%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP098^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n031.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 2.78s 1.55s
% Output : Refutation 2.78s
% Verified :
% SZS Type : Refutation
% Derivation depth : 9
% Number of leaves : 4
% Syntax : Number of formulae : 20 ( 12 unt; 0 typ; 0 def)
% Number of atoms : 49 ( 30 equ; 0 cnn)
% Maximal formula atoms : 3 ( 2 avg)
% Number of connectives : 147 ( 12 ~; 9 |; 0 &; 120 @)
% ( 0 <=>; 6 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 11 ( 10 usr)
% Number of type conns : 18 ( 18 >; 0 *; 0 +; 0 <<)
% Number of symbols : 111 ( 108 usr; 10 con; 0-4 aty)
% Number of variables : 18 ( 0 ^; 18 !; 0 ?; 18 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
a: $tType ).
thf(type_def_7,type,
c: $tType ).
thf(type_def_8,type,
b: $tType ).
thf(type_def_9,type,
set_a_a: $tType ).
thf(type_def_10,type,
set_a_b: $tType ).
thf(type_def_11,type,
set_a_c: $tType ).
thf(type_def_12,type,
set_c_a: $tType ).
thf(type_def_13,type,
set_a: $tType ).
thf(type_def_14,type,
set_b: $tType ).
thf(type_def_15,type,
set_c: $tType ).
thf(func_def_0,type,
set_c_a: $tType ).
thf(func_def_1,type,
set_a_c: $tType ).
thf(func_def_2,type,
set_a_b: $tType ).
thf(func_def_3,type,
set_a_a: $tType ).
thf(func_def_4,type,
set_c: $tType ).
thf(func_def_5,type,
set_b: $tType ).
thf(func_def_6,type,
set_a: $tType ).
thf(func_def_7,type,
c: $tType ).
thf(func_def_8,type,
b: $tType ).
thf(func_def_9,type,
a: $tType ).
thf(func_def_10,type,
comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
thf(func_def_11,type,
comp_a_a_c: ( a > a ) > ( c > a ) > c > a ).
thf(func_def_12,type,
comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
thf(func_def_13,type,
comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).
thf(func_def_14,type,
comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).
thf(func_def_15,type,
comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).
thf(func_def_16,type,
comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).
thf(func_def_17,type,
comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).
thf(func_def_18,type,
comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).
thf(func_def_19,type,
comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).
thf(func_def_20,type,
comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).
thf(func_def_21,type,
comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).
thf(func_def_22,type,
comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).
thf(func_def_23,type,
comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).
thf(func_def_24,type,
comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).
thf(func_def_25,type,
comp_c_a_b: ( c > a ) > ( b > c ) > b > a ).
thf(func_def_26,type,
comp_c_a_c: ( c > a ) > ( c > c ) > c > a ).
thf(func_def_27,type,
comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).
thf(func_def_28,type,
comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).
thf(func_def_29,type,
comp_c_b_c: ( c > b ) > ( c > c ) > c > b ).
thf(func_def_30,type,
comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).
thf(func_def_31,type,
comp_c_c_c: ( c > c ) > ( c > c ) > c > c ).
thf(func_def_32,type,
inj_on_a_a_a_b: ( ( a > a ) > a > b ) > set_a_a > $o ).
thf(func_def_33,type,
inj_on_a_b_a_b: ( ( a > b ) > a > b ) > set_a_b > $o ).
thf(func_def_34,type,
inj_on_a_c_a_b: ( ( a > c ) > a > b ) > set_a_c > $o ).
thf(func_def_35,type,
inj_on_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > $o ).
thf(func_def_36,type,
inj_on_c_a_c_b: ( ( c > a ) > c > b ) > set_c_a > $o ).
thf(func_def_37,type,
inj_on_c_a_c_c: ( ( c > a ) > c > c ) > set_c_a > $o ).
thf(func_def_38,type,
inj_on_c_a_a: ( ( c > a ) > a ) > set_c_a > $o ).
thf(func_def_39,type,
inj_on_a_a: ( a > a ) > set_a > $o ).
thf(func_def_40,type,
inj_on_a_b: ( a > b ) > set_a > $o ).
thf(func_def_41,type,
inj_on_a_c: ( a > c ) > set_a > $o ).
thf(func_def_42,type,
inj_on_b_a: ( b > a ) > set_b > $o ).
thf(func_def_43,type,
inj_on_b_b: ( b > b ) > set_b > $o ).
thf(func_def_44,type,
inj_on_b_c: ( b > c ) > set_b > $o ).
thf(func_def_45,type,
inj_on_c_a: ( c > a ) > set_c > $o ).
thf(func_def_46,type,
inj_on_c_b: ( c > b ) > set_c > $o ).
thf(func_def_47,type,
inj_on_c_c: ( c > c ) > set_c > $o ).
thf(func_def_48,type,
sup_sup_set_c_a: set_c_a > set_c_a > set_c_a ).
thf(func_def_49,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(func_def_50,type,
sup_sup_set_b: set_b > set_b > set_b ).
thf(func_def_51,type,
sup_sup_set_c: set_c > set_c > set_c ).
thf(func_def_52,type,
top_top_set_a_a: set_a_a ).
thf(func_def_53,type,
top_top_set_a_b: set_a_b ).
thf(func_def_54,type,
top_top_set_a_c: set_a_c ).
thf(func_def_55,type,
top_top_set_c_a: set_c_a ).
thf(func_def_56,type,
top_top_set_a: set_a ).
thf(func_def_57,type,
top_top_set_b: set_b ).
thf(func_def_58,type,
top_top_set_c: set_c ).
thf(func_def_59,type,
collect_a: ( a > $o ) > set_a ).
thf(func_def_60,type,
image_c_a_a: ( ( c > a ) > a ) > set_c_a > set_a ).
thf(func_def_61,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(func_def_62,type,
image_a_b: ( a > b ) > set_a > set_b ).
thf(func_def_63,type,
image_a_c: ( a > c ) > set_a > set_c ).
thf(func_def_64,type,
image_b_a: ( b > a ) > set_b > set_a ).
thf(func_def_65,type,
image_b_b: ( b > b ) > set_b > set_b ).
thf(func_def_66,type,
image_b_c: ( b > c ) > set_b > set_c ).
thf(func_def_67,type,
image_c_c_a: ( c > c > a ) > set_c > set_c_a ).
thf(func_def_68,type,
image_c_a: ( c > a ) > set_c > set_a ).
thf(func_def_69,type,
image_c_b: ( c > b ) > set_c > set_b ).
thf(func_def_70,type,
image_c_c: ( c > c ) > set_c > set_c ).
thf(func_def_71,type,
member_c_a: ( c > a ) > set_c_a > $o ).
thf(func_def_72,type,
member_a: a > set_a > $o ).
thf(func_def_73,type,
member_b: b > set_b > $o ).
thf(func_def_74,type,
member_c: c > set_c > $o ).
thf(func_def_75,type,
f: a > b ).
thf(func_def_76,type,
g: c > a ).
thf(func_def_77,type,
h: c > a ).
thf(func_def_89,type,
sK0: ( a > b ) > set_a > ( b > a ) > a ).
thf(func_def_90,type,
sK1: ( c > a ) > c ).
thf(func_def_91,type,
sK2: ( c > a ) > c ).
thf(func_def_92,type,
sK3: ( a > b ) > ( a > b ) > ( c > a ) > a ).
thf(func_def_93,type,
sK4: ( b > a ) > ( a > b ) > b ).
thf(func_def_94,type,
sK5: ( c > a ) > a > c ).
thf(func_def_95,type,
sK6: a > ( c > a ) > c ).
thf(func_def_96,type,
sK7: ( c > a ) > ( a > b ) > ( a > b ) > ( c > a ) > a ).
thf(func_def_97,type,
sK8: ( c > a ) > ( a > b ) > ( a > b ) > ( c > a ) > a ).
thf(func_def_98,type,
sK9: set_a > ( a > b ) > a ).
thf(func_def_99,type,
sK10: set_a > ( a > b ) > a ).
thf(func_def_100,type,
sK11: ( c > b ) > c ).
thf(func_def_101,type,
sK12: ( c > b ) > c ).
thf(func_def_102,type,
sK13: ( a > b ) > ( a > b ) > ( c > a ) > a ).
thf(func_def_103,type,
sK14: b > ( a > b ) > a ).
thf(func_def_104,type,
sK15: ( a > b ) > a ).
thf(func_def_105,type,
sK16: ( a > b ) > a ).
thf(func_def_106,type,
sK17: ( a > b ) > b ).
thf(func_def_107,type,
sK18: b > ( a > b ) > a ).
thf(func_def_108,type,
sK19: ( a > b ) > ( a > b ) > set_a > a ).
thf(func_def_109,type,
sK20: ( c > b ) > b > c ).
thf(func_def_110,type,
sK21: ( a > b ) > ( c > a ) > ( a > b ) > a ).
thf(func_def_111,type,
sK22: ( c > b ) > c ).
thf(func_def_112,type,
sK23: ( c > b ) > c ).
thf(func_def_113,type,
sK24: ( a > c ) > ( c > a ) > a ).
thf(func_def_114,type,
sK25: b > ( c > b ) > c ).
thf(func_def_115,type,
sK26: ( c > b ) > b ).
thf(func_def_116,type,
sK27: ( c > b ) > ( b > c ) > b ).
thf(func_def_117,type,
sK28: ( c > a ) > a > c ).
thf(func_def_118,type,
sK29: c > ( c > a ) > a > c ).
thf(func_def_119,type,
sK30: ( c > a ) > a ).
thf(func_def_120,type,
sK31: a > ( c > a ) > c ).
thf(func_def_121,type,
sK32: b > ( c > b ) > c ).
thf(func_def_122,type,
sK33: ( c > a ) > c ).
thf(func_def_123,type,
sK34: ( c > a ) > c ).
thf(func_def_124,type,
sK35: b > ( a > b ) > a ).
thf(func_def_125,type,
sK36: ( a > b ) > a ).
thf(func_def_126,type,
sK37: ( a > b ) > a ).
thf(func_def_128,type,
ph39:
!>[X0: $tType] : X0 ).
