf(fact_246_compl__le__swap2,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [Y2: A,X3: A] :
% 0.07/0.32 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 0.07/0.32 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % compl_le_swap2
% 0.07/0.32 thf(fact_247_Fpow__mono,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 0.07/0.32 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 0.07/0.32 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Fpow_mono
% 0.07/0.32 thf(fact_248_sup__cancel__left2,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,A3: A,B3: A] :
% 0.07/0.32 ( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ A3 ) @ ( sup_sup @ A @ X3 @ B3 ) )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % sup_cancel_left2
% 0.07/0.32 thf(fact_249_sup__cancel__left1,axiom,
% 0.07/0.32 ! [A: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A )
% 0.07/0.32 => ! [X3: A,A3: A,B3: A] :
% 0.07/0.32 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ B3 ) )
% 0.07/0.32 = ( top_top @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % sup_cancel_left1
% 0.07/0.32 thf(fact_250_Compl__partition2,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A] :
% 0.07/0.32 ( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ A2 )
% 0.07/0.32 = ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_partition2
% 0.07/0.32 thf(fact_251_Compl__partition,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A] :
% 0.07/0.32 ( ( sup_sup @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.07/0.32 = ( top_top @ ( set @ A ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % Compl_partition
% 0.07/0.32 thf(fact_252_fun__Compl__def,axiom,
% 0.07/0.32 ! [B: $tType,A: $tType] :
% 0.07/0.32 ( ( uminus @ B )
% 0.07/0.32 => ( ( uminus_uminus @ ( A > B ) )
% 0.07/0.32 = ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % fun_Compl_def
% 0.07/0.32 thf(fact_253_ComplD,axiom,
% 0.07/0.32 ! [A: $tType,C2: A,A2: set @ A] :
% 0.07/0.32 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.07/0.32 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 0.07/0.32
% 0.07/0.32 % ComplD
% 0.07/0.32 thf(fact_254_double__complement,axiom,
% 0.07/0.32 ! [A: $tType,A2: set @ A] :
% 0.07/0.32 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 0.07/0.32 = A2 ) ).
% 0.07/0.32
% 0.07/0.32 % double_complement
% 0.07/0.32 thf(fact_255_surj__Compl__image__subset,axiom,
% 0.07/0.32 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 0.07/0.32 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 0.07/0.32 = ( top_top @ ( set @ A ) ) )
% 0.07/0.32 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A2 ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 % surj_Compl_image_subset
% 0.07/0.32
% 0.07/0.32 % Type constructors (34)
% 0.07/0.32 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 0.07/0.32 bounded_lattice @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 0.07/0.32 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( bounded_lattice @ A9 )
% 0.07/0.32 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( bounded_lattice @ A9 )
% 0.07/0.32 => ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( semilattice_sup @ A9 )
% 0.07/0.32 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( boolean_algebra @ A9 )
% 0.07/0.32 => ( boolean_algebra @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( order_top @ A9 )
% 0.07/0.32 => ( order_top @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( preorder @ A9 )
% 0.07/0.32 => ( preorder @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Lattices_Olattice,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( lattice @ A9 )
% 0.07/0.32 => ( lattice @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( order @ A9 )
% 0.07/0.32 => ( order @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Otop,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( top @ A9 )
% 0.07/0.32 => ( top @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Orderings_Oord,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( ord @ A9 )
% 0.07/0.32 => ( ord @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.07/0.32 ! [A8: $tType,A9: $tType] :
% 0.07/0.32 ( ( uminus @ A9 )
% 0.07/0.32 => ( uminus @ ( A8 > A9 ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
% 0.07/0.32 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
% 0.07/0.32 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
% 0.07/0.32 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
% 0.07/0.32 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 0.07/0.32 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 0.07/0.32 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 0.07/0.32 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
% 0.07/0.32 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 0.07/0.32 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_Set_Oset___Groups_Ouminus_12,axiom,
% 0.07/0.32 ! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
% 0.07/0.32 bounded_lattice_top @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 0.07/0.32 semilattice_sup @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_15,axiom,
% 0.07/0.32 boolean_algebra @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Oorder__top_16,axiom,
% 0.07/0.32 order_top @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
% 0.07/0.32 preorder @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.07/0.32 linorder @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Lattices_Olattice_18,axiom,
% 0.07/0.32 lattice @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
% 0.07/0.32 order @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Otop_20,axiom,
% 0.07/0.32 top @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
% 0.07/0.32 ord @ $o ).
% 0.07/0.32
% 0.07/0.32 thf(tcon_HOL_Obool___Groups_Ouminus_22,axiom,
% 0.07/0.32 uminus @ $o ).
% 0.07/0.32
% 0.07/0.32 % Conjectures (3)
% 0.07/0.32 thf(conj_0,hypothesis,
% 0.07/0.32 inj_on @ a @ b @ f @ ( sup_sup @ ( set @ a ) @ ( image @ c @ a @ g @ ( top_top @ ( set @ c ) ) ) @ ( image @ c @ a @ h @ ( top_top @ ( set @ c ) ) ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(conj_1,hypothesis,
% 0.07/0.32 ( ( comp @ a @ b @ c @ f @ g )
% 0.07/0.32 = ( comp @ a @ b @ c @ f @ h ) ) ).
% 0.07/0.32
% 0.07/0.32 thf(conj_2,conjecture,
% 0.07/0.32 g = h ).
% 0.07/0.32
% 0.07/0.32 %------------------------------------------------------------------------------
% 0.07/0.32 ------------------------
% 2.09/1.18 ---- Embedded in THF ---
% 2.09/1.18 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.09/1.18 %%% Generated on Tue Feb 24 03:26:11 EST 2026
% 2.09/1.18 %%% using '$$dhol' embedding, version 1.3.0.
% 2.09/1.18 %%% Logic specification used:
% 2.09/1.18 %%% thf(spec, logic, $$dhol).
% 2.09/1.18
% 2.09/1.18 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2DTF32419.p
% 2.09/1.18 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.09/1.18 thf(ty_tf_c, type, c: $tType).
% 2.09/1.18 thf(ty_tf_b, type, b: $tType).
% 2.09/1.18 thf(ty_tf_a, type, a: $tType).
% 2.09/1.18 thf(sy_cl_Lattices_Obounded__lattice, type, bounded_lattice: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Lattices_Olattice, type, lattice: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_cl_Lattices_Obounded__lattice__top, type, bounded_lattice_top: (!> [A:$tType]: ($o))).
% 2.09/1.18 thf(sy_c_Finite__Set_OFpow, type, finite_Fpow: (!> [A:$tType]: (((set @ A) > (set @ (set @ A)))))).
% 2.09/1.18 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.09/1.18 thf(sy_c_Fun_Oinj__on, type, inj_on: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > $o))))).
% 2.09/1.18 thf(sy_c_Fun_Othe__inv__into, type, the_inv_into: (!> [A:$tType,B:$tType]: (((set @ A) > ((A > B) > (B > A)))))).
% 2.09/1.18 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.09/1.18 thf(sy_c_Lattices_Osup__class_Osup, type, sup_sup: (!> [A:$tType]: ((A > (A > A))))).
% 2.09/1.18 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.09/1.18 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.09/1.18 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.09/1.18 thf(sy_c_Set_Oimage, type, image: (!> [A:$tType,B:$tType]: (((A > B) > ((set @ A) > (set @ B)))))).
% 2.09/1.18 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.09/1.18 thf(sy_v_f, type, f: (a > b)).
% 2.09/1.18 thf(sy_v_g, type, g: (c > a)).
% 2.09/1.18 thf(sy_v_h, type, h: (c > a)).
% 2.09/1.18 thf(fact_0_o__ext, axiom, (! [C:$tType,A:$tType,B:$tType,H:(B > A),F:(A > C),G:(A > C)]: (((! [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ H) @ (top_top @ (set @ B)))) => ((F @ X) = (G @ X))))) => ((((((comp @ A) @ C) @ B) @ F) @ H) = (((((comp @ A) @ C) @ B) @ G) @ H)))))).
% 2.09/1.18 thf(fact_1_o__cong, axiom, (! [C:$tType,B:$tType,A:$tType,H:(A > B),I:(A > B),F:(B > C),G:(B > C)]: (((H = I) => ((! [X:B]: (((((member @ B) @ X) @ ((((image @ A) @ B) @ I) @ (top_top @ (set @ A)))) => ((F @ X) = (G @ X))))) => ((((((comp @ B) @ C) @ A) @ F) @ H) = (((((comp @ B) @ C) @ A) @ F) @ I))))))).
% 2.09/1.18 thf(fact_2_o__inj__on, axiom, (! [B:$tType,C:$tType,A:$tType,F:(C > B),G:(A > C),H:(A > C)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ C) @ B) @ A) @ F) @ H)) => (((((inj_on @ C) @ B) @ F) @ (((sup_sup @ (set @ C)) @ ((((image @ A) @ C) @ G) @ (top_top @ (set @ A)))) @ ((((image @ A) @ C) @ H) @ (top_top @ (set @ A))))) => (G = H)))))).
% 2.09/1.18 thf(fact_3_ex__o__conv, axiom, (! [B:$tType,A:$tType,C:$tType,G:(A > C),F:(B > C)]: (((? [H2:(A > B)]: ((G = (((((comp @ B) @ C) @ A) @ F) @ H2)))) = (! [X2:C]: (((((member @ C) @ X2) @ ((((image @ A) @ C) @ G) @ (top_top @ (set @ A)))) => (? [Y:B]: ((X2 = (F @ Y))))))))))).
% 2.09/1.18 thf(fact_4_o__eq__conv, axiom, (! [B:$tType,A:$tType,C:$tType,F:(C > B),H:(A > C),G:(C > B)]: ((((((((comp @ C) @ B) @ A) @ F) @ H) = (((((comp @ C) @ B) @ A) @ G) @ H)) = (! [X2:C]: (((((member @ C) @ X2) @ ((((image @ A) @ C) @ H) @ (top_top @ (set @ A)))) => ((F @ X2) = (G @ X2))))))))).
% 2.09/1.18 thf(fact_5_sup__top__left, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X3:A]: (((((sup_sup @ A) @ (top_top @ A)) @ X3) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_6_sup__top__right, axiom, (! [A:$tType]: (((bounded_lattice_top @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ (top_top @ A)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_7_inj__on__Un__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B2)) = (A2 = B2)))))).
% 2.09/1.18 thf(fact_8_inj__compose, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(C > A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((((inj_on @ C) @ A) @ G) @ (top_top @ (set @ C))) => ((((inj_on @ C) @ B) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ (top_top @ (set @ C)))))))).
% 2.09/1.18 thf(fact_9_fun_Oinj__map, axiom, (! [B:$tType,A:$tType,D:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ (D > A)) @ (D > B)) @ ((((comp @ A) @ B) @ D) @ F)) @ (top_top @ (set @ (D > A)))))))).
% 2.09/1.18 thf(fact_10_comp__inj__on, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A2:(set @ A),G:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((((inj_on @ B) @ C) @ G) @ ((((image @ A) @ B) @ F) @ A2)) => ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ A2)))))).
% 2.09/1.18 thf(fact_11_inj__on__imageI, axiom, (! [B:$tType,C:$tType,A:$tType,G:(C > B),F:(A > C),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ G) @ F)) @ A2) => ((((inj_on @ C) @ B) @ G) @ ((((image @ A) @ C) @ F) @ A2)))))).
% 2.09/1.18 thf(fact_12_comp__inj__on__iff, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > B),A2:(set @ A),F2:(B > C)]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((((inj_on @ B) @ C) @ F2) @ ((((image @ A) @ B) @ F) @ A2)) = ((((inj_on @ A) @ C) @ (((((comp @ B) @ C) @ A) @ F2) @ F)) @ A2)))))).
% 2.09/1.18 thf(fact_13_range__ex1__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),B3:B]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ (top_top @ (set @ A)))) = (? [X2:A]: (((B3 = (F @ X2)) & (! [Y:A]: (((B3 = (F @ Y)) => (Y = X2)))))))))))).
% 2.09/1.18 thf(fact_14_inj__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B2)) = (A2 = B2)))))).
% 2.09/1.18 thf(fact_15_inj__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A3:A,A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ A2)) = (((member @ A) @ A3) @ A2)))))).
% 2.09/1.18 thf(fact_16_comp__surj, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(A > C)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => ((((((image @ A) @ C) @ G) @ (top_top @ (set @ A))) = (top_top @ (set @ C))) => (((((image @ B) @ C) @ (((((comp @ A) @ C) @ B) @ G) @ F)) @ (top_top @ (set @ B))) = (top_top @ (set @ C)))))))).
% 2.09/1.18 thf(fact_17_sup_Oright__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ B3) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.09/1.18 thf(fact_18_sup__left__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)) = (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_19_sup_Oleft__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A]: (((((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)) = (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.09/1.18 thf(fact_20_sup__idem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ X3) = X3))))))).
% 2.09/1.18 thf(fact_21_sup_Oidem, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A]: (((((sup_sup @ A) @ A3) @ A3) = A3))))))).
% 2.09/1.18 thf(fact_22_sup__apply, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F3:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F3 @ X2)) @ (G2 @ X2))))))))).
% 2.09/1.18 thf(fact_23_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F3:(B > A),G2:(C > B),X2:C]: ((F3 @ (G2 @ X2)))))))).
% 2.09/1.18 thf(fact_24_sup__left__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z)) = (((sup_sup @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Z))))))))).
% 2.09/1.18 thf(fact_25_sup_Oleft__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A3:A,C2:A]: (((((sup_sup @ A) @ B3) @ (((sup_sup @ A) @ A3) @ C2)) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.09/1.18 thf(fact_26_sup__commute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 2.09/1.18 thf(fact_27_sup_Ocommute, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((sup_sup @ A) = (^ [A4:A,B4:A]: ((((sup_sup @ A) @ B4) @ A4)))))))).
% 2.09/1.18 thf(fact_28_sup__assoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z) = (((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z))))))))).
% 2.09/1.18 thf(fact_29_sup_Oassoc, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A,C2:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ C2) = (((sup_sup @ A) @ A3) @ (((sup_sup @ A) @ B3) @ C2))))))))).
% 2.09/1.18 thf(fact_30_boolean__algebra__cancel_Osup2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B2:A,K:A,B3:A,A3:A]: (((B2 = (((sup_sup @ A) @ K) @ B3)) => ((((sup_sup @ A) @ A3) @ B2) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B3)))))))))).
% 2.09/1.18 thf(fact_31_boolean__algebra__cancel_Osup1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A2:A,K:A,A3:A,B3:A]: (((A2 = (((sup_sup @ A) @ K) @ A3)) => ((((sup_sup @ A) @ A2) @ B3) = (((sup_sup @ A) @ K) @ (((sup_sup @ A) @ A3) @ B3)))))))))).
% 2.09/1.18 thf(fact_32_sup__fun__def, axiom, (! [B:$tType,A:$tType]: (((semilattice_sup @ B) => ((sup_sup @ (A > B)) = (^ [F3:(A > B),G2:(A > B),X2:A]: ((((sup_sup @ B) @ (F3 @ X2)) @ (G2 @ X2))))))))).
% 2.09/1.18 thf(fact_33_inf__sup__aci_I5_J, axiom, (! [A:$tType]: (((lattice @ A) => ((sup_sup @ A) = (^ [X2:A,Y:A]: ((((sup_sup @ A) @ Y) @ X2)))))))).
% 2.09/1.18 thf(fact_34_inf__sup__aci_I6_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A,Z:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z) = (((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z))))))))).
% 2.09/1.18 thf(fact_35_inf__sup__aci_I7_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A,Z:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ Y2) @ Z)) = (((sup_sup @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Z))))))))).
% 2.09/1.18 thf(fact_36_inf__sup__aci_I8_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)) = (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_37_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 2.09/1.18 thf(fact_38_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A3:(C > B),B3:(A > C),C2:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = C2) => ((A3 @ (B3 @ V)) = (C2 @ V)))))).
% 2.09/1.18 thf(fact_39_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B3:(A > C),C2:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => (! [V2:A]: (((A3 @ (B3 @ V2)) = (C2 @ (D2 @ V2))))))))).
% 2.09/1.18 thf(fact_40_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A3:(C > B),B3:(A > C),C2:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A3) @ B3) = (((((comp @ D) @ B) @ A) @ C2) @ D2)) => ((A3 @ (B3 @ V)) = (C2 @ (D2 @ V))))))).
% 2.09/1.18 thf(fact_41_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H)))))).
% 2.09/1.18 thf(fact_42_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F3:(B > C),G2:(A > B),X2:A]: ((F3 @ (G2 @ X2)))))))).
% 2.09/1.18 thf(fact_43_inj__on__inverseI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(B > A),F:(A > B)]: (((! [X:A]: (((((member @ A) @ X) @ A2) => ((G @ (F @ X)) = X)))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 2.09/1.18 thf(fact_44_inj__on__contraD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:A,Y2:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((X3 != Y2) => ((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => ((F @ X3) != (F @ Y2))))))))).
% 2.09/1.18 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A3:A,P:(A > $o)]: (((((member @ A) @ A3) @ ((collect @ A) @ P)) = (P @ A3))))).
% 2.09/1.18 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A2:(set @ A)]: ((((collect @ A) @ (^ [X2:A]: ((((member @ A) @ X2) @ A2)))) = A2)))).
% 2.09/1.18 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) = (Q @ X)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.09/1.18 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X:A]: (((F @ X) = (G @ X)))) => (F = G))))).
% 2.09/1.18 thf(fact_49_inj__on__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:A,Y2:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => (((F @ X3) = (F @ Y2)) = (X3 = Y2)))))))).
% 2.09/1.18 thf(fact_50_inj__on__cong, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B),G:(A > B)]: (((! [A5:A]: (((((member @ A) @ A5) @ A2) => ((F @ A5) = (G @ A5))))) => (((((inj_on @ A) @ B) @ F) @ A2) = ((((inj_on @ A) @ B) @ G) @ A2)))))).
% 2.09/1.18 thf(fact_51_inj__on__def, axiom, (! [B:$tType,A:$tType]: ((((inj_on @ A) @ B) = (^ [F3:(A > B),A6:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A6) => (! [Y:A]: (((((member @ A) @ Y) @ A6) => (((F3 @ X2) = (F3 @ Y)) => (X2 = Y)))))))))))))).
% 2.09/1.18 thf(fact_52_inj__onI, axiom, (! [B:$tType,A:$tType,A2:(set @ A),F:(A > B)]: (((! [X:A,Y3:A]: (((((member @ A) @ X) @ A2) => ((((member @ A) @ Y3) @ A2) => (((F @ X) = (F @ Y3)) => (X = Y3)))))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 2.09/1.18 thf(fact_53_inj__onD, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:A,Y2:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((F @ X3) = (F @ Y2)) => ((((member @ A) @ X3) @ A2) => ((((member @ A) @ Y2) @ A2) => (X3 = Y2)))))))).
% 2.09/1.18 thf(fact_54_surj__def, axiom, (! [B:$tType,A:$tType,F:(B > A)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) = (! [Y:A]: ((? [X2:B]: ((Y = (F @ X2)))))))))).
% 2.09/1.18 thf(fact_55_surjI, axiom, (! [B:$tType,A:$tType,G:(B > A),F:(A > B)]: (((! [X:A]: (((G @ (F @ X)) = X))) => (((((image @ B) @ A) @ G) @ (top_top @ (set @ B))) = (top_top @ (set @ A))))))).
% 2.09/1.18 thf(fact_56_surjE, axiom, (! [A:$tType,B:$tType,F:(B > A),Y2:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (~ ((! [X:B]: ((Y2 != (F @ X)))))))))).
% 2.09/1.18 thf(fact_57_surjD, axiom, (! [A:$tType,B:$tType,F:(B > A),Y2:A]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (? [X:B]: ((Y2 = (F @ X)))))))).
% 2.09/1.18 thf(fact_58_image__eq__imp__comp, axiom, (! [B:$tType,A:$tType,D:$tType,C:$tType,F:(B > A),A2:(set @ B),G:(C > A),B2:(set @ C),H:(A > D)]: (((((((image @ B) @ A) @ F) @ A2) = ((((image @ C) @ A) @ G) @ B2)) => (((((image @ B) @ D) @ (((((comp @ A) @ D) @ B) @ H) @ F)) @ A2) = ((((image @ C) @ D) @ (((((comp @ A) @ D) @ C) @ H) @ G)) @ B2)))))).
% 2.09/1.18 thf(fact_59_image__comp, axiom, (! [B:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),R:(set @ C)]: ((((((image @ B) @ A) @ F) @ ((((image @ C) @ B) @ G) @ R)) = ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ F) @ G)) @ R))))).
% 2.09/1.18 thf(fact_60_inj__on__image__iff, axiom, (! [B:$tType,A:$tType,A2:(set @ A),G:(A > B),F:(A > A)]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (! [Xa:A]: (((((member @ A) @ Xa) @ A2) => (((G @ (F @ X)) = (G @ (F @ Xa))) = ((G @ X) = (G @ Xa))))))))) => (((((inj_on @ A) @ A) @ F) @ A2) => (((((inj_on @ A) @ B) @ G) @ ((((image @ A) @ A) @ F) @ A2)) = ((((inj_on @ A) @ B) @ G) @ A2))))))).
% 2.09/1.18 thf(fact_61_inj__def, axiom, (! [B:$tType,A:$tType,F:(A > B)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) = (! [X2:A,Y:A]: ((((F @ X2) = (F @ Y)) => (X2 = Y)))))))).
% 2.09/1.18 thf(fact_62_inj__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),X3:A,Y2:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X3) = (F @ Y2)) = (X3 = Y2)))))).
% 2.09/1.18 thf(fact_63_injI, axiom, (! [B:$tType,A:$tType,F:(A > B)]: (((! [X:A,Y3:A]: ((((F @ X) = (F @ Y3)) => (X = Y3)))) => ((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))))))).
% 2.09/1.18 thf(fact_64_injD, axiom, (! [B:$tType,A:$tType,F:(A > B),X3:A,Y2:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((F @ X3) = (F @ Y2)) => (X3 = Y2)))))).
% 2.09/1.18 thf(fact_65_inj__on__imageI2, axiom, (! [B:$tType,C:$tType,A:$tType,F2:(C > B),F:(A > C),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ (((((comp @ C) @ B) @ A) @ F2) @ F)) @ A2) => ((((inj_on @ A) @ C) @ F) @ A2))))).
% 2.09/1.18 thf(fact_66_fun_Oinj__map__strong, axiom, (! [B:$tType,A:$tType,D:$tType,X3:(D > A),Xa2:(D > A),F:(A > B),Fa:(A > B)]: (((! [Z2:A,Za:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ X3) @ (top_top @ (set @ D)))) => ((((member @ A) @ Za) @ ((((image @ D) @ A) @ Xa2) @ (top_top @ (set @ D)))) => (((F @ Z2) = (Fa @ Za)) => (Z2 = Za)))))) => (((((((comp @ A) @ B) @ D) @ F) @ X3) = (((((comp @ A) @ B) @ D) @ Fa) @ Xa2)) => (X3 = Xa2)))))).
% 2.09/1.18 thf(fact_67_fun_Omap__cong0, axiom, (! [B:$tType,A:$tType,D:$tType,X3:(D > A),F:(A > B),G:(A > B)]: (((! [Z2:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ X3) @ (top_top @ (set @ D)))) => ((F @ Z2) = (G @ Z2))))) => ((((((comp @ A) @ B) @ D) @ F) @ X3) = (((((comp @ A) @ B) @ D) @ G) @ X3)))))).
% 2.09/1.18 thf(fact_68_fun_Omap__cong, axiom, (! [B:$tType,A:$tType,D:$tType,X3:(D > A),Ya:(D > A),F:(A > B),G:(A > B)]: (((X3 = Ya) => ((! [Z2:A]: (((((member @ A) @ Z2) @ ((((image @ D) @ A) @ Ya) @ (top_top @ (set @ D)))) => ((F @ Z2) = (G @ Z2))))) => ((((((comp @ A) @ B) @ D) @ F) @ X3) = (((((comp @ A) @ B) @ D) @ G) @ Ya))))))).
% 2.09/1.18 thf(fact_69_fun_Oset__map, axiom, (! [B:$tType,A:$tType,D:$tType,F:(A > B),V:(D > A)]: ((((((image @ D) @ B) @ (((((comp @ A) @ B) @ D) @ F) @ V)) @ (top_top @ (set @ D))) = ((((image @ A) @ B) @ F) @ ((((image @ D) @ A) @ V) @ (top_top @ (set @ D)))))))).
% 2.09/1.18 thf(fact_70_surj__fun__eq, axiom, (! [B:$tType,C:$tType,A:$tType,F:(B > A),X4:(set @ B),G1:(A > C),G22:(A > C)]: (((((((image @ B) @ A) @ F) @ X4) = (top_top @ (set @ A))) => ((! [X:B]: (((((member @ B) @ X) @ X4) => (((((((comp @ A) @ C) @ B) @ G1) @ F) @ X) = ((((((comp @ A) @ C) @ B) @ G22) @ F) @ X))))) => (G1 = G22)))))).
% 2.09/1.18 thf(fact_71_Un__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = ((((member @ A) @ C2) @ A2) | (((member @ A) @ C2) @ B2)))))).
% 2.09/1.18 thf(fact_72_UnCI, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: ((((~ ((((member @ A) @ C2) @ B2))) => (((member @ A) @ C2) @ A2)) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.09/1.18 thf(fact_73_iso__tuple__UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_74_UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_75_top__apply, axiom, (! [C:$tType,D:$tType]: (((top @ C) => ((top_top @ (D > C)) = (^ [X2:D]: ((top_top @ C)))))))).
% 2.09/1.18 thf(fact_76_image__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X3:B,A2:(set @ B)]: (((B3 = (F @ X3)) => ((((member @ B) @ X3) @ A2) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ A2))))))).
% 2.09/1.18 thf(fact_77_inj__imp__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((inj_on @ A) @ B) @ F) @ A2))))).
% 2.09/1.18 thf(fact_78_Un__UNIV__right, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_79_Un__UNIV__left, axiom, (! [A:$tType,B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ (top_top @ (set @ A))) @ B2) = (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_80_imageI, axiom, (! [B:$tType,A:$tType,X3:A,A2:(set @ A),F:(A > B)]: (((((member @ A) @ X3) @ A2) => (((member @ B) @ (F @ X3)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.09/1.18 thf(fact_81_image__iff, axiom, (! [A:$tType,B:$tType,Z:A,F:(B > A),A2:(set @ B)]: (((((member @ A) @ Z) @ ((((image @ B) @ A) @ F) @ A2)) = (? [X2:B]: (((((member @ B) @ X2) @ A2) & (Z = (F @ X2))))))))).
% 2.09/1.18 thf(fact_82_bex__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((? [X5:A]: (((((member @ A) @ X5) @ ((((image @ B) @ A) @ F) @ A2)) & (P @ X5)))) => (? [X:B]: (((((member @ B) @ X) @ A2) & (P @ (F @ X))))))))).
% 2.09/1.18 thf(fact_83_image__cong, axiom, (! [B:$tType,A:$tType,M:(set @ A),N:(set @ A),F:(A > B),G:(A > B)]: (((M = N) => ((! [X:A]: (((((member @ A) @ X) @ N) => ((F @ X) = (G @ X))))) => (((((image @ A) @ B) @ F) @ M) = ((((image @ A) @ B) @ G) @ N))))))).
% 2.09/1.18 thf(fact_84_ball__imageD, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:(A > $o)]: (((! [X:A]: (((((member @ A) @ X) @ ((((image @ B) @ A) @ F) @ A2)) => (P @ X)))) => (! [X5:B]: (((((member @ B) @ X5) @ A2) => (P @ (F @ X5))))))))).
% 2.09/1.18 thf(fact_85_rev__image__eqI, axiom, (! [B:$tType,A:$tType,X3:A,A2:(set @ A),B3:B,F:(A > B)]: (((((member @ A) @ X3) @ A2) => ((B3 = (F @ X3)) => (((member @ B) @ B3) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.09/1.18 thf(fact_86_UNIV__eq__I, axiom, (! [A:$tType,A2:(set @ A)]: (((! [X:A]: ((((member @ A) @ X) @ A2))) => ((top_top @ (set @ A)) = A2))))).
% 2.09/1.18 thf(fact_87_UNIV__witness, axiom, (! [A:$tType]: ((? [X:A]: ((((member @ A) @ X) @ (top_top @ (set @ A)))))))).
% 2.09/1.18 thf(fact_88_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.09/1.18 thf(fact_89_UnE, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => ((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ B2)))))).
% 2.09/1.18 thf(fact_90_UnI1, axiom, (! [A:$tType,C2:A,A2:(set @ A),B2:(set @ A)]: (((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.09/1.18 thf(fact_91_UnI2, axiom, (! [A:$tType,C2:A,B2:(set @ A),A2:(set @ A)]: (((((member @ A) @ C2) @ B2) => (((member @ A) @ C2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)))))).
% 2.09/1.18 thf(fact_92_bex__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((? [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) & (P @ X2)))) = ((? [X2:A]: (((((member @ A) @ X2) @ A2) & (P @ X2)))) | (? [X2:A]: (((((member @ A) @ X2) @ B2) & (P @ X2))))))))).
% 2.09/1.18 thf(fact_93_ball__Un, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),P:(A > $o)]: (((! [X2:A]: (((((member @ A) @ X2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (P @ X2)))) = ((! [X2:A]: (((((member @ A) @ X2) @ A2) => (P @ X2)))) & (! [X2:A]: (((((member @ A) @ X2) @ B2) => (P @ X2))))))))).
% 2.09/1.18 thf(fact_94_Un__assoc, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C3) = (((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C3)))))).
% 2.09/1.18 thf(fact_95_Un__absorb, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ A2) = A2)))).
% 2.09/1.18 thf(fact_96_Un__commute, axiom, (! [A:$tType]: (((sup_sup @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((((sup_sup @ (set @ A)) @ B5) @ A6))))))).
% 2.09/1.18 thf(fact_97_Un__left__absorb, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.09/1.18 thf(fact_98_Un__left__commute, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ B2) @ C3)) = (((sup_sup @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ C3)))))).
% 2.09/1.18 thf(fact_99_rangeI, axiom, (! [A:$tType,B:$tType,F:(B > A),X3:B]: ((((member @ A) @ (F @ X3)) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B))))))).
% 2.09/1.18 thf(fact_100_range__eqI, axiom, (! [A:$tType,B:$tType,B3:A,F:(B > A),X3:B]: (((B3 = (F @ X3)) => (((member @ A) @ B3) @ ((((image @ B) @ A) @ F) @ (top_top @ (set @ B)))))))).
% 2.09/1.18 thf(fact_101_image__Un, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ B)]: ((((((image @ B) @ A) @ F) @ (((sup_sup @ (set @ B)) @ A2) @ B2)) = (((sup_sup @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ ((((image @ B) @ A) @ F) @ B2)))))).
% 2.09/1.18 thf(fact_102_Inf_OINF__image, axiom, (! [B:$tType,A:$tType,C:$tType,Inf:((set @ A) > A),G:(B > A),F:(C > B),A2:(set @ C)]: (((Inf @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A2))) = (Inf @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A2)))))).
% 2.09/1.18 thf(fact_103_Sup_OSUP__image, axiom, (! [B:$tType,A:$tType,C:$tType,Sup:((set @ A) > A),G:(B > A),F:(C > B),A2:(set @ C)]: (((Sup @ ((((image @ B) @ A) @ G) @ ((((image @ C) @ B) @ F) @ A2))) = (Sup @ ((((image @ C) @ A) @ (((((comp @ B) @ A) @ C) @ G) @ F)) @ A2)))))).
% 2.09/1.18 thf(fact_104_the__inv__into__comp, axiom, (! [A:$tType,C:$tType,B:$tType,F:(A > B),G:(C > A),A2:(set @ C),X3:B]: ((((((inj_on @ A) @ B) @ F) @ ((((image @ C) @ A) @ G) @ A2)) => (((((inj_on @ C) @ A) @ G) @ A2) => ((((member @ B) @ X3) @ ((((image @ A) @ B) @ F) @ ((((image @ C) @ A) @ G) @ A2))) => ((((((the_inv_into @ C) @ B) @ A2) @ (((((comp @ A) @ B) @ C) @ F) @ G)) @ X3) = ((((((comp @ A) @ C) @ B) @ ((((the_inv_into @ C) @ A) @ A2) @ G)) @ ((((the_inv_into @ A) @ B) @ ((((image @ C) @ A) @ G) @ A2)) @ F)) @ X3)))))))).
% 2.09/1.18 thf(fact_105_the__inv__into__onto, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((((image @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A2) @ F)) @ ((((image @ A) @ B) @ F) @ A2)) = A2))))).
% 2.09/1.18 thf(fact_106_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G:(C > B),X3:C,F2:(D > A),G3:(E > D),X6:E]: ((((F @ (G @ X3)) = (F2 @ (G3 @ X6))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X3) = ((((((comp @ D) @ A) @ E) @ F2) @ G3) @ X6)))))).
% 2.09/1.18 thf(fact_107_comp2__conv, axiom, (! [A:$tType,B:$tType,C:$tType]: (((((comp @ C) @ B) @ A) = (^ [F1:(C > B),F22:(A > C),X2:A]: ((F1 @ (F22 @ X2)))))))).
% 2.09/1.18 thf(fact_108_top1I, axiom, (! [A:$tType,X3:A]: (((top_top @ (A > $o)) @ X3)))).
% 2.09/1.18 thf(fact_109_the__inv__into__f__eq, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:A,Y2:B]: ((((((inj_on @ A) @ B) @ F) @ A2) => (((F @ X3) = Y2) => ((((member @ A) @ X3) @ A2) => ((((((the_inv_into @ A) @ B) @ A2) @ F) @ Y2) = X3))))))).
% 2.09/1.18 thf(fact_110_the__inv__into__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ A) @ X3) @ A2) => ((((((the_inv_into @ A) @ B) @ A2) @ F) @ (F @ X3)) = X3)))))).
% 2.09/1.18 thf(fact_111_inj__on__the__inv__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((inj_on @ B) @ A) @ ((((the_inv_into @ A) @ B) @ A2) @ F)) @ ((((image @ A) @ B) @ F) @ A2)))))).
% 2.09/1.18 thf(fact_112_f__the__inv__into__f, axiom, (! [A:$tType,B:$tType,F:(A > B),A2:(set @ A),Y2:B]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ B) @ Y2) @ ((((image @ A) @ B) @ F) @ A2)) => ((F @ (((((the_inv_into @ A) @ B) @ A2) @ F) @ Y2)) = Y2)))))).
% 2.09/1.18 thf(fact_113_the__inv__f__f, axiom, (! [B:$tType,A:$tType,F:(A > B),X3:A]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((((the_inv_into @ A) @ B) @ (top_top @ (set @ A))) @ F) @ (F @ X3)) = X3))))).
% 2.09/1.18 thf(fact_114_Sup_OSUP__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C3:(B > A),D3:(B > A),Sup:((set @ A) > A)]: (((A2 = B2) => ((! [X:B]: (((((member @ B) @ X) @ B2) => ((C3 @ X) = (D3 @ X))))) => ((Sup @ ((((image @ B) @ A) @ C3) @ A2)) = (Sup @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.09/1.18 thf(fact_115_Inf_OINF__cong, axiom, (! [A:$tType,B:$tType,A2:(set @ B),B2:(set @ B),C3:(B > A),D3:(B > A),Inf:((set @ A) > A)]: (((A2 = B2) => ((! [X:B]: (((((member @ B) @ X) @ B2) => ((C3 @ X) = (D3 @ X))))) => ((Inf @ ((((image @ B) @ A) @ C3) @ A2)) = (Inf @ ((((image @ B) @ A) @ D3) @ B2)))))))).
% 2.09/1.18 thf(fact_116_comp3__conv, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,F12:(D > B),F23:(C > D),F32:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F12) @ F23)) @ F32) = (^ [X2:A]: ((F12 @ (F23 @ (F32 @ X2))))))))).
% 2.09/1.18 thf(fact_117_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A))))))))).
% 2.09/1.18 thf(fact_118_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X3:A]: (((P & ((top_top @ (A > $o)) @ X3)) = P)))).
% 2.09/1.18 thf(fact_119_top__conj_I1_J, axiom, (! [A:$tType,X3:A,P:$o]: (((((top_top @ (A > $o)) @ X3) & P) = P)))).
% 2.09/1.18 thf(fact_120_the__inv__into__into, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),X3:B,B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((member @ B) @ X3) @ ((((image @ A) @ B) @ F) @ A2)) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((member @ A) @ (((((the_inv_into @ A) @ B) @ A2) @ F) @ X3)) @ B2))))))).
% 2.09/1.18 thf(fact_121_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G:(C > B),H:(A > C),R1:(D > B),R2:(A > D),F:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R2)) => (((((((comp @ B) @ E) @ D) @ F) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R2))))))).
% 2.09/1.18 thf(fact_122_order__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.09/1.18 thf(fact_123_subsetI, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ A) @ X) @ B2)))) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.09/1.18 thf(fact_124_subset__antisym, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => (A2 = B2)))))).
% 2.09/1.18 thf(fact_125_le__sup__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ X3) @ Y2)) @ Z) = ((((ord_less_eq @ A) @ X3) @ Z) & (((ord_less_eq @ A) @ Y2) @ Z))))))))).
% 2.09/1.18 thf(fact_126_sup_Obounded__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) = ((((ord_less_eq @ A) @ B3) @ A3) & (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 2.09/1.18 thf(fact_127_Un__subset__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C3) = ((((ord_less_eq @ (set @ A)) @ A2) @ C3) & (((ord_less_eq @ (set @ A)) @ B2) @ C3)))))).
% 2.09/1.18 thf(fact_128_bex__cong2, axiom, (! [B:$tType,A:$tType,I2:(set @ A),A2:(set @ A),F:(A > B),G:(A > B),P:(B > $o)]: (((((ord_less_eq @ (set @ A)) @ I2) @ A2) => ((! [X:A]: (((((member @ A) @ X) @ I2) => ((F @ X) = (G @ X))))) => ((? [X2:A]: (((((member @ A) @ X2) @ I2) & (P @ (F @ X2))))) = (? [X2:A]: (((((member @ A) @ X2) @ I2) & (P @ (G @ X2))))))))))).