thf(func_def_129,type,
sK40: c ).
thf(f1153,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1152,f966]) ).
thf(f966,plain,
g != h,
inference(cnf_transformation,[status(thm)],[f787]) ).
thf(f787,plain,
g != h,
inference(flattening,[status(thm)],[f357]) ).
thf(f357,negated_conjecture,
g != h,
inference(negated_conjecture,[status(cth)],[f356]) ).
thf(f356,conjecture,
g = h,
file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_2) ).
thf(f1152,plain,
g = h,
inference(subsumption_resolution,[status(thm)],[f1151,f1029]) ).
thf(f1029,plain,
( ( comp_a_b_c @ f @ g )
= ( comp_a_b_c @ f @ h ) ),
inference(cnf_transformation,[status(thm)],[f355]) ).
thf(f355,axiom,
( ( comp_a_b_c @ f @ g )
= ( comp_a_b_c @ f @ h ) ),
file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_1) ).
thf(f1151,plain,
( ( ( comp_a_b_c @ f @ g )
!= ( comp_a_b_c @ f @ h ) )
| ( g = h ) ),
inference(trivial_inequality_removal,[status(thm)],[f1142]) ).
thf(f1142,plain,
( ( $true != $true )
| ( ( comp_a_b_c @ f @ g )
!= ( comp_a_b_c @ f @ h ) )
| ( g = h ) ),
inference(superposition,[status(thm)],[f1059,f986]) ).
thf(f986,plain,
( ( 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 ),
inference(cnf_transformation,[status(thm)],[f576]) ).
thf(f576,plain,
( ( 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 ),
inference(fool_elimination,[status(thm)],[f575]) ).
thf(f575,plain,
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 ) ),
inference(rectify,[status(thm)],[f354]) ).
thf(f354,axiom,
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 ) ),
file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_0) ).
thf(f1059,plain,
! [X2: c > a,X0: c > a,X1: a > b] :
( ( $true
!= ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X0 @ top_top_set_c ) @ ( image_c_a @ X2 @ top_top_set_c ) ) ) )
| ( ( comp_a_b_c @ X1 @ X0 )
!= ( comp_a_b_c @ X1 @ X2 ) )
| ( X0 = X2 ) ),
inference(cnf_transformation,[status(thm)],[f880]) ).
thf(f880,plain,
! [X0: c > a,X1: a > b,X2: c > a] :
( ( $true
!= ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X0 @ top_top_set_c ) @ ( image_c_a @ X2 @ top_top_set_c ) ) ) )
| ( ( comp_a_b_c @ X1 @ X0 )
!= ( comp_a_b_c @ X1 @ X2 ) )
| ( X0 = X2 ) ),
inference(flattening,[status(thm)],[f879]) ).
thf(f879,plain,
! [X0: c > a,X1: a > b,X2: c > a] :
( ( X0 = X2 )
| ( $true
!= ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X0 @ top_top_set_c ) @ ( image_c_a @ X2 @ top_top_set_c ) ) ) )
| ( ( comp_a_b_c @ X1 @ X0 )
!= ( comp_a_b_c @ X1 @ X2 ) ) ),
inference(ennf_transformation,[status(thm)],[f544]) ).
thf(f544,plain,
! [X0: c > a,X1: a > b,X2: c > a] :
( ( ( comp_a_b_c @ X1 @ X0 )
= ( comp_a_b_c @ X1 @ X2 ) )
=> ( ( $true
= ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X0 @ top_top_set_c ) @ ( image_c_a @ X2 @ top_top_set_c ) ) ) )
=> ( X0 = X2 ) ) ),
inference(fool_elimination,[status(thm)],[f543]) ).
thf(f543,plain,
! [X0: c > a,X1: a > b,X2: c > a] :
( ( ( comp_a_b_c @ X1 @ X0 )
= ( comp_a_b_c @ X1 @ X2 ) )
=> ( ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X0 @ top_top_set_c ) @ ( image_c_a @ X2 @ top_top_set_c ) ) )
=> ( X0 = X2 ) ) ),
inference(rectify,[status(thm)],[f9]) ).
thf(f9,axiom,
! [X2: c > a,X1: a > b,X0: c > a] :
( ( ( comp_a_b_c @ X1 @ X0 )
= ( comp_a_b_c @ X1 @ X2 ) )
=> ( ( inj_on_a_b @ X1 @ ( sup_sup_set_a @ ( image_c_a @ X2 @ top_top_set_c ) @ ( image_c_a @ X0 @ top_top_set_c ) ) )
=> ( X0 = X2 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',fact_8_o__inj__on) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP098^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n031.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:27:18 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.20/0.46 ---- Original DTF file ---
% 0.20/0.46 thf(spec,logic,$$dhol).
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % File : ITP098^1 : TPTP v9.2.1. Released v7.5.0.
% 0.20/0.46 % Domain : Interactive Theorem Proving
% 0.20/0.46 % Problem : Sledgehammer ListInf problem prob_167__5410008_1
% 0.20/0.46 % Version : Especial.
% 0.20/0.46 % English :
% 0.20/0.46
% 0.20/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.20/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.20/0.46 % Source : [Des21]
% 0.20/0.46 % Names : ListInf/prob_167__5410008_1 [Des21]
% 0.20/0.46
% 0.20/0.46 % Status : Theorem
% 0.20/0.46 % Rating : 0.33 v9.1.0, 0.25 v9.0.0, 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% 0.20/0.46 % Syntax : Number of formulae : 434 ( 110 unt; 78 typ; 0 def)
% 0.20/0.46 % Number of atoms : 1026 ( 619 equ; 0 cnn)
% 0.20/0.46 % Maximal formula atoms : 6 ( 2 avg)
% 0.20/0.46 % Number of connectives : 3023 ( 32 ~; 0 |; 8 &;2519 @)
% 0.20/0.46 % ( 0 <=>; 464 =>; 0 <=; 0 <~>)
% 0.20/0.46 % Maximal formula depth : 13 ( 6 avg)
% 0.20/0.46 % Number of types : 11 ( 10 usr)
% 0.20/0.46 % Number of type conns : 857 ( 857 >; 0 *; 0 +; 0 <<)
% 0.20/0.46 % Number of symbols : 69 ( 68 usr; 7 con; 0-3 aty)
% 0.20/0.46 % Number of variables : 1155 ( 49 ^;1070 !; 36 ?;1155 :)
% 0.20/0.46 % SPC : TH0_THM_EQU_NAR
% 0.20/0.46
% 0.20/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:31:01.240
% 0.20/0.46 %------------------------------------------------------------------------------
% 0.20/0.46 % Could-be-implicit typings (10)
% 0.20/0.46 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 0.20/0.46 set_c_a: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
% 0.20/0.46 set_a_c: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
% 0.20/0.46 set_a_b: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
% 0.20/0.46 set_a_a: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 0.20/0.46 set_c: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 0.20/0.46 set_b: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.20/0.46 set_a: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_tf__c,type,
% 0.20/0.46 c: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_tf__b,type,
% 0.20/0.46 b: $tType ).
% 0.20/0.46
% 0.20/0.46 thf(ty_n_tf__a,type,
% 0.20/0.46 a: $tType ).
% 0.20/0.46
% 0.20/0.46 % Explicit typings (68)
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
% 0.20/0.46 comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__c,type,
% 0.20/0.46 comp_a_a_c: ( a > a ) > ( c > a ) > c > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
% 0.20/0.46 comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
% 0.20/0.46 comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__c,type,
% 0.20/0.46 comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
% 0.20/0.46 comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__c,type,
% 0.20/0.46 comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
% 0.20/0.46 comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__c,type,
% 0.20/0.46 comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
% 0.20/0.46 comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__b,type,
% 0.20/0.46 comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__c,type,
% 0.20/0.46 comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__a,type,
% 0.20/0.46 comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__c,type,
% 0.20/0.46 comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__a,type,
% 0.20/0.46 comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__b,type,
% 0.20/0.46 comp_c_a_b: ( c > a ) > ( b > c ) > b > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__c,type,
% 0.20/0.46 comp_c_a_c: ( c > a ) > ( c > c ) > c > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
% 0.20/0.46 comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__b,type,
% 0.20/0.46 comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__c,type,
% 0.20/0.46 comp_c_b_c: ( c > b ) > ( c > c ) > c > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__a,type,
% 0.20/0.46 comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__c,type,
% 0.20/0.46 comp_c_c_c: ( c > c ) > ( c > c ) > c > c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
% 0.20/0.46 inj_on_a_a_a_b: ( ( a > a ) > a > b ) > set_a_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__b_J_001_062_Itf__a_Mtf__b_J,type,
% 0.20/0.46 inj_on_a_b_a_b: ( ( a > b ) > a > b ) > set_a_b > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__c_J_001_062_Itf__a_Mtf__b_J,type,
% 0.20/0.46 inj_on_a_c_a_b: ( ( a > c ) > a > b ) > set_a_c > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
% 0.20/0.46 inj_on_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
% 0.20/0.46 inj_on_c_a_c_b: ( ( c > a ) > c > b ) > set_c_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__c_J,type,
% 0.20/0.46 inj_on_c_a_c_c: ( ( c > a ) > c > c ) > set_c_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001tf__a,type,
% 0.20/0.46 inj_on_c_a_a: ( ( c > a ) > a ) > set_c_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
% 0.20/0.46 inj_on_a_a: ( a > a ) > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
% 0.20/0.46 inj_on_a_b: ( a > b ) > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__c,type,
% 0.20/0.46 inj_on_a_c: ( a > c ) > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
% 0.20/0.46 inj_on_b_a: ( b > a ) > set_b > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
% 0.20/0.46 inj_on_b_b: ( b > b ) > set_b > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__c,type,
% 0.20/0.46 inj_on_b_c: ( b > c ) > set_b > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__a,type,
% 0.20/0.46 inj_on_c_a: ( c > a ) > set_c > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__b,type,
% 0.20/0.46 inj_on_c_b: ( c > b ) > set_c > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__c,type,
% 0.20/0.46 inj_on_c_c: ( c > c ) > set_c > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 0.20/0.46 sup_sup_set_c_a: set_c_a > set_c_a > set_c_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 0.20/0.46 sup_sup_set_a: set_a > set_a > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
% 0.20/0.46 sup_sup_set_b: set_b > set_b > set_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__c_J,type,
% 0.20/0.46 sup_sup_set_c: set_c > set_c > set_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
% 0.20/0.46 top_top_set_a_a: set_a_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
% 0.20/0.46 top_top_set_a_b: set_a_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
% 0.20/0.46 top_top_set_a_c: set_a_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 0.20/0.46 top_top_set_c_a: set_c_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
% 0.20/0.46 top_top_set_a: set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
% 0.20/0.46 top_top_set_b: set_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
% 0.20/0.46 top_top_set_c: set_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.20/0.46 collect_a: ( a > $o ) > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001_062_Itf__c_Mtf__a_J_001tf__a,type,
% 0.20/0.46 image_c_a_a: ( ( c > a ) > a ) > set_c_a > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 0.20/0.46 image_a_a: ( a > a ) > set_a > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
% 0.20/0.46 image_a_b: ( a > b ) > set_a > set_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
% 0.20/0.46 image_a_c: ( a > c ) > set_a > set_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
% 0.20/0.46 image_b_a: ( b > a ) > set_b > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
% 0.20/0.46 image_b_b: ( b > b ) > set_b > set_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
% 0.20/0.46 image_b_c: ( b > c ) > set_b > set_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__c_001_062_Itf__c_Mtf__a_J,type,
% 0.20/0.46 image_c_c_a: ( c > c > a ) > set_c > set_c_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
% 0.20/0.46 image_c_a: ( c > a ) > set_c > set_a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
% 0.20/0.46 image_c_b: ( c > b ) > set_c > set_b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
% 0.20/0.46 image_c_c: ( c > c ) > set_c > set_c ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
% 0.20/0.46 member_c_a: ( c > a ) > set_c_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001tf__a,type,
% 0.20/0.46 member_a: a > set_a > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001tf__b,type,
% 0.20/0.46 member_b: b > set_b > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_c_member_001tf__c,type,
% 0.20/0.46 member_c: c > set_c > $o ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_f,type,
% 0.20/0.46 f: a > b ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_g,type,
% 0.20/0.46 g: c > a ).