% 2.09/1.18 thf(fact_129_ball__cong2, axiom, (! [B:$tType,A:$tType,I2:(set @ A),A2:(set @ A),F:(A > B),G:(A > B),P:(B > $o)]: (((((ord_less_eq @ (set @ A)) @ I2) @ A2) => ((! [X:A]: (((((member @ A) @ X) @ A2) => ((F @ X) = (G @ X))))) => ((! [X2:A]: (((((member @ A) @ X2) @ I2) => (P @ (F @ X2))))) = (! [X2:A]: (((((member @ A) @ X2) @ I2) => (P @ (G @ X2))))))))))).
% 2.09/1.18 thf(fact_130_bex__subset__imp__bex, axiom, (! [A:$tType,A2:(set @ A),P:(A > $o),B2:(set @ A)]: (((? [X5:A]: (((((member @ A) @ X5) @ A2) & (P @ X5)))) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (? [X:A]: (((((member @ A) @ X) @ B2) & (P @ X))))))))).
% 2.09/1.18 thf(fact_131_ball__subset__imp__ball, axiom, (! [A:$tType,B2:(set @ A),P:(A > $o),A2:(set @ A)]: (((! [X:A]: (((((member @ A) @ X) @ B2) => (P @ X)))) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (! [X5:A]: (((((member @ A) @ X5) @ A2) => (P @ X5))))))))).
% 2.09/1.18 thf(fact_132_top_Oextremum__uniqueI, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) => (A3 = (top_top @ A))))))))).
% 2.09/1.18 thf(fact_133_top_Oextremum__unique, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A3) = (A3 = (top_top @ A))))))))).
% 2.09/1.18 thf(fact_134_top__greatest, axiom, (! [A:$tType]: (((order_top @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ (top_top @ A)))))))).
% 2.09/1.18 thf(fact_135_subset__image__iff, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) = (? [AA:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ AA) @ A2) & (B2 = ((((image @ B) @ A) @ F) @ AA))))))))).
% 2.09/1.18 thf(fact_136_image__subset__iff, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2)) @ B2) = (! [X2:B]: (((((member @ B) @ X2) @ A2) => (((member @ A) @ (F @ X2)) @ B2)))))))).
% 2.09/1.18 thf(fact_137_subset__imageE, axiom, (! [A:$tType,B:$tType,B2:(set @ A),F:(B > A),A2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ B2) @ ((((image @ B) @ A) @ F) @ A2)) => (~ ((! [C4:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ C4) @ A2) => (B2 != ((((image @ B) @ A) @ F) @ C4))))))))))).
% 2.09/1.18 thf(fact_138_image__subsetI, axiom, (! [A:$tType,B:$tType,A2:(set @ A),F:(A > B),B2:(set @ B)]: (((! [X:A]: (((((member @ A) @ X) @ A2) => (((member @ B) @ (F @ X)) @ B2)))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ B2))))).
% 2.09/1.18 thf(fact_139_image__mono, axiom, (! [B:$tType,A:$tType,A2:(set @ A),B2:(set @ A),F:(A > B)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)))))).
% 2.09/1.18 thf(fact_140_subset__UNIV, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_141_sup_OcoboundedI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,B3:A,A3:A]: (((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_142_sup_OcoboundedI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ C2) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_143_sup_Oabsorb__iff2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [A4:A,B4:A]: (((((sup_sup @ A) @ A4) @ B4) = B4)))))))).
% 2.09/1.18 thf(fact_144_sup_Oabsorb__iff1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: (((((sup_sup @ A) @ A4) @ B4) = A4)))))))).
% 2.09/1.18 thf(fact_145_sup_Ocobounded2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A3:A]: ((((ord_less_eq @ A) @ B3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.09/1.18 thf(fact_146_sup_Ocobounded1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A]: ((((ord_less_eq @ A) @ A3) @ (((sup_sup @ A) @ A3) @ B3)))))))).
% 2.09/1.18 thf(fact_147_sup_Oorder__iff, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [B4:A,A4:A]: ((A4 = (((sup_sup @ A) @ A4) @ B4))))))))).
% 2.09/1.18 thf(fact_148_sup_OboundedI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ A3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3))))))))).
% 2.09/1.18 thf(fact_149_sup_OboundedE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,C2:A,A3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ B3) @ C2)) @ A3) => (~ (((((ord_less_eq @ A) @ B3) @ A3) => (~ ((((ord_less_eq @ A) @ C2) @ A3))))))))))))).
% 2.09/1.18 thf(fact_150_sup__absorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((sup_sup @ A) @ X3) @ Y2) = Y2)))))))).
% 2.09/1.18 thf(fact_151_sup__absorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => ((((sup_sup @ A) @ X3) @ Y2) = X3)))))))).
% 2.09/1.18 thf(fact_152_sup_Oabsorb2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((sup_sup @ A) @ A3) @ B3) = B3)))))))).
% 2.09/1.18 thf(fact_153_sup_Oabsorb1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((sup_sup @ A) @ A3) @ B3) = A3)))))))).
% 2.09/1.18 thf(fact_154_sup__unique, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [F:(A > (A > A)),X3:A,Y2:A]: (((! [X:A,Y3:A]: ((((ord_less_eq @ A) @ X) @ ((F @ X) @ Y3)))) => ((! [X:A,Y3:A]: ((((ord_less_eq @ A) @ Y3) @ ((F @ X) @ Y3)))) => ((! [X:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ Y3) @ X) => ((((ord_less_eq @ A) @ Z2) @ X) => (((ord_less_eq @ A) @ ((F @ Y3) @ Z2)) @ X))))) => ((((sup_sup @ A) @ X3) @ Y2) = ((F @ X3) @ Y2))))))))))).
% 2.09/1.18 thf(fact_155_sup_OorderI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A]: (((A3 = (((sup_sup @ A) @ A3) @ B3)) => (((ord_less_eq @ A) @ B3) @ A3)))))))).
% 2.09/1.18 thf(fact_156_sup_OorderE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => (A3 = (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_157_le__iff__sup, axiom, (! [A:$tType]: (((semilattice_sup @ A) => ((ord_less_eq @ A) = (^ [X2:A,Y:A]: (((((sup_sup @ A) @ X2) @ Y) = Y)))))))).
% 2.09/1.18 thf(fact_158_sup__least, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y2:A,X3:A,Z:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => ((((ord_less_eq @ A) @ Z) @ X3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ Y2) @ Z)) @ X3))))))))).
% 2.09/1.18 thf(fact_159_sup__mono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,C2:A,B3:A,D2:A]: (((((ord_less_eq @ A) @ A3) @ C2) => ((((ord_less_eq @ A) @ B3) @ D2) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ (((sup_sup @ A) @ C2) @ D2)))))))))).
% 2.09/1.18 thf(fact_160_sup_Omono, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [C2:A,A3:A,D2:A,B3:A]: (((((ord_less_eq @ A) @ C2) @ A3) => ((((ord_less_eq @ A) @ D2) @ B3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ C2) @ D2)) @ (((sup_sup @ A) @ A3) @ B3)))))))))).
% 2.09/1.18 thf(fact_161_le__supI2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,B3:A,A3:A]: (((((ord_less_eq @ A) @ X3) @ B3) => (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_162_le__supI1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,A3:A,B3:A]: (((((ord_less_eq @ A) @ X3) @ A3) => (((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_163_sup__ge2, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [Y2:A,X3:A]: ((((ord_less_eq @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_164_sup__ge1, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_165_le__supI, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,X3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ X3) => ((((ord_less_eq @ A) @ B3) @ X3) => (((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X3))))))))).
% 2.09/1.18 thf(fact_166_le__supE, axiom, (! [A:$tType]: (((semilattice_sup @ A) => (! [A3:A,B3:A,X3:A]: (((((ord_less_eq @ A) @ (((sup_sup @ A) @ A3) @ B3)) @ X3) => (~ (((((ord_less_eq @ A) @ A3) @ X3) => (~ ((((ord_less_eq @ A) @ B3) @ X3))))))))))))).
% 2.09/1.18 thf(fact_167_inf__sup__ord_I3_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [X3:A,Y2:A]: ((((ord_less_eq @ A) @ X3) @ (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_168_inf__sup__ord_I4_J, axiom, (! [A:$tType]: (((lattice @ A) => (! [Y2:A,X3:A]: ((((ord_less_eq @ A) @ Y2) @ (((sup_sup @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_169_in__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ X3) @ A2) => (((member @ A) @ X3) @ B2)))))).
% 2.09/1.18 thf(fact_170_subsetD, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ A) @ C2) @ A2) => (((member @ A) @ C2) @ B2)))))).
% 2.09/1.18 thf(fact_171_equalityE, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (~ (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (~ ((((ord_less_eq @ (set @ A)) @ B2) @ A2)))))))))).
% 2.09/1.18 thf(fact_172_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [X2:A]: (((((member @ A) @ X2) @ A6) => (((member @ A) @ X2) @ B5)))))))))).
% 2.09/1.18 thf(fact_173_equalityD1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ A2) @ B2))))).
% 2.09/1.18 thf(fact_174_equalityD2, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((A2 = B2) => (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.09/1.18 thf(fact_175_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: ((! [T:A]: (((((member @ A) @ T) @ A6) => (((member @ A) @ T) @ B5)))))))))).
% 2.09/1.18 thf(fact_176_subset__refl, axiom, (! [A:$tType,A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ A2)))).
% 2.09/1.18 thf(fact_177_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X:A]: (((P @ X) => (Q @ X)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.09/1.18 thf(fact_178_subset__trans, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ A2) @ C3)))))).
% 2.09/1.18 thf(fact_179_le__funD, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.09/1.18 thf(fact_180_le__funE, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F) @ G) => (((ord_less_eq @ B) @ (F @ X3)) @ (G @ X3))))))))).
% 2.09/1.18 thf(fact_181_le__funI, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => (! [F:(A > B),G:(A > B)]: (((! [X:A]: ((((ord_less_eq @ B) @ (F @ X)) @ (G @ X)))) => (((ord_less_eq @ (A > B)) @ F) @ G)))))))).
% 2.09/1.18 thf(fact_182_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z3:(set @ A)]: ((Y4 = Z3))) = (^ [A6:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A6) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A6)))))))).
% 2.09/1.18 thf(fact_183_le__fun__def, axiom, (! [B:$tType,A:$tType]: (((ord @ B) => ((ord_less_eq @ (A > B)) = (^ [F3:(A > B),G2:(A > B)]: ((! [X2:A]: ((((ord_less_eq @ B) @ (F3 @ X2)) @ (G2 @ X2))))))))))).
% 2.09/1.18 thf(fact_184_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X2:A]: (((P @ X2) => (Q @ X2)))))))).
% 2.09/1.18 thf(fact_185_order__subst1, axiom, (! [A:$tType,B:$tType]: ((((order @ B) & (order @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((((ord_less_eq @ A) @ A3) @ (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X:B,Y3:B]: (((((ord_less_eq @ B) @ X) @ Y3) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 2.09/1.18 thf(fact_186_order__subst2, axiom, (! [A:$tType,C:$tType]: ((((order @ C) & (order @ A)) => (! [A3:A,B3:A,F:(A > C),C2:C]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ C) @ (F @ B3)) @ C2) => ((! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ C) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ C) @ (F @ A3)) @ C2)))))))))).
% 2.09/1.18 thf(fact_187_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,F:(B > A),B3:B,C2:B]: (((A3 = (F @ B3)) => ((((ord_less_eq @ B) @ B3) @ C2) => ((! [X:B,Y3:B]: (((((ord_less_eq @ B) @ X) @ Y3) => (((ord_less_eq @ A) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ A) @ A3) @ (F @ C2))))))))))).
% 2.09/1.18 thf(fact_188_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: ((((ord @ B) & (ord @ A)) => (! [A3:A,B3:A,F:(A > B),C2:B]: (((((ord_less_eq @ A) @ A3) @ B3) => (((F @ B3) = C2) => ((! [X:A,Y3:A]: (((((ord_less_eq @ A) @ X) @ Y3) => (((ord_less_eq @ B) @ (F @ X)) @ (F @ Y3))))) => (((ord_less_eq @ B) @ (F @ A3)) @ C2)))))))))).
% 2.09/1.18 thf(fact_189_eq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [X2:A,Y:A]: (((((ord_less_eq @ A) @ X2) @ Y) & (((ord_less_eq @ A) @ Y) @ X2))))))))).
% 2.09/1.18 thf(fact_190_antisym, axiom, (! [A:$tType]: (((order @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ X3) => (X3 = Y2))))))))).
% 2.09/1.18 thf(fact_191_linear, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) | (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.09/1.18 thf(fact_192_eq__refl, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A]: (((X3 = Y2) => (((ord_less_eq @ A) @ X3) @ Y2)))))))).
% 2.09/1.18 thf(fact_193_le__cases, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y2))) => (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.09/1.18 thf(fact_194_order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.09/1.18 thf(fact_195_le__cases3, axiom, (! [A:$tType]: (((linorder @ A) => (! [X3:A,Y2:A,Z:A]: ((((((ord_less_eq @ A) @ X3) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ Z)))) => (((((ord_less_eq @ A) @ Y2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z)))) => (((((ord_less_eq @ A) @ X3) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ Y2)))) => (((((ord_less_eq @ A) @ Z) @ Y2) => (~ ((((ord_less_eq @ A) @ Y2) @ X3)))) => (((((ord_less_eq @ A) @ Y2) @ Z) => (~ ((((ord_less_eq @ A) @ Z) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y2))))))))))))))))).
% 2.09/1.18 thf(fact_196_antisym__conv, axiom, (! [A:$tType]: (((order @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y2) = (X3 = Y2))))))))).
% 2.09/1.18 thf(fact_197_order__class_Oorder_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ A4) @ B4) & (((ord_less_eq @ A) @ B4) @ A4))))))))).
% 2.09/1.18 thf(fact_198_ord__eq__le__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((A3 = B3) => ((((ord_less_eq @ A) @ B3) @ C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.09/1.18 thf(fact_199_ord__le__eq__trans, axiom, (! [A:$tType]: (((ord @ A) => (! [A3:A,B3:A,C2:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((B3 = C2) => (((ord_less_eq @ A) @ A3) @ C2))))))))).
% 2.09/1.18 thf(fact_200_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A,B3:A]: (((((ord_less_eq @ A) @ A3) @ B3) => ((((ord_less_eq @ A) @ B3) @ A3) => (A3 = B3))))))))).
% 2.09/1.18 thf(fact_201_order__trans, axiom, (! [A:$tType]: (((preorder @ A) => (! [X3:A,Y2:A,Z:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => ((((ord_less_eq @ A) @ Y2) @ Z) => (((ord_less_eq @ A) @ X3) @ Z))))))))).
% 2.09/1.18 thf(fact_202_dual__order_Orefl, axiom, (! [A:$tType]: (((order @ A) => (! [A3:A]: ((((ord_less_eq @ A) @ A3) @ A3))))))).
% 2.09/1.18 thf(fact_203_linorder__wlog, axiom, (! [A:$tType]: (((linorder @ A) => (! [P:(A > (A > $o)),A3:A,B3:A]: (((! [A5:A,B6:A]: (((((ord_less_eq @ A) @ A5) @ B6) => ((P @ A5) @ B6)))) => ((! [A5:A,B6:A]: ((((P @ B6) @ A5) => ((P @ A5) @ B6)))) => ((P @ A3) @ B3))))))))).
% 2.09/1.18 thf(fact_204_dual__order_Otrans, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A,C2:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ C2) @ B3) => (((ord_less_eq @ A) @ C2) @ A3))))))))).
% 2.09/1.18 thf(fact_205_dual__order_Oeq__iff, axiom, (! [A:$tType]: (((order @ A) => ((^ [Y4:A,Z3:A]: ((Y4 = Z3))) = (^ [A4:A,B4:A]: (((((ord_less_eq @ A) @ B4) @ A4) & (((ord_less_eq @ A) @ A4) @ B4))))))))).
% 2.09/1.18 thf(fact_206_dual__order_Oantisym, axiom, (! [A:$tType]: (((order @ A) => (! [B3:A,A3:A]: (((((ord_less_eq @ A) @ B3) @ A3) => ((((ord_less_eq @ A) @ A3) @ B3) => (A3 = B3))))))))).
% 2.09/1.18 thf(fact_207_Un__mono, axiom, (! [A:$tType,A2:(set @ A),C3:(set @ A),B2:(set @ A),D3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B2) @ D3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ (((sup_sup @ (set @ A)) @ C3) @ D3))))))).
% 2.09/1.18 thf(fact_208_Un__least, axiom, (! [A:$tType,A2:(set @ A),C3:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => (((ord_less_eq @ (set @ A)) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) @ C3)))))).
% 2.09/1.18 thf(fact_209_Un__upper1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.09/1.18 thf(fact_210_Un__upper2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ B2) @ (((sup_sup @ (set @ A)) @ A2) @ B2))))).
% 2.09/1.18 thf(fact_211_Un__absorb1, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = B2))))).
% 2.09/1.18 thf(fact_212_Un__absorb2, axiom, (! [A:$tType,B2:(set @ A),A2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((sup_sup @ (set @ A)) @ A2) @ B2) = A2))))).
% 2.09/1.18 thf(fact_213_subset__UnE, axiom, (! [A:$tType,C3:(set @ A),A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ C3) @ (((sup_sup @ (set @ A)) @ A2) @ B2)) => (~ ((! [A7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ A2) => (! [B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B7) @ B2) => (C3 != (((sup_sup @ (set @ A)) @ A7) @ B7)))))))))))))).
% 2.09/1.18 thf(fact_214_subset__Un__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A6:(set @ A),B5:(set @ A)]: (((((sup_sup @ (set @ A)) @ A6) @ B5) = B5))))))).
% 2.09/1.18 thf(fact_215_inj__on__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((inj_on @ A) @ B) @ F) @ B2)))))).
% 2.09/1.18 thf(fact_216_subset__inj__on, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:(set @ A),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((inj_on @ A) @ B) @ F) @ A2)))))).
% 2.09/1.18 thf(fact_217_SetInterval2_Oinj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A),A3:A]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((ord_less_eq @ (set @ A)) @ B2) @ A2) => ((((member @ A) @ A3) @ A2) => ((((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ B2)) = (((member @ A) @ A3) @ B2)))))))).
% 2.09/1.18 thf(fact_218_inj__on__image__eq__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),C3:(set @ A),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ C3) => ((((ord_less_eq @ (set @ A)) @ A2) @ C3) => ((((ord_less_eq @ (set @ A)) @ B2) @ C3) => ((((((image @ A) @ B) @ F) @ A2) = ((((image @ A) @ B) @ F) @ B2)) = (A2 = B2)))))))).
% 2.09/1.18 thf(fact_219_Fun_Oinj__on__image__mem__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),B2:(set @ A),A3:A,A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ B2) => ((((member @ A) @ A3) @ B2) => ((((ord_less_eq @ (set @ A)) @ A2) @ B2) => ((((member @ B) @ (F @ A3)) @ ((((image @ A) @ B) @ F) @ A2)) = (((member @ A) @ A3) @ A2)))))))).
% 2.09/1.18 thf(fact_220_inj__image__subset__iff, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A),B2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => ((((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2)) @ ((((image @ A) @ B) @ F) @ B2)) = (((ord_less_eq @ (set @ A)) @ A2) @ B2)))))).
% 2.09/1.18 thf(fact_221_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 2.09/1.18 thf(fact_222_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G:(C > B),H:(A > C),R:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G)) @ H) = (((((comp @ B) @ D) @ A) @ F) @ R)))))).
% 2.09/1.18 thf(fact_223_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 2.09/1.18 thf(fact_224_subset__image__inj, axiom, (! [A:$tType,B:$tType,S:(set @ A),F:(B > A),T2:(set @ B)]: (((((ord_less_eq @ (set @ A)) @ S) @ ((((image @ B) @ A) @ F) @ T2)) = (? [U:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ U) @ T2) & (((((inj_on @ B) @ A) @ F) @ U) & (S = ((((image @ B) @ A) @ F) @ U)))))))))).
% 2.09/1.18 thf(fact_225_ex__subset__image__inj, axiom, (! [A:$tType,B:$tType,F:(B > A),S:(set @ B),P:((set @ A) > $o)]: (((? [T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T3) @ ((((image @ B) @ A) @ F) @ S)) & (P @ T3)))) = (? [T3:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ T3) @ S) & (((((inj_on @ B) @ A) @ F) @ T3) & (P @ ((((image @ B) @ A) @ F) @ T3)))))))))).
% 2.09/1.18 thf(fact_226_all__subset__image__inj, axiom, (! [A:$tType,B:$tType,F:(B > A),S:(set @ B),P:((set @ A) > $o)]: (((! [T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ T3) @ ((((image @ B) @ A) @ F) @ S)) => (P @ T3)))) = (! [T3:(set @ B)]: ((((((ord_less_eq @ (set @ B)) @ T3) @ S) & ((((inj_on @ B) @ A) @ F) @ T3)) => (P @ ((((image @ B) @ A) @ F) @ T3))))))))).
% 2.09/1.18 thf(fact_227_all__subset__image, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B),P:((set @ A) > $o)]: (((! [B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ B5) @ ((((image @ B) @ A) @ F) @ A2)) => (P @ B5)))) = (! [B5:(set @ B)]: (((((ord_less_eq @ (set @ B)) @ B5) @ A2) => (P @ ((((image @ B) @ A) @ F) @ B5))))))))).
% 2.09/1.18 thf(fact_228_inj__image__Compl__subset, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ (top_top @ (set @ A))) => (((ord_less_eq @ (set @ B)) @ ((((image @ A) @ B) @ F) @ ((uminus_uminus @ (set @ A)) @ A2))) @ ((uminus_uminus @ (set @ B)) @ ((((image @ A) @ B) @ F) @ A2))))))).
% 2.09/1.18 thf(fact_229_inj__on__image__Fpow, axiom, (! [B:$tType,A:$tType,F:(A > B),A2:(set @ A)]: ((((((inj_on @ A) @ B) @ F) @ A2) => ((((inj_on @ (set @ A)) @ (set @ B)) @ (((image @ A) @ B) @ F)) @ ((finite_Fpow @ A) @ A2)))))).
% 2.09/1.18 thf(fact_230_uminus__apply, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X2:A]: (((uminus_uminus @ B) @ (A6 @ X2))))))))).
% 2.09/1.18 thf(fact_231_double__compl, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X3)) = X3))))))).
% 2.09/1.18 thf(fact_232_compl__eq__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y2:A]: (((((uminus_uminus @ A) @ X3) = ((uminus_uminus @ A) @ Y2)) = (X3 = Y2)))))))).
% 2.09/1.18 thf(fact_233_ComplI, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((~ ((((member @ A) @ C2) @ A2))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.09/1.18 thf(fact_234_Compl__iff, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (~ ((((member @ A) @ C2) @ A2))))))).
% 2.09/1.18 thf(fact_235_Compl__eq__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A2) = ((uminus_uminus @ (set @ A)) @ B2)) = (A2 = B2))))).
% 2.09/1.18 thf(fact_236_compl__le__compl__iff, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y2)) = (((ord_less_eq @ A) @ Y2) @ X3)))))))).
% 2.09/1.18 thf(fact_237_inj__uminus, axiom, (! [A:$tType]: (((ordered_ab_group_add @ A) => (! [A2:(set @ A)]: (((((inj_on @ A) @ A) @ (uminus_uminus @ A)) @ A2))))))).
% 2.09/1.18 thf(fact_238_Compl__subset__Compl__iff, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ ((uminus_uminus @ (set @ A)) @ B2)) = (((ord_less_eq @ (set @ A)) @ B2) @ A2))))).
% 2.09/1.18 thf(fact_239_Compl__anti__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B2)) @ ((uminus_uminus @ (set @ A)) @ A2)))))).
% 2.09/1.18 thf(fact_240_sup__compl__top__left2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ X3) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X3)) @ Y2)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_241_sup__compl__top__left1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y2:A]: (((((sup_sup @ A) @ ((uminus_uminus @ A) @ X3)) @ (((sup_sup @ A) @ X3) @ Y2)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_242_sup__compl__top, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A]: (((((sup_sup @ A) @ X3) @ ((uminus_uminus @ A) @ X3)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_243_compl__sup__top, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A]: (((((sup_sup @ A) @ ((uminus_uminus @ A) @ X3)) @ X3) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_244_compl__mono, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,Y2:A]: (((((ord_less_eq @ A) @ X3) @ Y2) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y2)) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.09/1.18 thf(fact_245_compl__le__swap1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ Y2) @ ((uminus_uminus @ A) @ X3)) => (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y2))))))))).
% 2.09/1.18 thf(fact_246_compl__le__swap2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [Y2:A,X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y2)) @ X3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y2)))))))).
% 2.09/1.18 thf(fact_247_Fpow__mono, axiom, (! [A:$tType,A2:(set @ A),B2:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A2) @ B2) => (((ord_less_eq @ (set @ (set @ A))) @ ((finite_Fpow @ A) @ A2)) @ ((finite_Fpow @ A) @ B2)))))).
% 2.09/1.18 thf(fact_248_sup__cancel__left2, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,A3:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X3)) @ A3)) @ (((sup_sup @ A) @ X3) @ B3)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_249_sup__cancel__left1, axiom, (! [A:$tType]: (((boolean_algebra @ A) => (! [X3:A,A3:A,B3:A]: (((((sup_sup @ A) @ (((sup_sup @ A) @ X3) @ A3)) @ (((sup_sup @ A) @ ((uminus_uminus @ A) @ X3)) @ B3)) = (top_top @ A)))))))).
% 2.09/1.18 thf(fact_250_Compl__partition2, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) @ A2) = (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_251_Compl__partition, axiom, (! [A:$tType,A2:(set @ A)]: (((((sup_sup @ (set @ A)) @ A2) @ ((uminus_uminus @ (set @ A)) @ A2)) = (top_top @ (set @ A)))))).
% 2.09/1.18 thf(fact_252_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: (((uminus @ B) => ((uminus_uminus @ (A > B)) = (^ [A6:(A > B),X2:A]: (((uminus_uminus @ B) @ (A6 @ X2))))))))).
% 2.09/1.18 thf(fact_253_ComplD, axiom, (! [A:$tType,C2:A,A2:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A2)) => (~ ((((member @ A) @ C2) @ A2))))))).
% 2.09/1.18 thf(fact_254_double__complement, axiom, (! [A:$tType,A2:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A2)) = A2)))).
% 2.09/1.18 thf(fact_255_surj__Compl__image__subset, axiom, (! [A:$tType,B:$tType,F:(B > A),A2:(set @ B)]: (((((((image @ B) @ A) @ F) @ (top_top @ (set @ B))) = (top_top @ (set @ A))) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ ((((image @ B) @ A) @ F) @ A2))) @ ((((image @ B) @ A) @ F) @ ((uminus_uminus @ (set @ B)) @ A2))))))).
% 2.09/1.18 thf(tcon_HOL_Obool___Lattices_Obounded__lattice, axiom, (bounded_lattice @ $o)).
% 2.09/1.18 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1, axiom, (! [A8:$tType]: ((bounded_lattice @ (set @ A8))))).
% 2.09/1.18 thf(tcon_fun___Lattices_Obounded__lattice_2, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Lattices_Obounded__lattice__top, axiom, (! [A8:$tType,A9:$tType]: (((bounded_lattice @ A9) => (bounded_lattice_top @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A8:$tType,A9:$tType]: (((semilattice_sup @ A9) => (semilattice_sup @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A8:$tType,A9:$tType]: (((boolean_algebra @ A9) => (boolean_algebra @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A8:$tType,A9:$tType]: (((order_top @ A9) => (order_top @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A8:$tType,A9:$tType]: (((preorder @ A9) => (preorder @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Lattices_Olattice, axiom, (! [A8:$tType,A9:$tType]: (((lattice @ A9) => (lattice @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Orderings_Oorder, axiom, (! [A8:$tType,A9:$tType]: (((order @ A9) => (order @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Orderings_Otop, axiom, (! [A8:$tType,A9:$tType]: (((top @ A9) => (top @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Orderings_Oord, axiom, (! [A8:$tType,A9:$tType]: (((ord @ A9) => (ord @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_fun___Groups_Ouminus, axiom, (! [A8:$tType,A9:$tType]: (((uminus @ A9) => (uminus @ (A8 > A9)))))).
% 2.09/1.18 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3, axiom, (! [A8:$tType]: ((bounded_lattice_top @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4, axiom, (! [A8:$tType]: ((semilattice_sup @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5, axiom, (! [A8:$tType]: ((boolean_algebra @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Orderings_Oorder__top_6, axiom, (! [A8:$tType]: ((order_top @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Orderings_Opreorder_7, axiom, (! [A8:$tType]: ((preorder @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Lattices_Olattice_8, axiom, (! [A8:$tType]: ((lattice @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Orderings_Oorder_9, axiom, (! [A8:$tType]: ((order @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Orderings_Otop_10, axiom, (! [A8:$tType]: ((top @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Orderings_Oord_11, axiom, (! [A8:$tType]: ((ord @ (set @ A8))))).
% 2.09/1.18 thf(tcon_Set_Oset___Groups_Ouminus_12, axiom, (! [A8:$tType]: ((uminus @ (set @ A8))))).
% 2.09/1.18 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13, axiom, (bounded_lattice_top @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14, axiom, (semilattice_sup @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_15, axiom, (boolean_algebra @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Oorder__top_16, axiom, (order_top @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Opreorder_17, axiom, (preorder @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, (linorder @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Lattices_Olattice_18, axiom, (lattice @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Oorder_19, axiom, (order @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Otop_20, axiom, (top @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Orderings_Oord_21, axiom, (ord @ $o)).
% 2.09/1.18 thf(tcon_HOL_Obool___Groups_Ouminus_22, axiom, (uminus @ $o)).
% 2.09/1.18 thf(conj_0, hypothesis, ((((inj_on @ a) @ b) @ f) @ (((sup_sup @ (set @ a)) @ ((((image @ c) @ a) @ g) @ (top_top @ (set @ c)))) @ ((((image @ c) @ a) @ h) @ (top_top @ (set @ c)))))).
% 2.09/1.18 thf(conj_1, hypothesis, ((((((comp @ a) @ b) @ c) @ f) @ g) = (((((comp @ a) @ b) @ c) @ f) @ h))).
% 2.09/1.18 thf(conj_2, conjecture, (g = h)).
% 2.09/1.18 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2DTF32419.p
% 2.09/1.18 ------------------------
% 2.25/1.20 ---- Cleaned THF ---
% 2.25/1.20 thf(ty_t_Set_Oset,type,
% 2.25/1.20 set: $tType > $tType ).
% 2.25/1.20
% 2.25/1.20 thf(ty_tf_c,type,
% 2.25/1.20 c: $tType ).
% 2.25/1.20
% 2.25/1.20 thf(ty_tf_b,type,
% 2.25/1.20 b: $tType ).
% 2.25/1.20
% 2.25/1.20 thf(ty_tf_a,type,
% 2.25/1.20 a: $tType ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Lattices_Obounded__lattice,type,
% 2.25/1.20 bounded_lattice:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_HOL_Otype,type,
% 2.25/1.20 type:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Groups_Ouminus,type,
% 2.25/1.20 uminus:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Oord,type,
% 2.25/1.20 ord:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Otop,type,
% 2.25/1.20 top:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Oorder,type,
% 2.25/1.20 order:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Lattices_Olattice,type,
% 2.25/1.20 lattice:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Olinorder,type,
% 2.25/1.20 linorder:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Opreorder,type,
% 2.25/1.20 preorder:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Orderings_Oorder__top,type,
% 2.25/1.20 order_top:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.25/1.20 boolean_algebra:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.25/1.20 semilattice_sup:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.25/1.20 ordered_ab_group_add:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_cl_Lattices_Obounded__lattice__top,type,
% 2.25/1.20 bounded_lattice_top:
% 2.25/1.20 !>[A: $tType] : $o ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Finite__Set_OFpow,type,
% 2.25/1.20 finite_Fpow:
% 2.25/1.20 !>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Fun_Ocomp,type,
% 2.25/1.20 comp:
% 2.25/1.20 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Fun_Oinj__on,type,
% 2.25/1.20 inj_on:
% 2.25/1.20 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Fun_Othe__inv__into,type,
% 2.25/1.20 the_inv_into:
% 2.25/1.20 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.25/1.20 uminus_uminus:
% 2.25/1.20 !>[A: $tType] : ( A > A ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Lattices_Osup__class_Osup,type,
% 2.25/1.20 sup_sup:
% 2.25/1.20 !>[A: $tType] : ( A > A > A ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.25/1.20 ord_less_eq:
% 2.25/1.20 !>[A: $tType] : ( A > A > $o ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.25/1.20 top_top:
% 2.25/1.20 !>[A: $tType] : A ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Set_OCollect,type,
% 2.25/1.20 collect:
% 2.25/1.20 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_Set_Oimage,type,
% 2.25/1.20 image:
% 2.25/1.20 !>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_c_member,type,
% 2.25/1.20 member:
% 2.25/1.20 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.25/1.20
% 2.25/1.20 thf(sy_v_f,type,
% 2.25/1.20 f: a > b ).
% 2.25/1.20
% 2.25/1.20 thf(sy_v_g,type,
% 2.25/1.20 g: c > a ).
% 2.25/1.20
% 2.25/1.20 thf(sy_v_h,type,
% 2.25/1.20 h: c > a ).