% 0.20/0.46
% 0.20/0.46 thf(sy_v_h,type,
% 0.20/0.46 h: c > a ).
% 0.20/0.46
% 0.20/0.46 % Relevant facts (353)
% 0.20/0.46 thf(fact_0_o__ext,axiom,
% 0.20/0.46 ! [H: a > c,F: c > b,G: c > b] :
% 0.20/0.46 ( ! [X: c] :
% 0.20/0.46 ( ( member_c @ X @ ( image_a_c @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_c_b_a @ F @ H )
% 0.20/0.46 = ( comp_c_b_a @ G @ H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_ext
% 0.20/0.46 thf(fact_1_o__ext,axiom,
% 0.20/0.46 ! [H: a > b,F: b > b,G: b > b] :
% 0.20/0.46 ( ! [X: b] :
% 0.20/0.46 ( ( member_b @ X @ ( image_a_b @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_b_b_a @ F @ H )
% 0.20/0.46 = ( comp_b_b_a @ G @ H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_ext
% 0.20/0.46 thf(fact_2_o__ext,axiom,
% 0.20/0.46 ! [H: a > a,F: a > b,G: a > b] :
% 0.20/0.46 ( ! [X: a] :
% 0.20/0.46 ( ( member_a @ X @ ( image_a_a @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_a_b_a @ F @ H )
% 0.20/0.46 = ( comp_a_b_a @ G @ H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_ext
% 0.20/0.46 thf(fact_3_o__ext,axiom,
% 0.20/0.46 ! [H: c > a,F: a > b,G: a > b] :
% 0.20/0.46 ( ! [X: a] :
% 0.20/0.46 ( ( member_a @ X @ ( image_c_a @ H @ top_top_set_c ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_a_b_c @ F @ H )
% 0.20/0.46 = ( comp_a_b_c @ G @ H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_ext
% 0.20/0.46 thf(fact_4_o__cong,axiom,
% 0.20/0.46 ! [H: a > c,I: a > c,F: c > b,G: c > b] :
% 0.20/0.46 ( ( H = I )
% 0.20/0.46 => ( ! [X: c] :
% 0.20/0.46 ( ( member_c @ X @ ( image_a_c @ I @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_c_b_a @ F @ H )
% 0.20/0.46 = ( comp_c_b_a @ F @ I ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_cong
% 0.20/0.46 thf(fact_5_o__cong,axiom,
% 0.20/0.46 ! [H: a > b,I: a > b,F: b > b,G: b > b] :
% 0.20/0.46 ( ( H = I )
% 0.20/0.46 => ( ! [X: b] :
% 0.20/0.46 ( ( member_b @ X @ ( image_a_b @ I @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_b_b_a @ F @ H )
% 0.20/0.46 = ( comp_b_b_a @ F @ I ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_cong
% 0.20/0.46 thf(fact_6_o__cong,axiom,
% 0.20/0.46 ! [H: a > a,I: a > a,F: a > b,G: a > b] :
% 0.20/0.46 ( ( H = I )
% 0.20/0.46 => ( ! [X: a] :
% 0.20/0.46 ( ( member_a @ X @ ( image_a_a @ I @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_a_b_a @ F @ H )
% 0.20/0.46 = ( comp_a_b_a @ F @ I ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_cong
% 0.20/0.46 thf(fact_7_o__cong,axiom,
% 0.20/0.46 ! [H: c > a,I: c > a,F: a > b,G: a > b] :
% 0.20/0.46 ( ( H = I )
% 0.20/0.46 => ( ! [X: a] :
% 0.20/0.46 ( ( member_a @ X @ ( image_c_a @ I @ top_top_set_c ) )
% 0.20/0.46 => ( ( F @ X )
% 0.20/0.46 = ( G @ X ) ) )
% 0.20/0.46 => ( ( comp_a_b_c @ F @ H )
% 0.20/0.46 = ( comp_a_b_c @ F @ I ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_cong
% 0.20/0.46 thf(fact_8_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > b,G: c > a,H: c > a] :
% 0.20/0.46 ( ( ( comp_a_b_c @ F @ G )
% 0.20/0.46 = ( comp_a_b_c @ F @ H ) )
% 0.20/0.46 => ( ( 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 ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_9_o__inj__on,axiom,
% 0.20/0.46 ! [F: b > b,G: c > b,H: c > b] :
% 0.20/0.46 ( ( ( comp_b_b_c @ F @ G )
% 0.20/0.46 = ( comp_b_b_c @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_c_b @ G @ top_top_set_c ) @ ( image_c_b @ H @ top_top_set_c ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_10_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > c,G: c > a,H: c > a] :
% 0.20/0.46 ( ( ( comp_a_c_c @ F @ G )
% 0.20/0.46 = ( comp_a_c_c @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_11_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > a,G: c > a,H: c > a] :
% 0.20/0.46 ( ( ( comp_a_a_c @ F @ G )
% 0.20/0.46 = ( comp_a_a_c @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_12_o__inj__on,axiom,
% 0.20/0.46 ! [F: c > b,G: c > c,H: c > c] :
% 0.20/0.46 ( ( ( comp_c_b_c @ F @ G )
% 0.20/0.46 = ( comp_c_b_c @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_c_b @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_13_o__inj__on,axiom,
% 0.20/0.46 ! [F: c > a,G: c > c,H: c > c] :
% 0.20/0.46 ( ( ( comp_c_a_c @ F @ G )
% 0.20/0.46 = ( comp_c_a_c @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_c_a @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_14_o__inj__on,axiom,
% 0.20/0.46 ! [F: b > b,G: a > b,H: a > b] :
% 0.20/0.46 ( ( ( comp_b_b_a @ F @ G )
% 0.20/0.46 = ( comp_b_b_a @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_a_b @ G @ top_top_set_a ) @ ( image_a_b @ H @ top_top_set_a ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_15_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > b,G: a > a,H: a > a] :
% 0.20/0.46 ( ( ( comp_a_b_a @ F @ G )
% 0.20/0.46 = ( comp_a_b_a @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_16_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > c,G: a > a,H: a > a] :
% 0.20/0.46 ( ( ( comp_a_c_a @ F @ G )
% 0.20/0.46 = ( comp_a_c_a @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_17_o__inj__on,axiom,
% 0.20/0.46 ! [F: a > a,G: a > a,H: a > a] :
% 0.20/0.46 ( ( ( comp_a_a_a @ F @ G )
% 0.20/0.46 = ( comp_a_a_a @ F @ H ) )
% 0.20/0.46 => ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 0.20/0.46 => ( G = H ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_inj_on
% 0.20/0.46 thf(fact_18_ex__o__conv,axiom,
% 0.20/0.46 ! [G: a > b,F: c > b] :
% 0.20/0.46 ( ( ? [H2: a > c] :
% 0.20/0.46 ( G
% 0.20/0.46 = ( comp_c_b_a @ F @ H2 ) ) )
% 0.20/0.46 = ( ! [X2: b] :
% 0.20/0.46 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 0.20/0.46 => ? [Y: c] :
% 0.20/0.46 ( X2
% 0.20/0.46 = ( F @ Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_o_conv
% 0.20/0.46 thf(fact_19_ex__o__conv,axiom,
% 0.20/0.46 ! [G: a > b,F: a > b] :
% 0.20/0.46 ( ( ? [H2: a > a] :
% 0.20/0.46 ( G
% 0.20/0.46 = ( comp_a_b_a @ F @ H2 ) ) )
% 0.20/0.46 = ( ! [X2: b] :
% 0.20/0.46 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 0.20/0.46 => ? [Y: a] :
% 0.20/0.46 ( X2
% 0.20/0.46 = ( F @ Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_o_conv
% 0.20/0.46 thf(fact_20_ex__o__conv,axiom,
% 0.20/0.46 ! [G: a > b,F: b > b] :
% 0.20/0.46 ( ( ? [H2: a > b] :
% 0.20/0.46 ( G
% 0.20/0.46 = ( comp_b_b_a @ F @ H2 ) ) )
% 0.20/0.46 = ( ! [X2: b] :
% 0.20/0.46 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 0.20/0.46 => ? [Y: b] :
% 0.20/0.46 ( X2
% 0.20/0.46 = ( F @ Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_o_conv
% 0.20/0.46 thf(fact_21_ex__o__conv,axiom,
% 0.20/0.46 ! [G: c > b,F: a > b] :
% 0.20/0.46 ( ( ? [H2: c > a] :
% 0.20/0.46 ( G
% 0.20/0.46 = ( comp_a_b_c @ F @ H2 ) ) )
% 0.20/0.46 = ( ! [X2: b] :
% 0.20/0.46 ( ( member_b @ X2 @ ( image_c_b @ G @ top_top_set_c ) )
% 0.20/0.46 => ? [Y: a] :
% 0.20/0.46 ( X2
% 0.20/0.46 = ( F @ Y ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % ex_o_conv
% 0.20/0.46 thf(fact_22_o__eq__conv,axiom,
% 0.20/0.46 ! [F: c > b,H: a > c,G: c > b] :
% 0.20/0.46 ( ( ( comp_c_b_a @ F @ H )
% 0.20/0.46 = ( comp_c_b_a @ G @ H ) )
% 0.20/0.46 = ( ! [X2: c] :
% 0.20/0.46 ( ( member_c @ X2 @ ( image_a_c @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_eq_conv
% 0.20/0.46 thf(fact_23_o__eq__conv,axiom,
% 0.20/0.46 ! [F: a > b,H: a > a,G: a > b] :
% 0.20/0.46 ( ( ( comp_a_b_a @ F @ H )
% 0.20/0.46 = ( comp_a_b_a @ G @ H ) )
% 0.20/0.46 = ( ! [X2: a] :
% 0.20/0.46 ( ( member_a @ X2 @ ( image_a_a @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_eq_conv
% 0.20/0.46 thf(fact_24_o__eq__conv,axiom,
% 0.20/0.46 ! [F: b > b,H: a > b,G: b > b] :
% 0.20/0.46 ( ( ( comp_b_b_a @ F @ H )
% 0.20/0.46 = ( comp_b_b_a @ G @ H ) )
% 0.20/0.46 = ( ! [X2: b] :
% 0.20/0.46 ( ( member_b @ X2 @ ( image_a_b @ H @ top_top_set_a ) )
% 0.20/0.46 => ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_eq_conv
% 0.20/0.46 thf(fact_25_o__eq__conv,axiom,
% 0.20/0.46 ! [F: a > b,H: c > a,G: a > b] :
% 0.20/0.46 ( ( ( comp_a_b_c @ F @ H )
% 0.20/0.46 = ( comp_a_b_c @ G @ H ) )
% 0.20/0.46 = ( ! [X2: a] :
% 0.20/0.46 ( ( member_a @ X2 @ ( image_c_a @ H @ top_top_set_c ) )
% 0.20/0.46 => ( ( F @ X2 )
% 0.20/0.46 = ( G @ X2 ) ) ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % o_eq_conv
% 0.20/0.46 thf(fact_26_sup__top__left,axiom,
% 0.20/0.46 ! [X3: set_b] :
% 0.20/0.46 ( ( sup_sup_set_b @ top_top_set_b @ X3 )
% 0.20/0.46 = top_top_set_b ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_left
% 0.20/0.46 thf(fact_27_sup__top__left,axiom,
% 0.20/0.46 ! [X3: set_c_a] :
% 0.20/0.46 ( ( sup_sup_set_c_a @ top_top_set_c_a @ X3 )
% 0.20/0.46 = top_top_set_c_a ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_left
% 0.20/0.46 thf(fact_28_sup__top__left,axiom,
% 0.20/0.46 ! [X3: set_a] :
% 0.20/0.46 ( ( sup_sup_set_a @ top_top_set_a @ X3 )
% 0.20/0.46 = top_top_set_a ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_left
% 0.20/0.46 thf(fact_29_sup__top__left,axiom,
% 0.20/0.46 ! [X3: set_c] :
% 0.20/0.46 ( ( sup_sup_set_c @ top_top_set_c @ X3 )
% 0.20/0.46 = top_top_set_c ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_left
% 0.20/0.46 thf(fact_30_sup__top__right,axiom,
% 0.20/0.46 ! [X3: set_b] :
% 0.20/0.46 ( ( sup_sup_set_b @ X3 @ top_top_set_b )
% 0.20/0.46 = top_top_set_b ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_right
% 0.20/0.46 thf(fact_31_sup__top__right,axiom,
% 0.20/0.46 ! [X3: set_c_a] :
% 0.20/0.46 ( ( sup_sup_set_c_a @ X3 @ top_top_set_c_a )
% 0.20/0.46 = top_top_set_c_a ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_right
% 0.20/0.46 thf(fact_32_sup__top__right,axiom,
% 0.20/0.46 ! [X3: set_a] :
% 0.20/0.46 ( ( sup_sup_set_a @ X3 @ top_top_set_a )
% 0.20/0.46 = top_top_set_a ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_right
% 0.20/0.46 thf(fact_33_sup__top__right,axiom,
% 0.20/0.46 ! [X3: set_c] :
% 0.20/0.46 ( ( sup_sup_set_c @ X3 @ top_top_set_c )
% 0.20/0.46 = top_top_set_c ) ).