% 2.25/1.20
% 2.25/1.20 thf(fact_0_o__ext,axiom,
% 2.25/1.20 ! [C: $tType,A: $tType,B: $tType,H: B > A,F: A > C,G: A > C] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ ( image @ B @ A @ H @ ( top_top @ ( set @ B ) ) ) )
% 2.25/1.20 => ( ( F @ X )
% 2.25/1.20 = ( G @ X ) ) )
% 2.25/1.20 => ( ( comp @ A @ C @ B @ F @ H )
% 2.25/1.20 = ( comp @ A @ C @ B @ G @ H ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_1_o__cong,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,A: $tType,H: A > B,I: A > B,F: B > C,G: B > C] :
% 2.25/1.20 ( ( H = I )
% 2.25/1.20 => ( ! [X: B] :
% 2.25/1.20 ( ( member @ B @ X @ ( image @ A @ B @ I @ ( top_top @ ( set @ A ) ) ) )
% 2.25/1.20 => ( ( F @ X )
% 2.25/1.20 = ( G @ X ) ) )
% 2.25/1.20 => ( ( comp @ B @ C @ A @ F @ H )
% 2.25/1.20 = ( comp @ B @ C @ A @ F @ I ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_2_o__inj__on,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,H: A > C] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.25/1.20 = ( comp @ C @ B @ A @ F @ H ) )
% 2.25/1.20 => ( ( inj_on @ C @ B @ F @ ( sup_sup @ ( set @ C ) @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) @ ( image @ A @ C @ H @ ( top_top @ ( set @ A ) ) ) ) )
% 2.25/1.20 => ( G = H ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_3_ex__o__conv,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,G: A > C,F: B > C] :
% 2.25/1.20 ( ( ? [H2: A > B] :
% 2.25/1.20 ( G
% 2.25/1.20 = ( comp @ B @ C @ A @ F @ H2 ) ) )
% 2.25/1.20 = ( ! [X2: C] :
% 2.25/1.20 ( ( member @ C @ X2 @ ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) ) )
% 2.25/1.20 => ? [Y: B] :
% 2.25/1.20 ( X2
% 2.25/1.20 = ( F @ Y ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_4_o__eq__conv,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,F: C > B,H: A > C,G: C > B] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ F @ H )
% 2.25/1.20 = ( comp @ C @ B @ A @ G @ H ) )
% 2.25/1.20 = ( ! [X2: C] :
% 2.25/1.20 ( ( member @ C @ X2 @ ( image @ A @ C @ H @ ( top_top @ ( set @ A ) ) ) )
% 2.25/1.20 => ( ( F @ X2 )
% 2.25/1.20 = ( G @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_5_sup__top__left,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( bounded_lattice_top @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( top_top @ A ) @ X3 )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_6_sup__top__right,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( bounded_lattice_top @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( top_top @ A ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_7_inj__on__Un__image__eq__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 => ( ( ( image @ A @ B @ F @ A2 )
% 2.25/1.20 = ( image @ A @ B @ F @ B2 ) )
% 2.25/1.20 = ( A2 = B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_8_inj__compose,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
% 2.25/1.20 => ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_9_fun_Oinj__map,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,F: A > B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_10_comp__inj__on,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,G: B > C] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( inj_on @ B @ C @ G @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 => ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_11_inj__on__imageI,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A2 )
% 2.25/1.20 => ( inj_on @ C @ B @ G @ ( image @ A @ C @ F @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_12_comp__inj__on__iff,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,F: A > B,A2: set @ A,F2: B > C] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( inj_on @ B @ C @ F2 @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 = ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F2 @ F ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_13_range__ex1__eq,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,B3: B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( member @ B @ B3 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
% 2.25/1.20 = ( ? [X2: A] :
% 2.25/1.20 ( ( B3
% 2.25/1.20 = ( F @ X2 ) )
% 2.25/1.20 & ! [Y: A] :
% 2.25/1.20 ( ( B3
% 2.25/1.20 = ( F @ Y ) )
% 2.25/1.20 => ( Y = X2 ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_14_inj__image__eq__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( ( image @ A @ B @ F @ A2 )
% 2.25/1.20 = ( image @ A @ B @ F @ B2 ) )
% 2.25/1.20 = ( A2 = B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_15_inj__image__mem__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A3: A,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 = ( member @ A @ A3 @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_16_comp__surj,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( ( image @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 = ( top_top @ ( set @ C ) ) )
% 2.25/1.20 => ( ( image @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ C ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_17_sup_Oright__idem,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ B3 )
% 2.25/1.20 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_18_sup__left__idem,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.25/1.20 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_19_sup_Oleft__idem,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) )
% 2.25/1.20 = ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_20_sup__idem,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ X3 )
% 2.25/1.20 = X3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_21_sup_Oidem,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ A3 @ A3 )
% 2.25/1.20 = A3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_22_sup__apply,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ B )
% 2.25/1.20 => ( ( sup_sup @ ( A > B ) )
% 2.25/1.20 = ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_23_comp__apply,axiom,
% 2.25/1.20 ! [C: $tType,A: $tType,B: $tType] :
% 2.25/1.20 ( ( comp @ B @ A @ C )
% 2.25/1.20 = ( ^ [F3: B > A,G2: C > B,X2: C] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_24_sup__left__commute,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
% 2.25/1.20 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_25_sup_Oleft__commute,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,A3: A,C2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A3 @ C2 ) )
% 2.25/1.20 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_26_sup__commute,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( sup_sup @ A )
% 2.25/1.20 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_27_sup_Ocommute,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( sup_sup @ A )
% 2.25/1.20 = ( ^ [A4: A,B4: A] : ( sup_sup @ A @ B4 @ A4 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_28_sup__assoc,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 2.25/1.20 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_29_sup_Oassoc,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A,C2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B3 ) @ C2 )
% 2.25/1.20 = ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_30_boolean__algebra__cancel_Osup2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B2: A,K: A,B3: A,A3: A] :
% 2.25/1.20 ( ( B2
% 2.25/1.20 = ( sup_sup @ A @ K @ B3 ) )
% 2.25/1.20 => ( ( sup_sup @ A @ A3 @ B2 )
% 2.25/1.20 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_31_boolean__algebra__cancel_Osup1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A2: A,K: A,A3: A,B3: A] :
% 2.25/1.20 ( ( A2
% 2.25/1.20 = ( sup_sup @ A @ K @ A3 ) )
% 2.25/1.20 => ( ( sup_sup @ A @ A2 @ B3 )
% 2.25/1.20 = ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_32_sup__fun__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ B )
% 2.25/1.20 => ( ( sup_sup @ ( A > B ) )
% 2.25/1.20 = ( ^ [F3: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_33_inf__sup__aci_I5_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ( ( sup_sup @ A )
% 2.25/1.20 = ( ^ [X2: A,Y: A] : ( sup_sup @ A @ Y @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_34_inf__sup__aci_I6_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 2.25/1.20 = ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_35_inf__sup__aci_I7_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ Y2 @ Z ) )
% 2.25/1.20 = ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X3 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_36_inf__sup__aci_I8_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.25/1.20 = ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_37_fun_Omap__comp,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 2.25/1.20 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 2.25/1.20 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_38_comp__eq__dest__lhs,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,A: $tType,A3: C > B,B3: A > C,C2: A > B,V: A] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.25/1.20 = C2 )
% 2.25/1.20 => ( ( A3 @ ( B3 @ V ) )
% 2.25/1.20 = ( C2 @ V ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_39_comp__eq__elim,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.25/1.20 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 2.25/1.20 => ! [V2: A] :
% 2.25/1.20 ( ( A3 @ ( B3 @ V2 ) )
% 2.25/1.20 = ( C2 @ ( D2 @ V2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_40_comp__eq__dest,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B3: A > C,C2: D > B,D2: A > D,V: A] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ A3 @ B3 )
% 2.25/1.20 = ( comp @ D @ B @ A @ C2 @ D2 ) )
% 2.25/1.20 => ( ( A3 @ ( B3 @ V ) )
% 2.25/1.20 = ( C2 @ ( D2 @ V ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_41_comp__assoc,axiom,
% 2.25/1.20 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 2.25/1.20 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 2.25/1.20 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_42_comp__def,axiom,
% 2.25/1.20 ! [A: $tType,C: $tType,B: $tType] :
% 2.25/1.20 ( ( comp @ B @ C @ A )
% 2.25/1.20 = ( ^ [F3: B > C,G2: A > B,X2: A] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_43_inj__on__inverseI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,A2: set @ A,G: B > A,F: A > B] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ( ( G @ ( F @ X ) )
% 2.25/1.20 = X ) )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_44_inj__on__contraD,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( X3 != Y2 )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( member @ A @ Y2 @ A2 )
% 2.25/1.20 => ( ( F @ X3 )
% 2.25/1.20 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_45_mem__Collect__eq,axiom,
% 2.25/1.20 ! [A: $tType,A3: A,P: A > $o] :
% 2.25/1.20 ( ( member @ A @ A3 @ ( collect @ A @ P ) )
% 2.25/1.20 = ( P @ A3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_46_Collect__mem__eq,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( collect @ A
% 2.25/1.20 @ ^ [X2: A] : ( member @ A @ X2 @ A2 ) )
% 2.25/1.20 = A2 ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_47_Collect__cong,axiom,
% 2.25/1.20 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( P @ X )
% 2.25/1.20 = ( Q @ X ) )
% 2.25/1.20 => ( ( collect @ A @ P )
% 2.25/1.20 = ( collect @ A @ Q ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_48_ext,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( F @ X )
% 2.25/1.20 = ( G @ X ) )
% 2.25/1.20 => ( F = G ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_49_inj__on__eq__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( member @ A @ Y2 @ A2 )
% 2.25/1.20 => ( ( ( F @ X3 )
% 2.25/1.20 = ( F @ Y2 ) )
% 2.25/1.20 = ( X3 = Y2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_50_inj__on__cong,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,A2: set @ A,F: A > B,G: A > B] :
% 2.25/1.20 ( ! [A5: A] :
% 2.25/1.20 ( ( member @ A @ A5 @ A2 )
% 2.25/1.20 => ( ( F @ A5 )
% 2.25/1.20 = ( G @ A5 ) ) )
% 2.25/1.20 => ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 = ( inj_on @ A @ B @ G @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_51_inj__on__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( inj_on @ A @ B )
% 2.25/1.20 = ( ^ [F3: A > B,A6: set @ A] :
% 2.25/1.20 ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ A6 )
% 2.25/1.20 => ! [Y: A] :
% 2.25/1.20 ( ( member @ A @ Y @ A6 )
% 2.25/1.20 => ( ( ( F3 @ X2 )
% 2.25/1.20 = ( F3 @ Y ) )
% 2.25/1.20 => ( X2 = Y ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_52_inj__onI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,A2: set @ A,F: A > B] :
% 2.25/1.20 ( ! [X: A,Y3: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ( ( member @ A @ Y3 @ A2 )
% 2.25/1.20 => ( ( ( F @ X )
% 2.25/1.20 = ( F @ Y3 ) )
% 2.25/1.20 => ( X = Y3 ) ) ) )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_53_inj__onD,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( ( F @ X3 )
% 2.25/1.20 = ( F @ Y2 ) )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( member @ A @ Y2 @ A2 )
% 2.25/1.20 => ( X3 = Y2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_54_surj__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: B > A] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 = ( ! [Y: A] :
% 2.25/1.20 ? [X2: B] :
% 2.25/1.20 ( Y
% 2.25/1.20 = ( F @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_55_surjI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,G: B > A,F: A > B] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( G @ ( F @ X ) )
% 2.25/1.20 = X )
% 2.25/1.20 => ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_56_surjE,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,Y2: A] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ~ ! [X: B] :
% 2.25/1.20 ( Y2
% 2.25/1.20 != ( F @ X ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_57_surjD,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,Y2: A] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ? [X: B] :
% 2.25/1.20 ( Y2
% 2.25/1.20 = ( F @ X ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_58_image__eq__imp__comp,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A2: set @ B,G: C > A,B2: set @ C,H: A > D] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ A2 )
% 2.25/1.20 = ( image @ C @ A @ G @ B2 ) )
% 2.25/1.20 => ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A2 )
% 2.25/1.20 = ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_59_image__comp,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
% 2.25/1.20 ( ( image @ B @ A @ F @ ( image @ C @ B @ G @ R ) )
% 2.25/1.20 = ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_60_inj__on__image__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,A2: set @ A,G: A > B,F: A > A] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ! [Xa: A] :
% 2.25/1.20 ( ( member @ A @ Xa @ A2 )
% 2.25/1.20 => ( ( ( G @ ( F @ X ) )
% 2.25/1.20 = ( G @ ( F @ Xa ) ) )
% 2.25/1.20 = ( ( G @ X )
% 2.25/1.20 = ( G @ Xa ) ) ) ) )
% 2.25/1.20 => ( ( inj_on @ A @ A @ F @ A2 )
% 2.25/1.20 => ( ( inj_on @ A @ B @ G @ ( image @ A @ A @ F @ A2 ) )
% 2.25/1.20 = ( inj_on @ A @ B @ G @ A2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_61_inj__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 = ( ! [X2: A,Y: A] :
% 2.25/1.20 ( ( ( F @ X2 )
% 2.25/1.20 = ( F @ Y ) )
% 2.25/1.20 => ( X2 = Y ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_62_inj__eq,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,X3: A,Y2: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( ( F @ X3 )
% 2.25/1.20 = ( F @ Y2 ) )
% 2.25/1.20 = ( X3 = Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_63_injI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B] :
% 2.25/1.20 ( ! [X: A,Y3: A] :
% 2.25/1.20 ( ( ( F @ X )
% 2.25/1.20 = ( F @ Y3 ) )
% 2.25/1.20 => ( X = Y3 ) )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_64_injD,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,X3: A,Y2: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( ( F @ X3 )
% 2.25/1.20 = ( F @ Y2 ) )
% 2.25/1.20 => ( X3 = Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_65_inj__on__imageI2,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,F2: C > B,F: A > C,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F2 @ F ) @ A2 )
% 2.25/1.20 => ( inj_on @ A @ C @ F @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_66_fun_Oinj__map__strong,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,X3: D > A,Xa2: D > A,F: A > B,Fa: A > B] :
% 2.25/1.20 ( ! [Z2: A,Za: A] :
% 2.25/1.20 ( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
% 2.25/1.20 => ( ( member @ A @ Za @ ( image @ D @ A @ Xa2 @ ( top_top @ ( set @ D ) ) ) )
% 2.25/1.20 => ( ( ( F @ Z2 )
% 2.25/1.20 = ( Fa @ Za ) )
% 2.25/1.20 => ( Z2 = Za ) ) ) )
% 2.25/1.20 => ( ( ( comp @ A @ B @ D @ F @ X3 )
% 2.25/1.20 = ( comp @ A @ B @ D @ Fa @ Xa2 ) )
% 2.25/1.20 => ( X3 = Xa2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_67_fun_Omap__cong0,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,X3: D > A,F: A > B,G: A > B] :
% 2.25/1.20 ( ! [Z2: A] :
% 2.25/1.20 ( ( member @ A @ Z2 @ ( image @ D @ A @ X3 @ ( top_top @ ( set @ D ) ) ) )
% 2.25/1.20 => ( ( F @ Z2 )
% 2.25/1.20 = ( G @ Z2 ) ) )
% 2.25/1.20 => ( ( comp @ A @ B @ D @ F @ X3 )
% 2.25/1.20 = ( comp @ A @ B @ D @ G @ X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_68_fun_Omap__cong,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,X3: D > A,Ya: D > A,F: A > B,G: A > B] :
% 2.25/1.20 ( ( X3 = Ya )
% 2.25/1.20 => ( ! [Z2: A] :
% 2.25/1.20 ( ( member @ A @ Z2 @ ( image @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
% 2.25/1.20 => ( ( F @ Z2 )
% 2.25/1.20 = ( G @ Z2 ) ) )
% 2.25/1.20 => ( ( comp @ A @ B @ D @ F @ X3 )
% 2.25/1.20 = ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_69_fun_Oset__map,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
% 2.25/1.20 ( ( image @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
% 2.25/1.20 = ( image @ A @ B @ F @ ( image @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_70_surj__fun__eq,axiom,
% 2.25/1.20 ! [B: $tType,C: $tType,A: $tType,F: B > A,X4: set @ B,G1: A > C,G22: A > C] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ X4 )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ! [X: B] :
% 2.25/1.20 ( ( member @ B @ X @ X4 )
% 2.25/1.20 => ( ( comp @ A @ C @ B @ G1 @ F @ X )
% 2.25/1.20 = ( comp @ A @ C @ B @ G22 @ F @ X ) ) )
% 2.25/1.20 => ( G1 = G22 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_71_Un__iff,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 = ( ( member @ A @ C2 @ A2 )
% 2.25/1.20 | ( member @ A @ C2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_72_UnCI,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.25/1.20 ( ( ~ ( member @ A @ C2 @ B2 )
% 2.25/1.20 => ( member @ A @ C2 @ A2 ) )
% 2.25/1.20 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_73_iso__tuple__UNIV__I,axiom,
% 2.25/1.20 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_74_UNIV__I,axiom,
% 2.25/1.20 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_75_top__apply,axiom,
% 2.25/1.20 ! [C: $tType,D: $tType] :
% 2.25/1.20 ( ( top @ C )
% 2.25/1.20 => ( ( top_top @ ( D > C ) )
% 2.25/1.20 = ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_76_image__eqI,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B,A2: set @ B] :
% 2.25/1.20 ( ( B3
% 2.25/1.20 = ( F @ X3 ) )
% 2.25/1.20 => ( ( member @ B @ X3 @ A2 )
% 2.25/1.20 => ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_77_inj__imp__inj__on,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_78_Un__UNIV__right,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_79_Un__UNIV__left,axiom,
% 2.25/1.20 ! [A: $tType,B2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ B2 )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_80_imageI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,X3: A,A2: set @ A,F: A > B] :
% 2.25/1.20 ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( member @ B @ ( F @ X3 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_81_image__iff,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,Z: A,F: B > A,A2: set @ B] :
% 2.25/1.20 ( ( member @ A @ Z @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 = ( ? [X2: B] :
% 2.25/1.20 ( ( member @ B @ X2 @ A2 )
% 2.25/1.20 & ( Z
% 2.25/1.20 = ( F @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_82_bex__imageD,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 2.25/1.20 ( ? [X5: A] :
% 2.25/1.20 ( ( member @ A @ X5 @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 & ( P @ X5 ) )
% 2.25/1.20 => ? [X: B] :
% 2.25/1.20 ( ( member @ B @ X @ A2 )
% 2.25/1.20 & ( P @ ( F @ X ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_83_image__cong,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,M: set @ A,N: set @ A,F: A > B,G: A > B] :
% 2.25/1.20 ( ( M = N )
% 2.25/1.20 => ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ N )
% 2.25/1.20 => ( ( F @ X )
% 2.25/1.20 = ( G @ X ) ) )
% 2.25/1.20 => ( ( image @ A @ B @ F @ M )
% 2.25/1.20 = ( image @ A @ B @ G @ N ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_84_ball__imageD,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: A > $o] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 => ( P @ X ) )
% 2.25/1.20 => ! [X5: B] :
% 2.25/1.20 ( ( member @ B @ X5 @ A2 )
% 2.25/1.20 => ( P @ ( F @ X5 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_85_rev__image__eqI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,X3: A,A2: set @ A,B3: B,F: A > B] :
% 2.25/1.20 ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( B3
% 2.25/1.20 = ( F @ X3 ) )
% 2.25/1.20 => ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_86_UNIV__eq__I,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ! [X: A] : ( member @ A @ X @ A2 )
% 2.25/1.20 => ( ( top_top @ ( set @ A ) )
% 2.25/1.20 = A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_87_UNIV__witness,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ? [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_88_top__set__def,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( top_top @ ( set @ A ) )
% 2.25/1.20 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_89_UnE,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 => ( ~ ( member @ A @ C2 @ A2 )
% 2.25/1.20 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_90_UnI1,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ A2 )
% 2.25/1.20 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_91_UnI2,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,B2: set @ A,A2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ B2 )
% 2.25/1.20 => ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_92_bex__Un,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.25/1.20 ( ( ? [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 & ( P @ X2 ) ) )
% 2.25/1.20 = ( ? [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ A2 )
% 2.25/1.20 & ( P @ X2 ) )
% 2.25/1.20 | ? [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ B2 )
% 2.25/1.20 & ( P @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_93_ball__Un,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,P: A > $o] :
% 2.25/1.20 ( ( ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 => ( P @ X2 ) ) )
% 2.25/1.20 = ( ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ A2 )
% 2.25/1.20 => ( P @ X2 ) )
% 2.25/1.20 & ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ B2 )
% 2.25/1.20 => ( P @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_94_Un__assoc,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 2.25/1.20 = ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_95_Un__absorb,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A2 @ A2 )
% 2.25/1.20 = A2 ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_96_Un__commute,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) )
% 2.25/1.20 = ( ^ [A6: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A6 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_97_Un__left__absorb,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 = ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_98_Un__left__commute,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
% 2.25/1.20 = ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_99_rangeI,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,X3: B] : ( member @ A @ ( F @ X3 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_100_range__eqI,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,B3: A,F: B > A,X3: B] :
% 2.25/1.20 ( ( B3
% 2.25/1.20 = ( F @ X3 ) )
% 2.25/1.20 => ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_101_image__Un,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ B] :
% 2.25/1.20 ( ( image @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A2 @ B2 ) )
% 2.25/1.20 = ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ ( image @ B @ A @ F @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_102_Inf_OINF__image,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
% 2.25/1.20 ( ( Inf @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
% 2.25/1.20 = ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_103_Sup_OSUP__image,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A2: set @ C] :
% 2.25/1.20 ( ( Sup @ ( image @ B @ A @ G @ ( image @ C @ B @ F @ A2 ) ) )
% 2.25/1.20 = ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_104_the__inv__into__comp,axiom,
% 2.25/1.20 ! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A2: set @ C,X3: B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) )
% 2.25/1.20 => ( ( inj_on @ C @ A @ G @ A2 )
% 2.25/1.20 => ( ( member @ B @ X3 @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A2 ) ) )
% 2.25/1.20 => ( ( the_inv_into @ C @ B @ A2 @ ( comp @ A @ B @ C @ F @ G ) @ X3 )
% 2.25/1.20 = ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A2 @ G ) @ ( the_inv_into @ A @ B @ ( image @ C @ A @ G @ A2 ) @ F ) @ X3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_105_the__inv__into__onto,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( image @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 = A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_106_comp__cong,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X3: C,F2: D > A,G3: E > D,X6: E] :
% 2.25/1.20 ( ( ( F @ ( G @ X3 ) )
% 2.25/1.20 = ( F2 @ ( G3 @ X6 ) ) )
% 2.25/1.20 => ( ( comp @ B @ A @ C @ F @ G @ X3 )
% 2.25/1.20 = ( comp @ D @ A @ E @ F2 @ G3 @ X6 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_107_comp2__conv,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,C: $tType] :
% 2.25/1.20 ( ( comp @ C @ B @ A )
% 2.25/1.20 = ( ^ [F1: C > B,F22: A > C,X2: A] : ( F1 @ ( F22 @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_108_top1I,axiom,
% 2.25/1.20 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_109_the__inv__into__f__eq,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A,Y2: B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( ( F @ X3 )
% 2.25/1.20 = Y2 )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( the_inv_into @ A @ B @ A2 @ F @ Y2 )
% 2.25/1.20 = X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_110_the__inv__into__f__f,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( ( the_inv_into @ A @ B @ A2 @ F @ ( F @ X3 ) )
% 2.25/1.20 = X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_111_inj__on__the__inv__into,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A2 @ F ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_112_f__the__inv__into__f,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: A > B,A2: set @ A,Y2: B] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( member @ B @ Y2 @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 => ( ( F @ ( the_inv_into @ A @ B @ A2 @ F @ Y2 ) )
% 2.25/1.20 = Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_113_the__inv__f__f,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,X3: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X3 ) )
% 2.25/1.20 = X3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_114_Sup_OSUP__cong,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
% 2.25/1.20 ( ( A2 = B2 )
% 2.25/1.20 => ( ! [X: B] :
% 2.25/1.20 ( ( member @ B @ X @ B2 )
% 2.25/1.20 => ( ( C3 @ X )
% 2.25/1.20 = ( D3 @ X ) ) )
% 2.25/1.20 => ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
% 2.25/1.20 = ( Sup @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_115_Inf_OINF__cong,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
% 2.25/1.20 ( ( A2 = B2 )
% 2.25/1.20 => ( ! [X: B] :
% 2.25/1.20 ( ( member @ B @ X @ B2 )
% 2.25/1.20 => ( ( C3 @ X )
% 2.25/1.20 = ( D3 @ X ) ) )
% 2.25/1.20 => ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
% 2.25/1.20 = ( Inf @ ( image @ B @ A @ D3 @ B2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_116_comp3__conv,axiom,
% 2.25/1.20 ! [C: $tType,D: $tType,B: $tType,A: $tType,F12: D > B,F23: C > D,F32: A > C] :
% 2.25/1.20 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F12 @ F23 ) @ F32 )
% 2.25/1.20 = ( ^ [X2: A] : ( F12 @ ( F23 @ ( F32 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_117_top__empty__eq,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( top_top @ ( A > $o ) )
% 2.25/1.20 = ( ^ [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_118_top__conj_I2_J,axiom,
% 2.25/1.20 ! [A: $tType,P: $o,X3: A] :
% 2.25/1.20 ( ( P
% 2.25/1.20 & ( top_top @ ( A > $o ) @ X3 ) )
% 2.25/1.20 = P ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_119_top__conj_I1_J,axiom,
% 2.25/1.20 ! [A: $tType,X3: A,P: $o] :
% 2.25/1.20 ( ( ( top_top @ ( A > $o ) @ X3 )
% 2.25/1.20 & P )
% 2.25/1.20 = P ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_120_the__inv__into__into,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,X3: B,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( member @ B @ X3 @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( member @ A @ ( the_inv_into @ A @ B @ A2 @ F @ X3 ) @ B2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_121_rewriteR__comp__comp2,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.25/1.20 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 2.25/1.20 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 2.25/1.20 = L )
% 2.25/1.20 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 2.25/1.20 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_122_order__refl,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( preorder @ A )