% 0.20/0.46
% 0.20/0.46 % sup_top_right
% 0.20/0.46 thf(fact_34_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: b > b,A: set_b,B: set_b] :
% 0.20/0.46 ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ A @ B ) )
% 0.20/0.46 => ( ( ( image_b_b @ F @ A )
% 0.20/0.46 = ( image_b_b @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_35_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: a > c,A: set_a,B: set_a] :
% 0.20/0.46 ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ A @ B ) )
% 0.20/0.46 => ( ( ( image_a_c @ F @ A )
% 0.20/0.46 = ( image_a_c @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_36_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: a > a,A: set_a,B: set_a] :
% 0.20/0.46 ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
% 0.20/0.46 => ( ( ( image_a_a @ F @ A )
% 0.20/0.46 = ( image_a_a @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_37_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: c > b,A: set_c,B: set_c] :
% 0.20/0.46 ( ( inj_on_c_b @ F @ ( sup_sup_set_c @ A @ B ) )
% 0.20/0.46 => ( ( ( image_c_b @ F @ A )
% 0.20/0.46 = ( image_c_b @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_38_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: c > a,A: set_c,B: set_c] :
% 0.20/0.46 ( ( inj_on_c_a @ F @ ( sup_sup_set_c @ A @ B ) )
% 0.20/0.46 => ( ( ( image_c_a @ F @ A )
% 0.20/0.46 = ( image_c_a @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_39_inj__on__Un__image__eq__iff,axiom,
% 0.20/0.46 ! [F: a > b,A: set_a,B: set_a] :
% 0.20/0.46 ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ A @ B ) )
% 0.20/0.46 => ( ( ( image_a_b @ F @ A )
% 0.20/0.46 = ( image_a_b @ F @ B ) )
% 0.20/0.46 = ( A = B ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_on_Un_image_eq_iff
% 0.20/0.46 thf(fact_40_inj__compose,axiom,
% 0.20/0.46 ! [F: b > b,G: a > b] :
% 0.20/0.46 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.20/0.46 => ( ( inj_on_a_b @ G @ top_top_set_a )
% 0.20/0.46 => ( inj_on_a_b @ ( comp_b_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_41_inj__compose,axiom,
% 0.20/0.46 ! [F: a > b,G: a > a] :
% 0.20/0.46 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.46 => ( ( inj_on_a_a @ G @ top_top_set_a )
% 0.20/0.46 => ( inj_on_a_b @ ( comp_a_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_42_inj__compose,axiom,
% 0.20/0.46 ! [F: a > b,G: c > a] :
% 0.20/0.46 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.46 => ( ( inj_on_c_a @ G @ top_top_set_c )
% 0.20/0.46 => ( inj_on_c_b @ ( comp_a_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_43_inj__compose,axiom,
% 0.20/0.46 ! [F: c > b,G: a > c] :
% 0.20/0.46 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 0.20/0.46 => ( inj_on_a_b @ ( comp_c_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_44_inj__compose,axiom,
% 0.20/0.46 ! [F: c > b,G: c > c] :
% 0.20/0.46 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_c_c @ G @ top_top_set_c )
% 0.20/0.46 => ( inj_on_c_b @ ( comp_c_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_45_inj__compose,axiom,
% 0.20/0.46 ! [F: c > a,G: c > c] :
% 0.20/0.46 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_c_c @ G @ top_top_set_c )
% 0.20/0.46 => ( inj_on_c_a @ ( comp_c_a_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_46_inj__compose,axiom,
% 0.20/0.46 ! [F: c > c,G: a > c] :
% 0.20/0.46 ( ( inj_on_c_c @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 0.20/0.46 => ( inj_on_a_c @ ( comp_c_c_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_47_inj__compose,axiom,
% 0.20/0.46 ! [F: c > a,G: a > c] :
% 0.20/0.46 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 0.20/0.46 => ( inj_on_a_a @ ( comp_c_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 0.20/0.46
% 0.20/0.46 % inj_compose
% 0.20/0.46 thf(fact_48_inj__compose,axiom,
% 0.20/0.46 ! [F: c > b,G: b > c] :
% 0.20/0.46 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.46 => ( ( inj_on_b_c @ G @ top_top_set_b )
% 0.20/0.47 => ( inj_on_b_b @ ( comp_c_b_b @ F @ G ) @ top_top_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_compose
% 0.20/0.47 thf(fact_49_inj__compose,axiom,
% 0.20/0.47 ! [F: c > a,G: b > c] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.47 => ( ( inj_on_b_c @ G @ top_top_set_b )
% 0.20/0.47 => ( inj_on_b_a @ ( comp_c_a_b @ F @ G ) @ top_top_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_compose
% 0.20/0.47 thf(fact_50_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: c > b] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.47 => ( inj_on_a_c_a_b @ ( comp_c_b_a @ F ) @ top_top_set_a_c ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_51_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: a > b] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.47 => ( inj_on_a_a_a_b @ ( comp_a_b_a @ F ) @ top_top_set_a_a ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_52_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: a > c] :
% 0.20/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.20/0.47 => ( inj_on_c_a_c_c @ ( comp_a_c_c @ F ) @ top_top_set_c_a ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_53_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: a > a] :
% 0.20/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.20/0.47 => ( inj_on_c_a_c_a @ ( comp_a_a_c @ F ) @ top_top_set_c_a ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_54_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: b > b] :
% 0.20/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.20/0.47 => ( inj_on_a_b_a_b @ ( comp_b_b_a @ F ) @ top_top_set_a_b ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_55_fun_Oinj__map,axiom,
% 0.20/0.47 ! [F: a > b] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.47 => ( inj_on_c_a_c_b @ ( comp_a_b_c @ F ) @ top_top_set_c_a ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.inj_map
% 0.20/0.47 thf(fact_56_comp__inj__on,axiom,
% 0.20/0.47 ! [F: b > a,A: set_b,G: a > b] :
% 0.20/0.47 ( ( inj_on_b_a @ F @ A )
% 0.20/0.47 => ( ( inj_on_a_b @ G @ ( image_b_a @ F @ A ) )
% 0.20/0.47 => ( inj_on_b_b @ ( comp_a_b_b @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_57_comp__inj__on,axiom,
% 0.20/0.47 ! [F: c > c,A: set_c,G: c > b] :
% 0.20/0.47 ( ( inj_on_c_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
% 0.20/0.47 => ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_58_comp__inj__on,axiom,
% 0.20/0.47 ! [F: b > c,A: set_b,G: c > b] :
% 0.20/0.47 ( ( inj_on_b_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_b @ G @ ( image_b_c @ F @ A ) )
% 0.20/0.47 => ( inj_on_b_b @ ( comp_c_b_b @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_59_comp__inj__on,axiom,
% 0.20/0.47 ! [F: c > c,A: set_c,G: c > a] :
% 0.20/0.47 ( ( inj_on_c_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
% 0.20/0.47 => ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_60_comp__inj__on,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,G: b > c] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_c @ G @ ( image_a_b @ F @ A ) )
% 0.20/0.47 => ( inj_on_a_c @ ( comp_b_c_a @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_61_comp__inj__on,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,G: b > a] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_a @ G @ ( image_a_b @ F @ A ) )
% 0.20/0.47 => ( inj_on_a_a @ ( comp_b_a_a @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_62_comp__inj__on,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,G: b > b] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) )
% 0.20/0.47 => ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_63_comp__inj__on,axiom,
% 0.20/0.47 ! [F: c > b,A: set_c,G: b > a] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) )
% 0.20/0.47 => ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_64_comp__inj__on,axiom,
% 0.20/0.47 ! [F: c > b,A: set_c,G: b > b] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) )
% 0.20/0.47 => ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_65_comp__inj__on,axiom,
% 0.20/0.47 ! [F: c > a,A: set_c,G: a > b] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ A )
% 0.20/0.47 => ( ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) )
% 0.20/0.47 => ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on
% 0.20/0.47 thf(fact_66_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: a > c,F: c > a,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_c @ ( comp_a_c_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_a_c @ G @ ( image_c_a @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_67_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: a > b,F: a > a,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_b @ ( comp_a_b_a @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_68_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: c > b,F: a > c,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_b @ ( comp_c_b_a @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_c_b @ G @ ( image_a_c @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_69_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: b > b,F: a > b,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_70_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: a > b,F: c > a,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_71_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: c > b,F: c > c,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_72_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: b > b,F: c > b,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_73_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: b > a,F: c > b,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_74_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: c > a,F: c > c,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_75_inj__on__imageI,axiom,
% 0.20/0.47 ! [G: a > a,F: c > a,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_a @ ( comp_a_a_c @ G @ F ) @ A )
% 0.20/0.47 => ( inj_on_a_a @ G @ ( image_c_a @ F @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_on_imageI
% 0.20/0.47 thf(fact_76_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: b > a,A: set_b,F2: a > b] :
% 0.20/0.47 ( ( inj_on_b_a @ F @ A )
% 0.20/0.47 => ( ( inj_on_a_b @ F2 @ ( image_b_a @ F @ A ) )
% 0.20/0.47 = ( inj_on_b_b @ ( comp_a_b_b @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_77_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: c > c,A: set_c,F2: c > b] :
% 0.20/0.47 ( ( inj_on_c_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_b @ F2 @ ( image_c_c @ F @ A ) )
% 0.20/0.47 = ( inj_on_c_b @ ( comp_c_b_c @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_78_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: b > c,A: set_b,F2: c > b] :
% 0.20/0.47 ( ( inj_on_b_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_b @ F2 @ ( image_b_c @ F @ A ) )
% 0.20/0.47 = ( inj_on_b_b @ ( comp_c_b_b @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_79_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: c > c,A: set_c,F2: c > a] :
% 0.20/0.47 ( ( inj_on_c_c @ F @ A )
% 0.20/0.47 => ( ( inj_on_c_a @ F2 @ ( image_c_c @ F @ A ) )
% 0.20/0.47 = ( inj_on_c_a @ ( comp_c_a_c @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_80_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,F2: b > c] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_c @ F2 @ ( image_a_b @ F @ A ) )
% 0.20/0.47 = ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_81_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,F2: b > a] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_a @ F2 @ ( image_a_b @ F @ A ) )
% 0.20/0.47 = ( inj_on_a_a @ ( comp_b_a_a @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_82_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,F2: b > b] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_b @ F2 @ ( image_a_b @ F @ A ) )
% 0.20/0.47 = ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_83_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: c > b,A: set_c,F2: b > a] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_a @ F2 @ ( image_c_b @ F @ A ) )
% 0.20/0.47 = ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_84_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: c > b,A: set_c,F2: b > b] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ A )
% 0.20/0.47 => ( ( inj_on_b_b @ F2 @ ( image_c_b @ F @ A ) )
% 0.20/0.47 = ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_85_comp__inj__on__iff,axiom,
% 0.20/0.47 ! [F: c > a,A: set_c,F2: a > b] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ A )
% 0.20/0.47 => ( ( inj_on_a_b @ F2 @ ( image_c_a @ F @ A ) )
% 0.20/0.47 = ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_inj_on_iff
% 0.20/0.47 thf(fact_86_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: c > b,B2: b] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.47 => ( ( member_b @ B2 @ ( image_c_b @ F @ top_top_set_c ) )
% 0.20/0.47 = ( ? [X2: c] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: c] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_87_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: c > a,B2: a] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.47 => ( ( member_a @ B2 @ ( image_c_a @ F @ top_top_set_c ) )
% 0.20/0.47 = ( ? [X2: c] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: c] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_88_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: a > b,B2: b] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_b @ B2 @ ( image_a_b @ F @ top_top_set_a ) )
% 0.20/0.47 = ( ? [X2: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_89_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: a > c,B2: c] :
% 0.20/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_c @ B2 @ ( image_a_c @ F @ top_top_set_a ) )
% 0.20/0.47 = ( ? [X2: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_90_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: a > a,B2: a] :
% 0.20/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) )
% 0.20/0.47 = ( ? [X2: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_91_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: b > b,B2: b] :
% 0.20/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.20/0.47 => ( ( member_b @ B2 @ ( image_b_b @ F @ top_top_set_b ) )
% 0.20/0.47 = ( ? [X2: b] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: b] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_92_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: b > a,B2: a] :
% 0.20/0.47 ( ( inj_on_b_a @ F @ top_top_set_b )
% 0.20/0.47 => ( ( member_a @ B2 @ ( image_b_a @ F @ top_top_set_b ) )
% 0.20/0.47 = ( ? [X2: b] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: b] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_93_range__ex1__eq,axiom,
% 0.20/0.47 ! [F: ( c > a ) > a,B2: a] :
% 0.20/0.47 ( ( inj_on_c_a_a @ F @ top_top_set_c_a )
% 0.20/0.47 => ( ( member_a @ B2 @ ( image_c_a_a @ F @ top_top_set_c_a ) )
% 0.20/0.47 = ( ? [X2: c > a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ X2 ) )
% 0.20/0.47 & ! [Y: c > a] :
% 0.20/0.47 ( ( B2
% 0.20/0.47 = ( F @ Y ) )
% 0.20/0.47 => ( Y = X2 ) ) ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % range_ex1_eq
% 0.20/0.47 thf(fact_94_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: c > b,A: set_c,B: set_c] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_b @ F @ A )
% 0.20/0.47 = ( image_c_b @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_95_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: c > a,A: set_c,B: set_c] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_a @ F @ A )
% 0.20/0.47 = ( image_c_a @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_96_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: a > b,A: set_a,B: set_a] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_b @ F @ A )
% 0.20/0.47 = ( image_a_b @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_97_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: a > c,A: set_a,B: set_a] :
% 0.20/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_c @ F @ A )
% 0.20/0.47 = ( image_a_c @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_98_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: a > a,A: set_a,B: set_a] :
% 0.20/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_a @ F @ A )
% 0.20/0.47 = ( image_a_a @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_99_inj__image__eq__iff,axiom,
% 0.20/0.47 ! [F: b > b,A: set_b,B: set_b] :
% 0.20/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.20/0.47 => ( ( ( image_b_b @ F @ A )
% 0.20/0.47 = ( image_b_b @ F @ B ) )
% 0.20/0.47 = ( A = B ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_eq_iff
% 0.20/0.47 thf(fact_100_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: c > b,A2: c,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.20/0.47 => ( ( member_b @ ( F @ A2 ) @ ( image_c_b @ F @ A ) )
% 0.20/0.47 = ( member_c @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_101_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: c > a,A2: c,A: set_c] :
% 0.20/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.20/0.47 => ( ( member_a @ ( F @ A2 ) @ ( image_c_a @ F @ A ) )
% 0.20/0.47 = ( member_c @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_102_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: a > b,A2: a,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_b @ ( F @ A2 ) @ ( image_a_b @ F @ A ) )
% 0.20/0.47 = ( member_a @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_103_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: a > c,A2: a,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_c @ ( F @ A2 ) @ ( image_a_c @ F @ A ) )
% 0.20/0.47 = ( member_a @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_104_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: a > a,A2: a,A: set_a] :
% 0.20/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.20/0.47 => ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
% 0.20/0.47 = ( member_a @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_105_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: b > b,A2: b,A: set_b] :
% 0.20/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.20/0.47 => ( ( member_b @ ( F @ A2 ) @ ( image_b_b @ F @ A ) )
% 0.20/0.47 = ( member_b @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_106_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: b > a,A2: b,A: set_b] :
% 0.20/0.47 ( ( inj_on_b_a @ F @ top_top_set_b )
% 0.20/0.47 => ( ( member_a @ ( F @ A2 ) @ ( image_b_a @ F @ A ) )
% 0.20/0.47 = ( member_b @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_107_inj__image__mem__iff,axiom,
% 0.20/0.47 ! [F: ( c > a ) > a,A2: c > a,A: set_c_a] :
% 0.20/0.47 ( ( inj_on_c_a_a @ F @ top_top_set_c_a )
% 0.20/0.47 => ( ( member_a @ ( F @ A2 ) @ ( image_c_a_a @ F @ A ) )
% 0.20/0.47 = ( member_c_a @ A2 @ A ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inj_image_mem_iff
% 0.20/0.47 thf(fact_108_comp__surj,axiom,
% 0.20/0.47 ! [F: c > c,G: c > c] :
% 0.20/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_c @ G @ top_top_set_c )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( image_c_c @ ( comp_c_c_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_c ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_109_comp__surj,axiom,
% 0.20/0.47 ! [F: c > c,G: c > a] :
% 0.20/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_a @ G @ top_top_set_c )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( image_c_a @ ( comp_c_a_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_a ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_110_comp__surj,axiom,
% 0.20/0.47 ! [F: c > c,G: c > b] :
% 0.20/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_b @ G @ top_top_set_c )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( image_c_b @ ( comp_c_b_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_111_comp__surj,axiom,