% 2.25/1.20 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_123_subsetI,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ( member @ A @ X @ B2 ) )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_124_subset__antisym,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.25/1.20 => ( A2 = B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_125_le__sup__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ Z )
% 2.25/1.20 = ( ( ord_less_eq @ A @ X3 @ Z )
% 2.25/1.20 & ( ord_less_eq @ A @ Y2 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_126_sup_Obounded__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,C2: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 2.25/1.20 = ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 & ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_127_Un__subset__iff,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 )
% 2.25/1.20 = ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.25/1.20 & ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_128_bex__cong2,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
% 2.25/1.20 => ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ I2 )
% 2.25/1.20 => ( ( F @ X )
% 2.25/1.20 = ( G @ X ) ) )
% 2.25/1.20 => ( ( ? [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ I2 )
% 2.25/1.20 & ( P @ ( F @ X2 ) ) ) )
% 2.25/1.20 = ( ? [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ I2 )
% 2.25/1.20 & ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_129_ball__cong2,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,I2: set @ A,A2: set @ A,F: A > B,G: A > B,P: B > $o] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ I2 @ A2 )
% 2.25/1.20 => ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ( ( F @ X )
% 2.25/1.20 = ( G @ X ) ) )
% 2.25/1.20 => ( ( ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ I2 )
% 2.25/1.20 => ( P @ ( F @ X2 ) ) ) )
% 2.25/1.20 = ( ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ I2 )
% 2.25/1.20 => ( P @ ( G @ X2 ) ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_130_bex__subset__imp__bex,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,P: A > $o,B2: set @ A] :
% 2.25/1.20 ( ? [X5: A] :
% 2.25/1.20 ( ( member @ A @ X5 @ A2 )
% 2.25/1.20 & ( P @ X5 ) )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ? [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ B2 )
% 2.25/1.20 & ( P @ X ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_131_ball__subset__imp__ball,axiom,
% 2.25/1.20 ! [A: $tType,B2: set @ A,P: A > $o,A2: set @ A] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ B2 )
% 2.25/1.20 => ( P @ X ) )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ! [X5: A] :
% 2.25/1.20 ( ( member @ A @ X5 @ A2 )
% 2.25/1.20 => ( P @ X5 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_132_top_Oextremum__uniqueI,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order_top @ A )
% 2.25/1.20 => ! [A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.25/1.20 => ( A3
% 2.25/1.20 = ( top_top @ A ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_133_top_Oextremum__unique,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order_top @ A )
% 2.25/1.20 => ! [A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
% 2.25/1.20 = ( A3
% 2.25/1.20 = ( top_top @ A ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_134_top__greatest,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order_top @ A )
% 2.25/1.20 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_135_subset__image__iff,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 = ( ? [AA: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
% 2.25/1.20 & ( B2
% 2.25/1.20 = ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_136_image__subset__iff,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
% 2.25/1.20 = ( ! [X2: B] :
% 2.25/1.20 ( ( member @ B @ X2 @ A2 )
% 2.25/1.20 => ( member @ A @ ( F @ X2 ) @ B2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_137_subset__imageE,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 => ~ ! [C4: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
% 2.25/1.20 => ( B2
% 2.25/1.20 != ( image @ B @ A @ F @ C4 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_138_image__subsetI,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( member @ A @ X @ A2 )
% 2.25/1.20 => ( member @ B @ ( F @ X ) @ B2 ) )
% 2.25/1.20 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_139_image__mono,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_140_subset__UNIV,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_141_sup_OcoboundedI2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [C2: A,B3: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.25/1.20 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_142_sup_OcoboundedI1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [C2: A,A3: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.25/1.20 => ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_143_sup_Oabsorb__iff2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( ord_less_eq @ A )
% 2.25/1.20 = ( ^ [A4: A,B4: A] :
% 2.25/1.20 ( ( sup_sup @ A @ A4 @ B4 )
% 2.25/1.20 = B4 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_144_sup_Oabsorb__iff1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( ord_less_eq @ A )
% 2.25/1.20 = ( ^ [B4: A,A4: A] :
% 2.25/1.20 ( ( sup_sup @ A @ A4 @ B4 )
% 2.25/1.20 = A4 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_145_sup_Ocobounded2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,A3: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_146_sup_Ocobounded1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_147_sup_Oorder__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( ord_less_eq @ A )
% 2.25/1.20 = ( ^ [B4: A,A4: A] :
% 2.25/1.20 ( A4
% 2.25/1.20 = ( sup_sup @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_148_sup_OboundedI,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,A3: A,C2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_149_sup_OboundedE,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,C2: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A3 )
% 2.25/1.20 => ~ ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_150_sup__absorb2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ( ( sup_sup @ A @ X3 @ Y2 )
% 2.25/1.20 = Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_151_sup__absorb1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.25/1.20 => ( ( sup_sup @ A @ X3 @ Y2 )
% 2.25/1.20 = X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_152_sup_Oabsorb2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( sup_sup @ A @ A3 @ B3 )
% 2.25/1.20 = B3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_153_sup_Oabsorb1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( ( sup_sup @ A @ A3 @ B3 )
% 2.25/1.20 = A3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_154_sup__unique,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [F: A > A > A,X3: A,Y2: A] :
% 2.25/1.20 ( ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( F @ X @ Y3 ) )
% 2.25/1.20 => ( ! [X: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F @ X @ Y3 ) )
% 2.25/1.20 => ( ! [X: A,Y3: A,Z2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ Y3 @ X )
% 2.25/1.20 => ( ( ord_less_eq @ A @ Z2 @ X )
% 2.25/1.20 => ( ord_less_eq @ A @ ( F @ Y3 @ Z2 ) @ X ) ) )
% 2.25/1.20 => ( ( sup_sup @ A @ X3 @ Y2 )
% 2.25/1.20 = ( F @ X3 @ Y2 ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_155_sup_OorderI,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A] :
% 2.25/1.20 ( ( A3
% 2.25/1.20 = ( sup_sup @ A @ A3 @ B3 ) )
% 2.25/1.20 => ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_156_sup_OorderE,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [B3: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( A3
% 2.25/1.20 = ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_157_le__iff__sup,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ( ( ord_less_eq @ A )
% 2.25/1.20 = ( ^ [X2: A,Y: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X2 @ Y )
% 2.25/1.20 = Y ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_158_sup__least,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [Y2: A,X3: A,Z: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ Z @ X3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z ) @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_159_sup__mono,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,C2: A,B3: A,D2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ C2 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ B3 @ D2 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_160_sup_Omono,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [C2: A,A3: A,D2: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ C2 @ A3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ D2 @ B3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_161_le__supI2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,B3: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ B3 )
% 2.25/1.20 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_162_le__supI1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,A3: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ A3 )
% 2.25/1.20 => ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_163_sup__ge2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_164_sup__ge1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_165_le__supI,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,X3: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ X3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ B3 @ X3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_166_le__supE,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A )
% 2.25/1.20 => ! [A3: A,B3: A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B3 ) @ X3 )
% 2.25/1.20 => ~ ( ( ord_less_eq @ A @ A3 @ X3 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ B3 @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_167_inf__sup__ord_I3_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_168_inf__sup__ord_I4_J,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( lattice @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_169_in__mono,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( member @ A @ X3 @ A2 )
% 2.25/1.20 => ( member @ A @ X3 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_170_subsetD,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( member @ A @ C2 @ A2 )
% 2.25/1.20 => ( member @ A @ C2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_171_equalityE,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( A2 = B2 )
% 2.25/1.20 => ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_172_subset__eq,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) )
% 2.25/1.20 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.25/1.20 ! [X2: A] :
% 2.25/1.20 ( ( member @ A @ X2 @ A6 )
% 2.25/1.20 => ( member @ A @ X2 @ B5 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_173_equalityD1,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( A2 = B2 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_174_equalityD2,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( A2 = B2 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_175_subset__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) )
% 2.25/1.20 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.25/1.20 ! [T: A] :
% 2.25/1.20 ( ( member @ A @ T @ A6 )
% 2.25/1.20 => ( member @ A @ T @ B5 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_176_subset__refl,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_177_Collect__mono,axiom,
% 2.25/1.20 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.25/1.20 ( ! [X: A] :
% 2.25/1.20 ( ( P @ X )
% 2.25/1.20 => ( Q @ X ) )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_178_subset__trans,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_179_le__funD,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( ord @ B )
% 2.25/1.20 => ! [F: A > B,G: A > B,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.25/1.20 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_180_le__funE,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( ord @ B )
% 2.25/1.20 => ! [F: A > B,G: A > B,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ ( A > B ) @ F @ G )
% 2.25/1.20 => ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_181_le__funI,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( ord @ B )
% 2.25/1.20 => ! [F: A > B,G: A > B] :
% 2.25/1.20 ( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
% 2.25/1.20 => ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_182_set__eq__subset,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ^ [Y4: set @ A,Z3: set @ A] : ( Y4 = Z3 ) )
% 2.25/1.20 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
% 2.25/1.20 & ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_183_le__fun__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( ord @ B )
% 2.25/1.20 => ( ( ord_less_eq @ ( A > B ) )
% 2.25/1.20 = ( ^ [F3: A > B,G2: A > B] :
% 2.25/1.20 ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_184_Collect__mono__iff,axiom,
% 2.25/1.20 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.25/1.20 = ( ! [X2: A] :
% 2.25/1.20 ( ( P @ X2 )
% 2.25/1.20 => ( Q @ X2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_185_order__subst1,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType] :
% 2.25/1.20 ( ( ( order @ B )
% 2.25/1.20 & ( order @ A ) )
% 2.25/1.20 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
% 2.25/1.20 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.25/1.20 => ( ! [X: B,Y3: B] :
% 2.25/1.20 ( ( ord_less_eq @ B @ X @ Y3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.25/1.20 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_186_order__subst2,axiom,
% 2.25/1.20 ! [A: $tType,C: $tType] :
% 2.25/1.20 ( ( ( order @ C )
% 2.25/1.20 & ( order @ A ) )
% 2.25/1.20 => ! [A3: A,B3: A,F: A > C,C2: C] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
% 2.25/1.20 => ( ! [X: A,Y3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.25/1.20 => ( ord_less_eq @ C @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.25/1.20 => ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_187_ord__eq__le__subst,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType] :
% 2.25/1.20 ( ( ( ord @ B )
% 2.25/1.20 & ( ord @ A ) )
% 2.25/1.20 => ! [A3: A,F: B > A,B3: B,C2: B] :
% 2.25/1.20 ( ( A3
% 2.25/1.20 = ( F @ B3 ) )
% 2.25/1.20 => ( ( ord_less_eq @ B @ B3 @ C2 )
% 2.25/1.20 => ( ! [X: B,Y3: B] :
% 2.25/1.20 ( ( ord_less_eq @ B @ X @ Y3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.25/1.20 => ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_188_ord__le__eq__subst,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType] :
% 2.25/1.20 ( ( ( ord @ B )
% 2.25/1.20 & ( ord @ A ) )
% 2.25/1.20 => ! [A3: A,B3: A,F: A > B,C2: B] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( ( F @ B3 )
% 2.25/1.20 = C2 )
% 2.25/1.20 => ( ! [X: A,Y3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X @ Y3 )
% 2.25/1.20 => ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) )
% 2.25/1.20 => ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_189_eq__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.25/1.20 = ( ^ [X2: A,Y: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X2 @ Y )
% 2.25/1.20 & ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_190_antisym,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.25/1.20 => ( X3 = Y2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_191_linear,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( linorder @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 | ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_192_eq__refl,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( preorder @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( X3 = Y2 )
% 2.25/1.20 => ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_193_le__cases,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( linorder @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ~ ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_194_order_Otrans,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [A3: A,B3: A,C2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.25/1.20 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_195_le__cases3,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( linorder @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ Y2 @ Z ) )
% 2.25/1.20 => ( ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ X3 @ Z ) )
% 2.25/1.20 => ( ( ( ord_less_eq @ A @ X3 @ Z )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ Z @ Y2 ) )
% 2.25/1.20 => ( ( ( ord_less_eq @ A @ Z @ Y2 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
% 2.25/1.20 => ( ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ Z @ X3 ) )
% 2.25/1.20 => ~ ( ( ord_less_eq @ A @ Z @ X3 )
% 2.25/1.20 => ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_196_antisym__conv,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ Y2 @ X3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 = ( X3 = Y2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_197_order__class_Oorder_Oeq__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.25/1.20 = ( ^ [A4: A,B4: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A4 @ B4 )
% 2.25/1.20 & ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_198_ord__eq__le__trans,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ord @ A )
% 2.25/1.20 => ! [A3: A,B3: A,C2: A] :
% 2.25/1.20 ( ( A3 = B3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ B3 @ C2 )
% 2.25/1.20 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_199_ord__le__eq__trans,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ord @ A )
% 2.25/1.20 => ! [A3: A,B3: A,C2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( B3 = C2 )
% 2.25/1.20 => ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_200_order__class_Oorder_Oantisym,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [A3: A,B3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( A3 = B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_201_order__trans,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( preorder @ A )
% 2.25/1.20 => ! [X3: A,Y2: A,Z: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ Y2 @ Z )
% 2.25/1.20 => ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_202_dual__order_Orefl,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_203_linorder__wlog,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( linorder @ A )
% 2.25/1.20 => ! [P: A > A > $o,A3: A,B3: A] :
% 2.25/1.20 ( ! [A5: A,B6: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ A5 @ B6 )
% 2.25/1.20 => ( P @ A5 @ B6 ) )
% 2.25/1.20 => ( ! [A5: A,B6: A] :
% 2.25/1.20 ( ( P @ B6 @ A5 )
% 2.25/1.20 => ( P @ A5 @ B6 ) )
% 2.25/1.20 => ( P @ A3 @ B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_204_dual__order_Otrans,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [B3: A,A3: A,C2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ C2 @ B3 )
% 2.25/1.20 => ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_205_dual__order_Oeq__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ( ( ^ [Y4: A,Z3: A] : ( Y4 = Z3 ) )
% 2.25/1.20 = ( ^ [A4: A,B4: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B4 @ A4 )
% 2.25/1.20 & ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_206_dual__order_Oantisym,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( order @ A )
% 2.25/1.20 => ! [B3: A,A3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ B3 @ A3 )
% 2.25/1.20 => ( ( ord_less_eq @ A @ A3 @ B3 )
% 2.25/1.20 => ( A3 = B3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_207_Un__mono,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A,D3: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ D3 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_208_Un__least,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,C3: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) @ C3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_209_Un__upper1,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_210_Un__upper2,axiom,
% 2.25/1.20 ! [A: $tType,B2: set @ A,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_211_Un__absorb1,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 = B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_212_Un__absorb2,axiom,
% 2.25/1.20 ! [A: $tType,B2: set @ A,A2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.25/1.20 => ( ( sup_sup @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 = A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_213_subset__UnE,axiom,
% 2.25/1.20 ! [A: $tType,C3: set @ A,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ C3 @ ( sup_sup @ ( set @ A ) @ A2 @ B2 ) )
% 2.25/1.20 => ~ ! [A7: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A7 @ A2 )
% 2.25/1.20 => ! [B7: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ B7 @ B2 )
% 2.25/1.20 => ( C3
% 2.25/1.20 != ( sup_sup @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_214_subset__Un__eq,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) )
% 2.25/1.20 = ( ^ [A6: set @ A,B5: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
% 2.25/1.20 = B5 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_215_inj__on__subset,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_216_subset__inj__on,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ B2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( inj_on @ A @ B @ F @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_217_SetInterval2_Oinj__on__image__mem__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A,A3: A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
% 2.25/1.20 => ( ( member @ A @ A3 @ A2 )
% 2.25/1.20 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ B2 ) )
% 2.25/1.20 = ( member @ A @ A3 @ B2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_218_inj__on__image__eq__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,C3: set @ A,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ C3 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ C3 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
% 2.25/1.20 => ( ( ( image @ A @ B @ F @ A2 )
% 2.25/1.20 = ( image @ A @ B @ F @ B2 ) )
% 2.25/1.20 = ( A2 = B2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_219_Fun_Oinj__on__image__mem__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,B2: set @ A,A3: A,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ B2 )
% 2.25/1.20 => ( ( member @ A @ A3 @ B2 )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ( member @ B @ ( F @ A3 ) @ ( image @ A @ B @ F @ A2 ) )
% 2.25/1.20 = ( member @ A @ A3 @ A2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_220_inj__image__subset__iff,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) )
% 2.25/1.20 = ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_221_rewriteL__comp__comp,axiom,
% 2.25/1.20 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.25/1.20 = L )
% 2.25/1.20 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 2.25/1.20 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_222_rewriteR__comp__comp,axiom,
% 2.25/1.20 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.25/1.20 = R )
% 2.25/1.20 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 2.25/1.20 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_223_rewriteL__comp__comp2,axiom,
% 2.25/1.20 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 2.25/1.20 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.25/1.20 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 2.25/1.20 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 2.25/1.20 = R )
% 2.25/1.20 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 2.25/1.20 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_224_subset__image__inj,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,S: set @ A,F: B > A,T2: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ S @ ( image @ B @ A @ F @ T2 ) )
% 2.25/1.20 = ( ? [U: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ B ) @ U @ T2 )
% 2.25/1.20 & ( inj_on @ B @ A @ F @ U )
% 2.25/1.20 & ( S
% 2.25/1.20 = ( image @ B @ A @ F @ U ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_225_ex__subset__image__inj,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
% 2.25/1.20 ( ( ? [T3: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
% 2.25/1.20 & ( P @ T3 ) ) )
% 2.25/1.20 = ( ? [T3: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
% 2.25/1.20 & ( inj_on @ B @ A @ F @ T3 )
% 2.25/1.20 & ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_226_all__subset__image__inj,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,S: set @ B,P: ( set @ A ) > $o] :
% 2.25/1.20 ( ( ! [T3: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ T3 @ ( image @ B @ A @ F @ S ) )
% 2.25/1.20 => ( P @ T3 ) ) )
% 2.25/1.20 = ( ! [T3: set @ B] :
% 2.25/1.20 ( ( ( ord_less_eq @ ( set @ B ) @ T3 @ S )
% 2.25/1.20 & ( inj_on @ B @ A @ F @ T3 ) )
% 2.25/1.20 => ( P @ ( image @ B @ A @ F @ T3 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_227_all__subset__image,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B,P: ( set @ A ) > $o] :
% 2.25/1.20 ( ( ! [B5: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F @ A2 ) )
% 2.25/1.20 => ( P @ B5 ) ) )
% 2.25/1.20 = ( ! [B5: set @ B] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ B ) @ B5 @ A2 )
% 2.25/1.20 => ( P @ ( image @ B @ A @ F @ B5 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_228_inj__image__Compl__subset,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_229_inj__on__image__Fpow,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType,F: A > B,A2: set @ A] :
% 2.25/1.20 ( ( inj_on @ A @ B @ F @ A2 )
% 2.25/1.20 => ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F ) @ ( finite_Fpow @ A @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_230_uminus__apply,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( uminus @ B )
% 2.25/1.20 => ( ( uminus_uminus @ ( A > B ) )
% 2.25/1.20 = ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_231_double__compl,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 2.25/1.20 = X3 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_232_compl__eq__compl__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ( uminus_uminus @ A @ X3 )
% 2.25/1.20 = ( uminus_uminus @ A @ Y2 ) )
% 2.25/1.20 = ( X3 = Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_233_ComplI,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A] :
% 2.25/1.20 ( ~ ( member @ A @ C2 @ A2 )
% 2.25/1.20 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_234_Compl__iff,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.25/1.20 = ( ~ ( member @ A @ C2 @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_235_Compl__eq__Compl__iff,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ( uminus_uminus @ ( set @ A ) @ A2 )
% 2.25/1.20 = ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.25/1.20 = ( A2 = B2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_236_compl__le__compl__iff,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y2 ) )
% 2.25/1.20 = ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_237_inj__uminus,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( ordered_ab_group_add @ A )
% 2.25/1.20 => ! [A2: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_238_Compl__subset__Compl__iff,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) )
% 2.25/1.20 = ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_239_Compl__anti__mono,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B2 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_240_sup__compl__top__left2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_241_sup__compl__top__left1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ ( sup_sup @ A @ X3 @ Y2 ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_242_sup__compl__top,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ X3 @ ( uminus_uminus @ A @ X3 ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_243_compl__sup__top,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ X3 )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_244_compl__mono,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,Y2: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ X3 @ Y2 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_245_compl__le__swap1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X3 ) )
% 2.25/1.20 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_246_compl__le__swap2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [Y2: A,X3: A] :
% 2.25/1.20 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X3 )
% 2.25/1.20 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_247_Fpow__mono,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A,B2: set @ A] :
% 2.25/1.20 ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
% 2.25/1.20 => ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_248_sup__cancel__left2,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,A3: A,B3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ A3 ) @ ( sup_sup @ A @ X3 @ B3 ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_249_sup__cancel__left1,axiom,
% 2.25/1.20 ! [A: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A )
% 2.25/1.20 => ! [X3: A,A3: A,B3: A] :
% 2.25/1.20 ( ( sup_sup @ A @ ( sup_sup @ A @ X3 @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X3 ) @ B3 ) )
% 2.25/1.20 = ( top_top @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_250_Compl__partition2,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ A2 )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_251_Compl__partition,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( sup_sup @ ( set @ A ) @ A2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_252_fun__Compl__def,axiom,
% 2.25/1.20 ! [B: $tType,A: $tType] :
% 2.25/1.20 ( ( uminus @ B )
% 2.25/1.20 => ( ( uminus_uminus @ ( A > B ) )
% 2.25/1.20 = ( ^ [A6: A > B,X2: A] : ( uminus_uminus @ B @ ( A6 @ X2 ) ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_253_ComplD,axiom,
% 2.25/1.20 ! [A: $tType,C2: A,A2: set @ A] :
% 2.25/1.20 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.25/1.20 => ~ ( member @ A @ C2 @ A2 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_254_double__complement,axiom,
% 2.25/1.20 ! [A: $tType,A2: set @ A] :
% 2.25/1.20 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
% 2.25/1.20 = A2 ) ).