% 0.20/0.47 ! [F: c > a,G: a > c] :
% 0.20/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_c @ G @ top_top_set_a )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( image_c_c @ ( comp_a_c_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_c ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_112_comp__surj,axiom,
% 0.20/0.47 ! [F: c > a,G: a > a] :
% 0.20/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_a @ G @ top_top_set_a )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( image_c_a @ ( comp_a_a_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_a ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_113_comp__surj,axiom,
% 0.20/0.47 ! [F: c > a,G: a > b] :
% 0.20/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( ( image_a_b @ G @ top_top_set_a )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( image_c_b @ ( comp_a_b_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_114_comp__surj,axiom,
% 0.20/0.47 ! [F: c > b,G: b > c] :
% 0.20/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( ( image_b_c @ G @ top_top_set_b )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( image_c_c @ ( comp_b_c_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_c ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_115_comp__surj,axiom,
% 0.20/0.47 ! [F: c > b,G: b > a] :
% 0.20/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( ( image_b_a @ G @ top_top_set_b )
% 0.20/0.47 = top_top_set_a )
% 0.20/0.47 => ( ( image_c_a @ ( comp_b_a_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_a ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_116_comp__surj,axiom,
% 0.20/0.47 ! [F: c > b,G: b > b] :
% 0.20/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( ( image_b_b @ G @ top_top_set_b )
% 0.20/0.47 = top_top_set_b )
% 0.20/0.47 => ( ( image_c_b @ ( comp_b_b_c @ G @ F ) @ top_top_set_c )
% 0.20/0.47 = top_top_set_b ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_117_comp__surj,axiom,
% 0.20/0.47 ! [F: a > c,G: c > c] :
% 0.20/0.47 ( ( ( image_a_c @ F @ top_top_set_a )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( ( image_c_c @ G @ top_top_set_c )
% 0.20/0.47 = top_top_set_c )
% 0.20/0.47 => ( ( image_a_c @ ( comp_c_c_a @ G @ F ) @ top_top_set_a )
% 0.20/0.47 = top_top_set_c ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_surj
% 0.20/0.47 thf(fact_118_sup_Oright__idem,axiom,
% 0.20/0.47 ! [A2: set_a,B2: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
% 0.20/0.47 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.right_idem
% 0.20/0.47 thf(fact_119_sup_Oright__idem,axiom,
% 0.20/0.47 ! [A2: set_c,B2: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ B2 )
% 0.20/0.47 = ( sup_sup_set_c @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.right_idem
% 0.20/0.47 thf(fact_120_sup__left__idem,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 0.20/0.47 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_left_idem
% 0.20/0.47 thf(fact_121_sup__left__idem,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
% 0.20/0.47 = ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_left_idem
% 0.20/0.47 thf(fact_122_sup_Oleft__idem,axiom,
% 0.20/0.47 ! [A2: set_a,B2: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
% 0.20/0.47 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.left_idem
% 0.20/0.47 thf(fact_123_sup_Oleft__idem,axiom,
% 0.20/0.47 ! [A2: set_c,B2: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ A2 @ B2 ) )
% 0.20/0.47 = ( sup_sup_set_c @ A2 @ B2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.left_idem
% 0.20/0.47 thf(fact_124_sup__idem,axiom,
% 0.20/0.47 ! [X3: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ X3 @ X3 )
% 0.20/0.47 = X3 ) ).
% 0.20/0.47
% 0.20/0.47 % sup_idem
% 0.20/0.47 thf(fact_125_sup__idem,axiom,
% 0.20/0.47 ! [X3: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ X3 @ X3 )
% 0.20/0.47 = X3 ) ).
% 0.20/0.47
% 0.20/0.47 % sup_idem
% 0.20/0.47 thf(fact_126_sup_Oidem,axiom,
% 0.20/0.47 ! [A2: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ A2 @ A2 )
% 0.20/0.47 = A2 ) ).
% 0.20/0.47
% 0.20/0.47 % sup.idem
% 0.20/0.47 thf(fact_127_sup_Oidem,axiom,
% 0.20/0.47 ! [A2: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ A2 @ A2 )
% 0.20/0.47 = A2 ) ).
% 0.20/0.47
% 0.20/0.47 % sup.idem
% 0.20/0.47 thf(fact_128_comp__apply,axiom,
% 0.20/0.47 ( comp_a_b_c
% 0.20/0.47 = ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_apply
% 0.20/0.47 thf(fact_129_comp__apply,axiom,
% 0.20/0.47 ( comp_c_b_a
% 0.20/0.47 = ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_apply
% 0.20/0.47 thf(fact_130_comp__apply,axiom,
% 0.20/0.47 ( comp_a_b_a
% 0.20/0.47 = ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_apply
% 0.20/0.47 thf(fact_131_comp__apply,axiom,
% 0.20/0.47 ( comp_b_b_a
% 0.20/0.47 = ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_apply
% 0.20/0.47 thf(fact_132_sup__left__commute,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
% 0.20/0.47 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_left_commute
% 0.20/0.47 thf(fact_133_sup__left__commute,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
% 0.20/0.47 = ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_left_commute
% 0.20/0.47 thf(fact_134_sup_Oleft__commute,axiom,
% 0.20/0.47 ! [B2: set_a,A2: set_a,C: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
% 0.20/0.47 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.left_commute
% 0.20/0.47 thf(fact_135_sup_Oleft__commute,axiom,
% 0.20/0.47 ! [B2: set_c,A2: set_c,C: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ B2 @ ( sup_sup_set_c @ A2 @ C ) )
% 0.20/0.47 = ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.left_commute
% 0.20/0.47 thf(fact_136_sup__commute,axiom,
% 0.20/0.47 ( sup_sup_set_a
% 0.20/0.47 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_commute
% 0.20/0.47 thf(fact_137_sup__commute,axiom,
% 0.20/0.47 ( sup_sup_set_c
% 0.20/0.47 = ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_commute
% 0.20/0.47 thf(fact_138_sup_Ocommute,axiom,
% 0.20/0.47 ( sup_sup_set_a
% 0.20/0.47 = ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.commute
% 0.20/0.47 thf(fact_139_sup_Ocommute,axiom,
% 0.20/0.47 ( sup_sup_set_c
% 0.20/0.47 = ( ^ [A3: set_c,B3: set_c] : ( sup_sup_set_c @ B3 @ A3 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.commute
% 0.20/0.47 thf(fact_140_sup__assoc,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
% 0.20/0.47 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_assoc
% 0.20/0.47 thf(fact_141_sup__assoc,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
% 0.20/0.47 = ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup_assoc
% 0.20/0.47 thf(fact_142_sup_Oassoc,axiom,
% 0.20/0.47 ! [A2: set_a,B2: set_a,C: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
% 0.20/0.47 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.assoc
% 0.20/0.47 thf(fact_143_sup_Oassoc,axiom,
% 0.20/0.47 ! [A2: set_c,B2: set_c,C: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ C )
% 0.20/0.47 = ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % sup.assoc
% 0.20/0.47 thf(fact_144_boolean__algebra__cancel_Osup2,axiom,
% 0.20/0.47 ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
% 0.20/0.47 ( ( B
% 0.20/0.47 = ( sup_sup_set_a @ K @ B2 ) )
% 0.20/0.47 => ( ( sup_sup_set_a @ A2 @ B )
% 0.20/0.47 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % boolean_algebra_cancel.sup2
% 0.20/0.47 thf(fact_145_boolean__algebra__cancel_Osup2,axiom,
% 0.20/0.47 ! [B: set_c,K: set_c,B2: set_c,A2: set_c] :
% 0.20/0.47 ( ( B
% 0.20/0.47 = ( sup_sup_set_c @ K @ B2 ) )
% 0.20/0.47 => ( ( sup_sup_set_c @ A2 @ B )
% 0.20/0.47 = ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % boolean_algebra_cancel.sup2
% 0.20/0.47 thf(fact_146_boolean__algebra__cancel_Osup1,axiom,
% 0.20/0.47 ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
% 0.20/0.47 ( ( A
% 0.20/0.47 = ( sup_sup_set_a @ K @ A2 ) )
% 0.20/0.47 => ( ( sup_sup_set_a @ A @ B2 )
% 0.20/0.47 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % boolean_algebra_cancel.sup1
% 0.20/0.47 thf(fact_147_boolean__algebra__cancel_Osup1,axiom,
% 0.20/0.47 ! [A: set_c,K: set_c,A2: set_c,B2: set_c] :
% 0.20/0.47 ( ( A
% 0.20/0.47 = ( sup_sup_set_c @ K @ A2 ) )
% 0.20/0.47 => ( ( sup_sup_set_c @ A @ B2 )
% 0.20/0.47 = ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % boolean_algebra_cancel.sup1
% 0.20/0.47 thf(fact_148_inf__sup__aci_I5_J,axiom,
% 0.20/0.47 ( sup_sup_set_a
% 0.20/0.47 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(5)
% 0.20/0.47 thf(fact_149_inf__sup__aci_I5_J,axiom,
% 0.20/0.47 ( sup_sup_set_c
% 0.20/0.47 = ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(5)
% 0.20/0.47 thf(fact_150_inf__sup__aci_I6_J,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
% 0.20/0.47 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(6)
% 0.20/0.47 thf(fact_151_inf__sup__aci_I6_J,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
% 0.20/0.47 = ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(6)
% 0.20/0.47 thf(fact_152_inf__sup__aci_I7_J,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
% 0.20/0.47 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(7)
% 0.20/0.47 thf(fact_153_inf__sup__aci_I7_J,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
% 0.20/0.47 = ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(7)
% 0.20/0.47 thf(fact_154_inf__sup__aci_I8_J,axiom,
% 0.20/0.47 ! [X3: set_a,Y2: set_a] :
% 0.20/0.47 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 0.20/0.47 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(8)
% 0.20/0.47 thf(fact_155_inf__sup__aci_I8_J,axiom,
% 0.20/0.47 ! [X3: set_c,Y2: set_c] :
% 0.20/0.47 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
% 0.20/0.47 = ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% 0.20/0.47
% 0.20/0.47 % inf_sup_aci(8)