% 2.25/1.20
% 2.25/1.20 thf(fact_255_surj__Compl__image__subset,axiom,
% 2.25/1.20 ! [A: $tType,B: $tType,F: B > A,A2: set @ B] :
% 2.25/1.20 ( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
% 2.25/1.20 = ( top_top @ ( set @ A ) ) )
% 2.25/1.20 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A2 ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
% 2.25/1.20 bounded_lattice @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
% 2.25/1.20 ! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( bounded_lattice @ A9 )
% 2.25/1.20 => ( bounded_lattice @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( bounded_lattice @ A9 )
% 2.25/1.20 => ( bounded_lattice_top @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( semilattice_sup @ A9 )
% 2.25/1.20 => ( semilattice_sup @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( boolean_algebra @ A9 )
% 2.25/1.20 => ( boolean_algebra @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( order_top @ A9 )
% 2.25/1.20 => ( order_top @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( preorder @ A9 )
% 2.25/1.20 => ( preorder @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Lattices_Olattice,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( lattice @ A9 )
% 2.25/1.20 => ( lattice @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( order @ A9 )
% 2.25/1.20 => ( order @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Orderings_Otop,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( top @ A9 )
% 2.25/1.20 => ( top @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Orderings_Oord,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( ord @ A9 )
% 2.25/1.20 => ( ord @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.25/1.20 ! [A8: $tType,A9: $tType] :
% 2.25/1.20 ( ( uminus @ A9 )
% 2.25/1.20 => ( uminus @ ( A8 > A9 ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_3,axiom,
% 2.25/1.20 ! [A8: $tType] : ( bounded_lattice_top @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_4,axiom,
% 2.25/1.20 ! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_5,axiom,
% 2.25/1.20 ! [A8: $tType] : ( boolean_algebra @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Orderings_Oorder__top_6,axiom,
% 2.25/1.20 ! [A8: $tType] : ( order_top @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
% 2.25/1.20 ! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
% 2.25/1.20 ! [A8: $tType] : ( lattice @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
% 2.25/1.20 ! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Orderings_Otop_10,axiom,
% 2.25/1.20 ! [A8: $tType] : ( top @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Orderings_Oord_11,axiom,
% 2.25/1.20 ! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_Set_Oset___Groups_Ouminus_12,axiom,
% 2.25/1.20 ! [A8: $tType] : ( uminus @ ( set @ A8 ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_13,axiom,
% 2.25/1.20 bounded_lattice_top @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
% 2.25/1.20 semilattice_sup @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_15,axiom,
% 2.25/1.20 boolean_algebra @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Oorder__top_16,axiom,
% 2.25/1.20 order_top @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Opreorder_17,axiom,
% 2.25/1.20 preorder @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.25/1.20 linorder @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Lattices_Olattice_18,axiom,
% 2.25/1.20 lattice @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Oorder_19,axiom,
% 2.25/1.20 order @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Otop_20,axiom,
% 2.25/1.20 top @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Orderings_Oord_21,axiom,
% 2.25/1.20 ord @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(tcon_HOL_Obool___Groups_Ouminus_22,axiom,
% 2.25/1.20 uminus @ $o ).
% 2.25/1.20
% 2.25/1.20 thf(conj_0,hypothesis,
% 2.25/1.20 inj_on @ a @ b @ f @ ( sup_sup @ ( set @ a ) @ ( image @ c @ a @ g @ ( top_top @ ( set @ c ) ) ) @ ( image @ c @ a @ h @ ( top_top @ ( set @ c ) ) ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(conj_1,hypothesis,
% 2.25/1.20 ( ( comp @ a @ b @ c @ f @ g )
% 2.25/1.20 = ( comp @ a @ b @ c @ f @ h ) ) ).
% 2.25/1.20
% 2.25/1.20 thf(conj_2,conjecture,
% 2.25/1.20 g = h ).
% 2.25/1.20 ------------------------
% 2.25/1.22 % (32613)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/275Mi)
% 2.25/1.22 % (32609)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/4Mi)
% 2.25/1.22 % (32611)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/2Mi)
% 2.25/1.22 % (32612)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/2Mi)
% 2.25/1.22 % (32608)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/183Mi)
% 2.25/1.22 % (32610)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/27Mi)
% 2.25/1.22 % (32614)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/18Mi)
% 2.25/1.22 % (32609)Instruction limit reached!
% 2.25/1.22 % (32609)------------------------------
% 2.25/1.22 % (32609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.22 % (32609)Termination reason: Unknown
% 2.25/1.22 % (32609)Termination phase: shuffling
% 2.25/1.22
% 2.25/1.22 % (32609)Memory used [KB]: 1407
% 2.25/1.22 % (32609)Time elapsed: 0.003 s
% 2.25/1.22 % (32609)Instructions burned: 4 (million)
% 2.25/1.22 % (32609)------------------------------
% 2.25/1.22 % (32609)------------------------------
% 2.25/1.22 % (32611)Instruction limit reached!
% 2.25/1.22 % (32611)------------------------------
% 2.25/1.22 % (32611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.22 % (32611)Termination reason: Unknown
% 2.25/1.22 % (32611)Termination phase: shuffling
% 2.25/1.22
% 2.25/1.22 % (32611)Memory used [KB]: 1407
% 2.25/1.22 % (32611)Time elapsed: 0.003 s
% 2.25/1.22 % (32611)Instructions burned: 4 (million)
% 2.25/1.22 % (32611)------------------------------
% 2.25/1.22 % (32611)------------------------------
% 2.25/1.22 % (32612)Instruction limit reached!
% 2.25/1.22 % (32612)------------------------------
% 2.25/1.22 % (32612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.22 % (32612)Termination reason: Unknown
% 2.25/1.22 % (32612)Termination phase: shuffling
% 2.25/1.22
% 2.25/1.22 % (32612)Memory used [KB]: 1407
% 2.25/1.22 % (32612)Time elapsed: 0.003 s
% 2.25/1.22 % (32612)Instructions burned: 5 (million)
% 2.25/1.22 % (32612)------------------------------
% 2.25/1.22 % (32612)------------------------------
% 2.25/1.23 % (32614)Instruction limit reached!
% 2.25/1.23 % (32614)------------------------------
% 2.25/1.23 % (32614)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.23 % (32614)Termination reason: Unknown
% 2.25/1.23 % (32614)Termination phase: shuffling
% 2.25/1.23
% 2.25/1.23 % (32614)Memory used [KB]: 1663
% 2.25/1.23 % (32614)Time elapsed: 0.008 s
% 2.25/1.23 % (32614)Instructions burned: 19 (million)
% 2.25/1.23 % (32614)------------------------------
% 2.25/1.23 % (32614)------------------------------
% 2.25/1.23 % (32610)Instruction limit reached!
% 2.25/1.23 % (32610)------------------------------
% 2.25/1.23 % (32610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.23 % (32610)Termination reason: Unknown
% 2.25/1.23 % (32610)Termination phase: Property scanning
% 2.25/1.23
% 2.25/1.23 % (32610)Memory used [KB]: 1791
% 2.25/1.23 % (32610)Time elapsed: 0.011 s
% 2.25/1.23 % (32610)Instructions burned: 30 (million)
% 2.25/1.23 % (32610)------------------------------
% 2.25/1.23 % (32610)------------------------------
% 2.25/1.23 % (32615)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/3Mi)
% 2.25/1.23 % (32616)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/37Mi)
% 2.25/1.23 % (32617)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/15Mi)
% 2.25/1.23 % (32615)Instruction limit reached!
% 2.25/1.23 % (32615)------------------------------
% 2.25/1.23 % (32615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.23 % (32615)Termination reason: Unknown
% 2.25/1.23 % (32615)Termination phase: shuffling
% 2.25/1.23
% 2.25/1.23 % (32615)Memory used [KB]: 1407
% 2.25/1.23 % (32615)Time elapsed: 0.002 s
% 2.25/1.23 % (32615)Instructions burned: 3 (million)
% 2.25/1.23 % (32615)------------------------------
% 2.25/1.23 % (32615)------------------------------
% 2.25/1.24 % (32617)Instruction limit reached!
% 2.25/1.24 % (32617)------------------------------
% 2.25/1.24 % (32617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.24 % (32617)Termination reason: Unknown
% 2.25/1.24 % (32617)Termination phase: shuffling
% 2.25/1.24
% 2.25/1.24 % (32617)Memory used [KB]: 1663
% 2.25/1.24 % (32617)Time elapsed: 0.006 s
% 2.25/1.24 % (32617)Instructions burned: 15 (million)
% 2.25/1.24 % (32617)------------------------------
% 2.25/1.24 % (32617)------------------------------
% 2.25/1.24 % (32618)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/3Mi)
% 2.25/1.24 % (32618)Instruction limit reached!
% 2.25/1.24 % (32618)------------------------------
% 2.25/1.24 % (32618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.24 % (32618)Termination reason: Unknown
% 2.25/1.24 % (32618)Termination phase: shuffling
% 2.25/1.24
% 2.25/1.24 % (32618)Memory used [KB]: 1407
% 2.25/1.24 % (32618)Time elapsed: 0.002 s
% 2.25/1.24 % (32618)Instructions burned: 3 (million)
% 2.25/1.24 % (32618)------------------------------
% 2.25/1.24 % (32618)------------------------------
% 2.25/1.24 % (32619)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/1041Mi)
% 2.25/1.24 % (32620)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/7Mi)
% 2.25/1.24 % (32616)Instruction limit reached!
% 2.25/1.24 % (32616)------------------------------
% 2.25/1.24 % (32616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.24 % (32616)Termination reason: Unknown
% 2.25/1.24 % (32616)Termination phase: shuffling
% 2.25/1.24
% 2.25/1.24 % (32616)Memory used [KB]: 2174
% 2.25/1.24 % (32616)Time elapsed: 0.012 s
% 2.25/1.24 % (32616)Instructions burned: 40 (million)
% 2.25/1.24 % (32616)------------------------------
% 2.25/1.24 % (32616)------------------------------
% 2.25/1.24 % (32620)Instruction limit reached!
% 2.25/1.24 % (32620)------------------------------
% 2.25/1.24 % (32620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.24 % (32620)Termination reason: Unknown
% 2.25/1.24 % (32620)Termination phase: shuffling
% 2.25/1.24
% 2.25/1.24 % (32620)Memory used [KB]: 1407
% 2.25/1.24 % (32620)Time elapsed: 0.003 s
% 2.25/1.24 % (32620)Instructions burned: 7 (million)
% 2.25/1.24 % (32620)------------------------------
% 2.25/1.24 % (32620)------------------------------
% 2.25/1.25 % (32621)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/16Mi)
% 2.25/1.25 % (32622)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/3Mi)
% 2.25/1.25 % (32622)Instruction limit reached!
% 2.25/1.25 % (32622)------------------------------
% 2.25/1.25 % (32622)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.25 % (32622)Termination reason: Unknown
% 2.25/1.25 % (32622)Termination phase: shuffling
% 2.25/1.25
% 2.25/1.25 % (32622)Memory used [KB]: 1407
% 2.25/1.25 % (32622)Time elapsed: 0.002 s
% 2.25/1.25 % (32622)Instructions burned: 3 (million)
% 2.25/1.25 % (32622)------------------------------
% 2.25/1.25 % (32622)------------------------------
% 2.25/1.25 % (32621)Instruction limit reached!
% 2.25/1.25 % (32621)------------------------------
% 2.25/1.25 % (32621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.25 % (32621)Termination reason: Unknown
% 2.25/1.25 % (32621)Termination phase: shuffling
% 2.25/1.25
% 2.25/1.25 % (32621)Memory used [KB]: 1791
% 2.25/1.25 % (32621)Time elapsed: 0.007 s
% 2.25/1.25 % (32621)Instructions burned: 18 (million)
% 2.25/1.25 % (32621)------------------------------
% 2.25/1.25 % (32621)------------------------------
% 2.25/1.25 % (32623)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/3Mi)
% 2.25/1.25 % (32623)Instruction limit reached!
% 2.25/1.25 % (32623)------------------------------
% 2.25/1.25 % (32623)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.25 % (32623)Termination reason: Unknown
% 2.25/1.25 % (32623)Termination phase: shuffling
% 2.25/1.25
% 2.25/1.25 % (32623)Memory used [KB]: 1407
% 2.25/1.25 % (32623)Time elapsed: 0.002 s
% 2.25/1.25 % (32623)Instructions burned: 3 (million)
% 2.25/1.25 % (32623)------------------------------
% 2.25/1.25 % (32623)------------------------------
% 2.25/1.25 % (32624)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/7Mi)
% 2.25/1.26 % (32624)Instruction limit reached!
% 2.25/1.26 % (32624)------------------------------
% 2.25/1.26 % (32624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.26 % (32624)Termination reason: Unknown
% 2.25/1.26 % (32624)Termination phase: shuffling
% 2.25/1.26
% 2.25/1.26 % (32624)Memory used [KB]: 1535
% 2.25/1.26 % (32624)Time elapsed: 0.004 s
% 2.25/1.26 % (32624)Instructions burned: 9 (million)
% 2.25/1.26 % (32624)------------------------------
% 2.25/1.26 % (32624)------------------------------
% 2.25/1.26 % (32625)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/3Mi)
% 2.25/1.26 % (32625)Instruction limit reached!
% 2.25/1.26 % (32625)------------------------------
% 2.25/1.26 % (32625)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.26 % (32625)Termination reason: Unknown
% 2.25/1.26 % (32625)Termination phase: shuffling
% 2.25/1.26
% 2.25/1.26 % (32625)Memory used [KB]: 1407
% 2.25/1.26 % (32625)Time elapsed: 0.002 s
% 2.25/1.26 % (32625)Instructions burned: 3 (million)
% 2.25/1.26 % (32625)------------------------------
% 2.25/1.26 % (32625)------------------------------
% 2.25/1.26 % (32626)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/4Mi)
% 2.25/1.26 % (32627)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/18Mi)
% 2.25/1.26 % (32626)Instruction limit reached!
% 2.25/1.26 % (32626)------------------------------
% 2.25/1.26 % (32626)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.26 % (32626)Termination reason: Unknown
% 2.25/1.26 % (32626)Termination phase: shuffling
% 2.25/1.26
% 2.25/1.26 % (32626)Memory used [KB]: 1407
% 2.25/1.26 % (32626)Time elapsed: 0.003 s
% 2.25/1.26 % (32626)Instructions burned: 6 (million)
% 2.25/1.26 % (32626)------------------------------
% 2.25/1.26 % (32626)------------------------------
% 2.25/1.27 % (32628)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/710Mi)
% 2.25/1.27 % (32629)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/6Mi)
% 2.25/1.27 % (32627)Instruction limit reached!