% 0.20/0.47 thf(fact_156_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: c > b,F: a > c,V: c > a] :
% 0.20/0.47 ( ( comp_c_b_c @ G @ ( comp_a_c_c @ F @ V ) )
% 0.20/0.47 = ( comp_a_b_c @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_157_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: b > b,F: a > b,V: c > a] :
% 0.20/0.47 ( ( comp_b_b_c @ G @ ( comp_a_b_c @ F @ V ) )
% 0.20/0.47 = ( comp_a_b_c @ ( comp_b_b_a @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_158_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: a > b,F: c > a,V: c > c] :
% 0.20/0.47 ( ( comp_a_b_c @ G @ ( comp_c_a_c @ F @ V ) )
% 0.20/0.47 = ( comp_c_b_c @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_159_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: a > b,F: a > a,V: c > a] :
% 0.20/0.47 ( ( comp_a_b_c @ G @ ( comp_a_a_c @ F @ V ) )
% 0.20/0.47 = ( comp_a_b_c @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_160_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: c > b,F: c > c,V: a > c] :
% 0.20/0.47 ( ( comp_c_b_a @ G @ ( comp_c_c_a @ F @ V ) )
% 0.20/0.47 = ( comp_c_b_a @ ( comp_c_b_c @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_161_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: c > b,F: a > c,V: a > a] :
% 0.20/0.47 ( ( comp_c_b_a @ G @ ( comp_a_c_a @ F @ V ) )
% 0.20/0.47 = ( comp_a_b_a @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_162_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: c > b,F: b > c,V: a > b] :
% 0.20/0.47 ( ( comp_c_b_a @ G @ ( comp_b_c_a @ F @ V ) )
% 0.20/0.47 = ( comp_b_b_a @ ( comp_c_b_b @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_163_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: a > b,F: c > a,V: a > c] :
% 0.20/0.47 ( ( comp_a_b_a @ G @ ( comp_c_a_a @ F @ V ) )
% 0.20/0.47 = ( comp_c_b_a @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_164_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: a > b,F: a > a,V: a > a] :
% 0.20/0.47 ( ( comp_a_b_a @ G @ ( comp_a_a_a @ F @ V ) )
% 0.20/0.47 = ( comp_a_b_a @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_165_fun_Omap__comp,axiom,
% 0.20/0.47 ! [G: a > b,F: b > a,V: a > b] :
% 0.20/0.47 ( ( comp_a_b_a @ G @ ( comp_b_a_a @ F @ V ) )
% 0.20/0.47 = ( comp_b_b_a @ ( comp_a_b_b @ G @ F ) @ V ) ) ).
% 0.20/0.47
% 0.20/0.47 % fun.map_comp
% 0.20/0.47 thf(fact_166_comp__eq__dest__lhs,axiom,
% 0.20/0.47 ! [A2: a > b,B2: c > a,C: c > b,V: c] :
% 0.20/0.47 ( ( ( comp_a_b_c @ A2 @ B2 )
% 0.20/0.47 = C )
% 0.20/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.20/0.47 = ( C @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest_lhs
% 0.20/0.47 thf(fact_167_comp__eq__dest__lhs,axiom,
% 0.20/0.47 ! [A2: c > b,B2: a > c,C: a > b,V: a] :
% 0.20/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.20/0.47 = C )
% 0.20/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.20/0.47 = ( C @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest_lhs
% 0.20/0.47 thf(fact_168_comp__eq__dest__lhs,axiom,
% 0.20/0.47 ! [A2: a > b,B2: a > a,C: a > b,V: a] :
% 0.20/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.20/0.47 = C )
% 0.20/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.20/0.47 = ( C @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest_lhs
% 0.20/0.47 thf(fact_169_comp__eq__dest__lhs,axiom,
% 0.20/0.47 ! [A2: b > b,B2: a > b,C: a > b,V: a] :
% 0.20/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.20/0.47 = C )
% 0.20/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.20/0.47 = ( C @ V ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_dest_lhs
% 0.20/0.47 thf(fact_170_comp__eq__elim,axiom,
% 0.20/0.47 ! [A2: a > b,B2: c > a,C: a > b,D: c > a] :
% 0.20/0.47 ( ( ( comp_a_b_c @ A2 @ B2 )
% 0.20/0.47 = ( comp_a_b_c @ C @ D ) )
% 0.20/0.47 => ! [V2: c] :
% 0.20/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.20/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_elim
% 0.20/0.47 thf(fact_171_comp__eq__elim,axiom,
% 0.20/0.47 ! [A2: c > b,B2: a > c,C: c > b,D: a > c] :
% 0.20/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.20/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.20/0.47 => ! [V2: a] :
% 0.20/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.20/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_elim
% 0.20/0.47 thf(fact_172_comp__eq__elim,axiom,
% 0.20/0.47 ! [A2: c > b,B2: a > c,C: a > b,D: a > a] :
% 0.20/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.20/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.20/0.47 => ! [V2: a] :
% 0.20/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.20/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_elim
% 0.20/0.47 thf(fact_173_comp__eq__elim,axiom,
% 0.20/0.47 ! [A2: c > b,B2: a > c,C: b > b,D: a > b] :
% 0.20/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.20/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.20/0.47 => ! [V2: a] :
% 0.20/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.20/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.20/0.47
% 0.20/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_174_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: c > b,D: a > c] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_175_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: a > b,D: a > a] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_176_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: b > b,D: a > b] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_177_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: c > b,D: a > c] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_178_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: a > b,D: a > a] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_179_comp__eq__elim,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: b > b,D: a > b] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.32/0.47 => ! [V2: a] :
% 0.32/0.47 ( ( A2 @ ( B2 @ V2 ) )
% 0.32/0.47 = ( C @ ( D @ V2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_elim
% 0.32/0.47 thf(fact_180_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: a > b,B2: c > a,C: a > b,D: c > a,V: c] :
% 0.32/0.47 ( ( ( comp_a_b_c @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_c @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_181_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: c > b,B2: a > c,C: c > b,D: a > c,V: a] :
% 0.32/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_182_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: c > b,B2: a > c,C: a > b,D: a > a,V: a] :
% 0.32/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_183_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: c > b,B2: a > c,C: b > b,D: a > b,V: a] :
% 0.32/0.47 ( ( ( comp_c_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_184_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: c > b,D: a > c,V: a] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_185_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: a > b,D: a > a,V: a] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_186_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: a > b,B2: a > a,C: b > b,D: a > b,V: a] :
% 0.32/0.47 ( ( ( comp_a_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_187_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: c > b,D: a > c,V: a] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_c_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_188_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: a > b,D: a > a,V: a] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_a_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_189_comp__eq__dest,axiom,
% 0.32/0.47 ! [A2: b > b,B2: a > b,C: b > b,D: a > b,V: a] :
% 0.32/0.47 ( ( ( comp_b_b_a @ A2 @ B2 )
% 0.32/0.47 = ( comp_b_b_a @ C @ D ) )
% 0.32/0.47 => ( ( A2 @ ( B2 @ V ) )
% 0.32/0.47 = ( C @ ( D @ V ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_eq_dest
% 0.32/0.47 thf(fact_190_comp__assoc,axiom,
% 0.32/0.47 ! [F: a > b,G: c > a,H: c > c] :
% 0.32/0.47 ( ( comp_c_b_c @ ( comp_a_b_c @ F @ G ) @ H )
% 0.32/0.47 = ( comp_a_b_c @ F @ ( comp_c_a_c @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_191_comp__assoc,axiom,
% 0.32/0.47 ! [F: c > b,G: a > c,H: c > a] :
% 0.32/0.47 ( ( comp_a_b_c @ ( comp_c_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_c_b_c @ F @ ( comp_a_c_c @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_192_comp__assoc,axiom,
% 0.32/0.47 ! [F: a > b,G: a > a,H: c > a] :
% 0.32/0.47 ( ( comp_a_b_c @ ( comp_a_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_a_b_c @ F @ ( comp_a_a_c @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_193_comp__assoc,axiom,
% 0.32/0.47 ! [F: b > b,G: a > b,H: c > a] :
% 0.32/0.47 ( ( comp_a_b_c @ ( comp_b_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_b_b_c @ F @ ( comp_a_b_c @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_194_comp__assoc,axiom,
% 0.32/0.47 ! [F: c > b,G: c > c,H: a > c] :
% 0.32/0.47 ( ( comp_c_b_a @ ( comp_c_b_c @ F @ G ) @ H )
% 0.32/0.47 = ( comp_c_b_a @ F @ ( comp_c_c_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_195_comp__assoc,axiom,
% 0.32/0.47 ! [F: b > b,G: c > b,H: a > c] :
% 0.32/0.47 ( ( comp_c_b_a @ ( comp_b_b_c @ F @ G ) @ H )
% 0.32/0.47 = ( comp_b_b_a @ F @ ( comp_c_b_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_196_comp__assoc,axiom,
% 0.32/0.47 ! [F: a > b,G: c > a,H: a > c] :
% 0.32/0.47 ( ( comp_c_b_a @ ( comp_a_b_c @ F @ G ) @ H )
% 0.32/0.47 = ( comp_a_b_a @ F @ ( comp_c_a_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_197_comp__assoc,axiom,
% 0.32/0.47 ! [F: c > b,G: a > c,H: a > a] :
% 0.32/0.47 ( ( comp_a_b_a @ ( comp_c_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_c_b_a @ F @ ( comp_a_c_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_198_comp__assoc,axiom,
% 0.32/0.47 ! [F: a > b,G: a > a,H: a > a] :