% 2.25/1.27 % (32627)------------------------------
% 2.25/1.27 % (32627)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.27 % (32627)Termination reason: Unknown
% 2.25/1.27 % (32627)Termination phase: shuffling
% 2.25/1.27
% 2.25/1.27 % (32627)Memory used [KB]: 1791
% 2.25/1.27 % (32627)Time elapsed: 0.007 s
% 2.25/1.27 % (32627)Instructions burned: 18 (million)
% 2.25/1.27 % (32627)------------------------------
% 2.25/1.27 % (32627)------------------------------
% 2.25/1.27 % (32629)Instruction limit reached!
% 2.25/1.27 % (32629)------------------------------
% 2.25/1.27 % (32629)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.27 % (32629)Termination reason: Unknown
% 2.25/1.27 % (32629)Termination phase: shuffling
% 2.25/1.27
% 2.25/1.27 % (32629)Memory used [KB]: 1407
% 2.25/1.27 % (32629)Time elapsed: 0.003 s
% 2.25/1.27 % (32629)Instructions burned: 6 (million)
% 2.25/1.27 % (32629)------------------------------
% 2.25/1.27 % (32629)------------------------------
% 2.25/1.27 % (32608)Instruction limit reached!
% 2.25/1.27 % (32608)------------------------------
% 2.25/1.27 % (32608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.27 % (32608)Termination reason: Unknown
% 2.25/1.27 % (32608)Termination phase: Saturation
% 2.25/1.27
% 2.25/1.27 % (32608)Memory used [KB]: 7291
% 2.25/1.27 % (32608)Time elapsed: 0.055 s
% 2.25/1.27 % (32608)Instructions burned: 186 (million)
% 2.25/1.27 % (32608)------------------------------
% 2.25/1.27 % (32608)------------------------------
% 2.25/1.28 % (32631)dis+21_1:8_apa=on:cnfonf=off:fd=off:fsr=off:hud=0:ins=1:kws=inv_frequency:nwc=10.0:ss=axioms:st=5.0:i=21:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/21Mi)
% 2.25/1.28 % (32630)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=902:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/902Mi)
% 2.25/1.28 % (32632)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/5Mi)
% 2.25/1.28 % (32632)Instruction limit reached!
% 2.25/1.28 % (32632)------------------------------
% 2.25/1.28 % (32632)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.28 % (32632)Termination reason: Unknown
% 2.25/1.28 % (32632)Termination phase: shuffling
% 2.25/1.28
% 2.25/1.28 % (32632)Memory used [KB]: 1407
% 2.25/1.28 % (32632)Time elapsed: 0.003 s
% 2.25/1.28 % (32632)Instructions burned: 7 (million)
% 2.25/1.28 % (32632)------------------------------
% 2.25/1.28 % (32632)------------------------------
% 2.25/1.28 % (32613)First to succeed.
% 2.25/1.28 % (32633)lrs+2_1:1_cnfonf=lazy_not_gen_be_off:cs=on:fe=off:hud=10:inj=on:ins=3:plsq=on:plsqc=1:sd=10:ss=axioms:tnu=1:i=6:si=on:rtra=on_0 on DTF2THF_32419 for (2999ds/6Mi)
% 2.25/1.28 % (32631)Instruction limit reached!
% 2.25/1.28 % (32631)------------------------------
% 2.25/1.28 % (32631)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.28 % (32631)Termination reason: Unknown
% 2.25/1.28 % (32631)Termination phase: shuffling
% 2.25/1.28
% 2.25/1.28 % (32631)Memory used [KB]: 1918
% 2.25/1.28 % (32631)Time elapsed: 0.008 s
% 2.25/1.28 % (32631)Instructions burned: 21 (million)
% 2.25/1.28 % (32631)------------------------------
% 2.25/1.28 % (32631)------------------------------
% 2.25/1.28 % (32613)Refutation found. Thanks to Tanya!
% 2.25/1.28 % SZS status Theorem for DTF2THF_32419
% 2.25/1.28 % SZS output start Proof for DTF2THF_32419
% 2.25/1.28 thf(type_def_5, type, set: ($tType) > $tType).
% 2.25/1.28 thf(type_def_7, type, a: $tType).
% 2.25/1.28 thf(type_def_8, type, b: $tType).
% 2.25/1.28 thf(type_def_9, type, c: $tType).
% 2.25/1.28 thf(func_def_0, type, set: ($tType) > $tType).
% 2.25/1.28 thf(func_def_1, type, c: $tType).
% 2.25/1.28 thf(func_def_2, type, b: $tType).
% 2.25/1.28 thf(func_def_3, type, a: $tType).
% 2.25/1.28 thf(func_def_4, type, bounded_lattice: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_5, type, type: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_6, type, uminus: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_7, type, ord: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_8, type, top: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_9, type, order: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_10, type, lattice: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_11, type, linorder: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_12, type, preorder: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_13, type, order_top: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_14, type, boolean_algebra: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_15, type, semilattice_sup: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_16, type, ordered_ab_group_add: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_17, type, bounded_lattice_top: !>[X0: $tType]:($o)).
% 2.25/1.28 thf(func_def_18, type, finite_Fpow: !>[X0: $tType]:(set @ X0 > set @ set @ X0)).
% 2.25/1.28 thf(func_def_19, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 2.25/1.28 thf(func_def_20, type, inj_on: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > $o)).
% 2.25/1.28 thf(func_def_21, type, the_inv_into: !>[X0: $tType, X1: $tType]:(set @ X0 > (X0 > X1) > X1 > X0)).
% 2.25/1.28 thf(func_def_22, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.25/1.28 thf(func_def_23, type, sup_sup: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.25/1.28 thf(func_def_24, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.25/1.28 thf(func_def_25, type, top_top: !>[X0: $tType]:(X0)).
% 2.25/1.28 thf(func_def_26, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.25/1.28 thf(func_def_27, type, image: !>[X0: $tType, X1: $tType]:((X0 > X1) > set @ X0 > set @ X1)).
% 2.25/1.28 thf(func_def_28, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.25/1.28 thf(func_def_29, type, f: a > b).
% 2.25/1.28 thf(func_def_30, type, g: c > a).
% 2.25/1.28 thf(func_def_31, type, h: c > a).
% 2.25/1.28 thf(func_def_44, type, sK0: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_45, type, sK1: !>[X0: $tType, X1: $tType]:((X1 > X1) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_46, type, sK2: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 2.25/1.28 thf(func_def_47, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X2) > (X1 > X2) > X1 > X0)).
% 2.25/1.28 thf(func_def_48, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X2) > (X1 > X2) > X2)).
% 2.25/1.28 thf(func_def_49, type, sK5: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_50, type, sK6: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_51, type, sK7: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_52, type, sK8: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_53, type, sK9: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_54, type, sK10: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_55, type, sK11: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_56, type, sK12: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_57, type, sK13: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_58, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_59, type, sK15: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.25/1.28 thf(func_def_60, type, sK16: !>[X0: $tType]:(set @ X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.25/1.28 thf(func_def_61, type, sK17: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > X1)).
% 2.25/1.28 thf(func_def_62, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X0 > X1) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_63, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_64, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_65, type, sK21: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_66, type, sK22: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_67, type, sK23: !>[X0: $tType, X1: $tType]:(X1 > X0 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_68, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X1) > (X2 > X1) > set @ X0 > (X0 > X2) > X0)).
% 2.25/1.28 thf(func_def_69, type, sK25: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_70, type, sK26: !>[X0: $tType, X1: $tType]:((X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_71, type, sK27: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X1)).
% 2.25/1.28 thf(func_def_72, type, sK28: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X0)).
% 2.25/1.28 thf(func_def_73, type, sK29: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_74, type, sK30: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_75, type, sK31: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > X0 > X1)).
% 2.25/1.28 thf(func_def_76, type, sK32: !>[X0: $tType, X1: $tType]:(X1 > (X0 > X1) > X0)).
% 2.25/1.28 thf(func_def_77, type, sK33: !>[X0: $tType, X1: $tType]:((X0 > X1) > X1)).
% 2.25/1.28 thf(func_def_78, type, sK34: !>[X0: $tType]:(set @ X0 > X0)).
% 2.25/1.28 thf(func_def_79, type, sK35: !>[X0: $tType, X1: $tType]:((X0 > $o) > set @ X1 > (X1 > X0) > X0)).
% 2.25/1.28 thf(func_def_80, type, sK36: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.25/1.28 thf(func_def_81, type, sK37: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_82, type, sK38: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.25/1.28 thf(func_def_83, type, sK39: !>[X0: $tType, X1: $tType]:(set @ X1 > (X1 > X0) > set @ X0 > set @ X1)).
% 2.25/1.28 thf(func_def_84, type, sK40: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_85, type, sK41: !>[X0: $tType, X1: $tType]:((X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_86, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_87, type, sK43: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_88, type, sK44: !>[X0: $tType, X1: $tType]:((X0 > $o) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_89, type, sK45: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_90, type, sK46: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_91, type, sK47: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_92, type, sK48: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_93, type, sK49: !>[X0: $tType, X1: $tType]:(set @ X1 > (X0 > X1) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_94, type, sK50: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_95, type, sK51: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > set @ X1 > X1)).
% 2.25/1.28 thf(func_def_96, type, sK52: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X0)).
% 2.25/1.28 thf(func_def_97, type, sK53: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X1)).
% 2.25/1.28 thf(func_def_98, type, sK54: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_99, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType]:((X2 > X0) > (X1 > X2) > (X2 > X0) > X2)).
% 2.25/1.28 thf(func_def_100, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType]:((X1 > X0) > (X1 > X0) > (X2 > X1) > X1)).
% 2.25/1.28 thf(func_def_101, type, sK57: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X1)).
% 2.25/1.28 thf(func_def_102, type, sK58: !>[X0: $tType, X1: $tType]:((set @ X0 > $o) > set @ X1 > (X1 > X0) > set @ X0)).
% 2.25/1.28 thf(func_def_103, type, sK59: !>[X0: $tType, X1: $tType]:(X0 > (X1 > X0) > X1)).
% 2.25/1.28 thf(func_def_104, type, sK60: !>[X0: $tType]:(set @ X0 > (X0 > $o) > X0)).
% 2.25/1.28 thf(func_def_105, type, sK61: !>[X0: $tType]:(X0)).
% 2.25/1.28 thf(func_def_106, type, sK62: !>[X0: $tType]:((X0 > $o) > set @ X0 > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_107, type, sK63: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_108, type, sK64: !>[X0: $tType]:((X0 > $o) > set @ X0 > X0)).
% 2.25/1.28 thf(func_def_110, type, ph66: !>[X0: $tType]:(X0)).
% 2.25/1.28 thf(func_def_112, type, sK67: c).
% 2.25/1.28 thf(f1599,plain,(
% 2.25/1.28 $false),
% 2.25/1.28 inference(subsumption_resolution,[status(thm)],[f1598,f1257])).
% 2.25/1.28 thf(f1257,plain,(
% 2.25/1.28 ((comp @ a @ b @ c @ f @ g) = (comp @ a @ b @ c @ f @ h))),
% 2.25/1.28 inference(cnf_transformation,[status(thm)],[f292])).
% 2.25/1.28 thf(f292,axiom,(
% 2.25/1.28 ((comp @ a @ b @ c @ f @ g) = (comp @ a @ b @ c @ f @ h))),
% 2.25/1.28 file('/export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2THF_32419.p',conj_1)).
% 2.25/1.28 thf(f1598,plain,(
% 2.25/1.28 ((comp @ a @ b @ c @ f @ g) != (comp @ a @ b @ c @ f @ h))),
% 2.25/1.28 inference(subsumption_resolution,[status(thm)],[f1540,f1313])).
% 2.25/1.28 thf(f1313,plain,(
% 2.25/1.28 (g != h)),
% 2.25/1.28 inference(cnf_transformation,[status(thm)],[f810])).
% 2.25/1.28 thf(f810,plain,(
% 2.25/1.28 (g != h)),
% 2.25/1.28 inference(flattening,[status(thm)],[f294])).
% 2.25/1.28 thf(f294,negated_conjecture,(
% 2.25/1.28 ~(g = h)),
% 2.25/1.28 inference(negated_conjecture,[status(cth)],[f293])).
% 2.25/1.28 thf(f293,conjecture,(
% 2.25/1.28 (g = h)),
% 2.25/1.28 file('/export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2THF_32419.p',conj_2)).
% 2.25/1.28 thf(f1540,plain,(
% 2.25/1.28 (g = h) | ((comp @ a @ b @ c @ f @ g) != (comp @ a @ b @ c @ f @ h))),
% 2.25/1.28 inference(trivial_inequality_removal,[status(thm)],[f1520])).
% 2.25/1.28 thf(f1520,plain,(
% 2.25/1.28 ($true != $true) | ((comp @ a @ b @ c @ f @ g) != (comp @ a @ b @ c @ f @ h)) | (g = h)),
% 2.25/1.28 inference(superposition,[status(thm)],[f1238,f1334])).
% 2.25/1.28 thf(f1334,plain,(
% 2.25/1.28 ((inj_on @ a @ b @ f @ (sup_sup @ set @ a @ (image @ c @ a @ g @ top_top @ set @ c) @ (image @ c @ a @ h @ top_top @ set @ c))) = $true)),
% 2.25/1.28 inference(cnf_transformation,[status(thm)],[f547])).
% 2.25/1.28 thf(f547,plain,(
% 2.25/1.28 ((inj_on @ a @ b @ f @ (sup_sup @ set @ a @ (image @ c @ a @ g @ top_top @ set @ c) @ (image @ c @ a @ h @ top_top @ set @ c))) = $true)),
% 2.25/1.28 inference(fool_elimination,[status(thm)],[f546])).
% 2.25/1.28 thf(f546,plain,(
% 2.25/1.28 (inj_on @ a @ b @ f @ (sup_sup @ set @ a @ (image @ c @ a @ g @ top_top @ set @ c) @ (image @ c @ a @ h @ top_top @ set @ c)))),
% 2.25/1.28 inference(rectify,[status(thm)],[f291])).
% 2.25/1.28 thf(f291,axiom,(
% 2.25/1.28 (inj_on @ a @ b @ f @ (sup_sup @ set @ a @ (image @ c @ a @ g @ top_top @ set @ c) @ (image @ c @ a @ h @ top_top @ set @ c)))),
% 2.25/1.28 file('/export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2THF_32419.p',conj_0)).
% 2.25/1.28 thf(f1238,plain,(
% 2.25/1.28 ( ! [X1 : $tType,X0 : $tType,X2 : $tType,X3 : X1 > X0,X4 : X2 > X1,X5 : X2 > X1] : (($true != (inj_on @ X1 @ X0 @ X3 @ (sup_sup @ set @ X1 @ (image @ X2 @ X1 @ X4 @ top_top @ set @ X2) @ (image @ X2 @ X1 @ X5 @ top_top @ set @ X2)))) | ((comp @ X1 @ X0 @ X2 @ X3 @ X5) != (comp @ X1 @ X0 @ X2 @ X3 @ X4)) | (X4 = X5)) )),
% 2.25/1.28 inference(cnf_transformation,[status(thm)],[f904])).
% 2.25/1.28 thf(f904,plain,(
% 2.25/1.28 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1 > X0,X4 : X2 > X1,X5 : X2 > X1] : ((X4 = X5) | ((comp @ X1 @ X0 @ X2 @ X3 @ X5) != (comp @ X1 @ X0 @ X2 @ X3 @ X4)) | ($true != (inj_on @ X1 @ X0 @ X3 @ (sup_sup @ set @ X1 @ (image @ X2 @ X1 @ X4 @ top_top @ set @ X2) @ (image @ X2 @ X1 @ X5 @ top_top @ set @ X2)))))),
% 2.25/1.28 inference(flattening,[status(thm)],[f903])).
% 2.25/1.28 thf(f903,plain,(
% 2.25/1.28 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1 > X0,X4 : X2 > X1,X5 : X2 > X1] : (((X4 = X5) | ($true != (inj_on @ X1 @ X0 @ X3 @ (sup_sup @ set @ X1 @ (image @ X2 @ X1 @ X4 @ top_top @ set @ X2) @ (image @ X2 @ X1 @ X5 @ top_top @ set @ X2))))) | ((comp @ X1 @ X0 @ X2 @ X3 @ X5) != (comp @ X1 @ X0 @ X2 @ X3 @ X4)))),
% 2.25/1.28 inference(ennf_transformation,[status(thm)],[f781])).
% 2.25/1.28 thf(f781,plain,(
% 2.25/1.28 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1 > X0,X4 : X2 > X1,X5 : X2 > X1] : (((comp @ X1 @ X0 @ X2 @ X3 @ X5) = (comp @ X1 @ X0 @ X2 @ X3 @ X4)) => (($true = (inj_on @ X1 @ X0 @ X3 @ (sup_sup @ set @ X1 @ (image @ X2 @ X1 @ X4 @ top_top @ set @ X2) @ (image @ X2 @ X1 @ X5 @ top_top @ set @ X2)))) => (X4 = X5)))),
% 2.25/1.28 inference(fool_elimination,[status(thm)],[f780])).
% 2.25/1.28 thf(f780,plain,(
% 2.25/1.28 ! [X0 : $tType,X1 : $tType,X2 : $tType,X3 : X1 > X0,X4 : X2 > X1,X5 : X2 > X1] : (((comp @ X1 @ X0 @ X2 @ X3 @ X5) = (comp @ X1 @ X0 @ X2 @ X3 @ X4)) => ((inj_on @ X1 @ X0 @ X3 @ (sup_sup @ set @ X1 @ (image @ X2 @ X1 @ X4 @ top_top @ set @ X2) @ (image @ X2 @ X1 @ X5 @ top_top @ set @ X2))) => (X4 = X5)))),
% 2.25/1.28 inference(rectify,[status(thm)],[f3])).
% 2.25/1.28 thf(f3,axiom,(
% 2.25/1.28 ! [X1 : $tType,X2 : $tType,X0 : $tType,X4 : X2 > X1,X5 : X0 > X2,X3 : X0 > X2] : (((comp @ X2 @ X1 @ X0 @ X4 @ X5) = (comp @ X2 @ X1 @ X0 @ X4 @ X3)) => ((inj_on @ X2 @ X1 @ X4 @ (sup_sup @ set @ X2 @ (image @ X0 @ X2 @ X5 @ top_top @ set @ X0) @ (image @ X0 @ X2 @ X3 @ top_top @ set @ X0))) => (X3 = X5)))),
% 2.25/1.28 file('/export/starexec/sandbox2/tmp/tmp.qDS5KMCuZu/DTF2THF_32419.p',fact_2_o__inj__on)).
% 2.25/1.28 % SZS output end Proof for DTF2THF_32419
% 2.25/1.28 % (32613)------------------------------
% 2.25/1.28 % (32613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.25/1.28 % (32613)Termination reason: Refutation
% 2.25/1.28
% 2.25/1.28 % (32613)Memory used [KB]: 7291
% 2.25/1.28 % (32613)Time elapsed: 0.064 s
% 2.25/1.28 % (32613)Instructions burned: 197 (million)
% 2.25/1.28 % (32613)------------------------------
% 2.25/1.28 % (32613)------------------------------
% 2.25/1.28 % (32607)Success in time 0.072 s
%------------------------------------------------------------------------------