% 0.32/0.47 ( ( comp_a_b_a @ ( comp_a_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_a_b_a @ F @ ( comp_a_a_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_199_comp__assoc,axiom,
% 0.32/0.47 ! [F: b > b,G: a > b,H: a > a] :
% 0.32/0.47 ( ( comp_a_b_a @ ( comp_b_b_a @ F @ G ) @ H )
% 0.32/0.47 = ( comp_b_b_a @ F @ ( comp_a_b_a @ G @ H ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_assoc
% 0.32/0.47 thf(fact_200_comp__def,axiom,
% 0.32/0.47 ( comp_a_b_c
% 0.32/0.47 = ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_def
% 0.32/0.47 thf(fact_201_comp__def,axiom,
% 0.32/0.47 ( comp_c_b_a
% 0.32/0.47 = ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_def
% 0.32/0.47 thf(fact_202_comp__def,axiom,
% 0.32/0.47 ( comp_a_b_a
% 0.32/0.47 = ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_def
% 0.32/0.47 thf(fact_203_comp__def,axiom,
% 0.32/0.47 ( comp_b_b_a
% 0.32/0.47 = ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % comp_def
% 0.32/0.47 thf(fact_204_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_c,G: b > c,F: c > b] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_c_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_205_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_c,G: a > c,F: c > a] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_c_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_206_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_b,G: b > b,F: b > b] :
% 0.32/0.47 ( ! [X: b] :
% 0.32/0.47 ( ( member_b @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_b_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_207_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_a,G: b > a,F: a > b] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_a_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_208_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_a,G: c > a,F: a > c] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_a_c @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_209_inj__on__inverseI,axiom,
% 0.32/0.47 ! [A: set_a,G: a > a,F: a > a] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( G @ ( F @ X ) )
% 0.32/0.47 = X ) )
% 0.32/0.47 => ( inj_on_a_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_inverseI
% 0.32/0.47 thf(fact_210_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_211_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_212_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_b @ X3 @ A )
% 0.32/0.47 => ( ( member_b @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_213_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_214_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_215_inj__on__contraD,axiom,
% 0.32/0.47 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( X3 != Y2 )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( F @ X3 )
% 0.32/0.47 != ( F @ Y2 ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_contraD
% 0.32/0.47 thf(fact_216_mem__Collect__eq,axiom,
% 0.32/0.47 ! [A2: a,P: a > $o] :
% 0.32/0.47 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 0.32/0.47 = ( P @ A2 ) ) ).
% 0.32/0.47
% 0.32/0.47 % mem_Collect_eq
% 0.32/0.47 thf(fact_217_Collect__mem__eq,axiom,
% 0.32/0.47 ! [A: set_a] :
% 0.32/0.47 ( ( collect_a
% 0.32/0.47 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 0.32/0.47 = A ) ).
% 0.32/0.47
% 0.32/0.47 % Collect_mem_eq
% 0.32/0.47 thf(fact_218_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ A )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_219_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ A )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_220_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ A )
% 0.32/0.47 => ( ( member_b @ X3 @ A )
% 0.32/0.47 => ( ( member_b @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_221_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ A )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_222_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ A )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_223_inj__on__eq__iff,axiom,
% 0.32/0.47 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_eq_iff
% 0.32/0.47 thf(fact_224_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_c,F: c > b,G: c > b] :
% 0.32/0.47 ( ! [A4: c] :
% 0.32/0.47 ( ( member_c @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_c_b @ F @ A )
% 0.32/0.47 = ( inj_on_c_b @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_225_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_c,F: c > a,G: c > a] :
% 0.32/0.47 ( ! [A4: c] :
% 0.32/0.47 ( ( member_c @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_c_a @ F @ A )
% 0.32/0.47 = ( inj_on_c_a @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_226_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_b,F: b > b,G: b > b] :
% 0.32/0.47 ( ! [A4: b] :
% 0.32/0.47 ( ( member_b @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_b_b @ F @ A )
% 0.32/0.47 = ( inj_on_b_b @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_227_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_a,F: a > b,G: a > b] :
% 0.32/0.47 ( ! [A4: a] :
% 0.32/0.47 ( ( member_a @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_a_b @ F @ A )
% 0.32/0.47 = ( inj_on_a_b @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_228_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_a,F: a > c,G: a > c] :
% 0.32/0.47 ( ! [A4: a] :
% 0.32/0.47 ( ( member_a @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_a_c @ F @ A )
% 0.32/0.47 = ( inj_on_a_c @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_229_inj__on__cong,axiom,
% 0.32/0.47 ! [A: set_a,F: a > a,G: a > a] :
% 0.32/0.47 ( ! [A4: a] :
% 0.32/0.47 ( ( member_a @ A4 @ A )
% 0.32/0.47 => ( ( F @ A4 )
% 0.32/0.47 = ( G @ A4 ) ) )
% 0.32/0.47 => ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 = ( inj_on_a_a @ G @ A ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_cong
% 0.32/0.47 thf(fact_230_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_a_b
% 0.32/0.47 = ( ^ [F3: a > b,A5: set_a] :
% 0.32/0.47 ! [X2: a] :
% 0.32/0.47 ( ( member_a @ X2 @ A5 )
% 0.32/0.47 => ! [Y: a] :
% 0.32/0.47 ( ( member_a @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_231_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_c_b
% 0.32/0.47 = ( ^ [F3: c > b,A5: set_c] :
% 0.32/0.47 ! [X2: c] :
% 0.32/0.47 ( ( member_c @ X2 @ A5 )
% 0.32/0.47 => ! [Y: c] :
% 0.32/0.47 ( ( member_c @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_232_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_c_a
% 0.32/0.47 = ( ^ [F3: c > a,A5: set_c] :
% 0.32/0.47 ! [X2: c] :
% 0.32/0.47 ( ( member_c @ X2 @ A5 )
% 0.32/0.47 => ! [Y: c] :
% 0.32/0.47 ( ( member_c @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_233_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_b_b
% 0.32/0.47 = ( ^ [F3: b > b,A5: set_b] :
% 0.32/0.47 ! [X2: b] :
% 0.32/0.47 ( ( member_b @ X2 @ A5 )
% 0.32/0.47 => ! [Y: b] :
% 0.32/0.47 ( ( member_b @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_234_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_a_c
% 0.32/0.47 = ( ^ [F3: a > c,A5: set_a] :
% 0.32/0.47 ! [X2: a] :
% 0.32/0.47 ( ( member_a @ X2 @ A5 )
% 0.32/0.47 => ! [Y: a] :
% 0.32/0.47 ( ( member_a @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_235_inj__on__def,axiom,
% 0.32/0.47 ( inj_on_a_a
% 0.32/0.47 = ( ^ [F3: a > a,A5: set_a] :
% 0.32/0.47 ! [X2: a] :
% 0.32/0.47 ( ( member_a @ X2 @ A5 )
% 0.32/0.47 => ! [Y: a] :
% 0.32/0.47 ( ( member_a @ Y @ A5 )
% 0.32/0.47 => ( ( ( F3 @ X2 )
% 0.32/0.47 = ( F3 @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_def
% 0.32/0.47 thf(fact_236_inj__onI,axiom,
% 0.32/0.47 ! [A: set_c,F: c > b] :
% 0.32/0.47 ( ! [X: c,Y3: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ( ( member_c @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_c_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_237_inj__onI,axiom,
% 0.32/0.47 ! [A: set_c,F: c > a] :
% 0.32/0.47 ( ! [X: c,Y3: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ( ( member_c @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_c_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_238_inj__onI,axiom,
% 0.32/0.47 ! [A: set_b,F: b > b] :
% 0.32/0.47 ( ! [X: b,Y3: b] :
% 0.32/0.47 ( ( member_b @ X @ A )
% 0.32/0.47 => ( ( member_b @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_b_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_239_inj__onI,axiom,
% 0.32/0.47 ! [A: set_a,F: a > b] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( member_a @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_a_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_240_inj__onI,axiom,
% 0.32/0.47 ! [A: set_a,F: a > c] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( member_a @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_a_c @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_241_inj__onI,axiom,
% 0.32/0.47 ! [A: set_a,F: a > a] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ( ( member_a @ Y3 @ A )
% 0.32/0.47 => ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) ) ) )
% 0.32/0.47 => ( inj_on_a_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onI
% 0.32/0.47 thf(fact_242_inj__onD,axiom,
% 0.32/0.47 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_243_inj__onD,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_c @ X3 @ A )
% 0.32/0.47 => ( ( member_c @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_244_inj__onD,axiom,
% 0.32/0.47 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_b @ X3 @ A )
% 0.32/0.47 => ( ( member_b @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_245_inj__onD,axiom,
% 0.32/0.47 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_246_inj__onD,axiom,
% 0.32/0.47 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_247_inj__onD,axiom,
% 0.32/0.47 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( ( member_a @ X3 @ A )
% 0.32/0.47 => ( ( member_a @ Y2 @ A )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_onD
% 0.32/0.47 thf(fact_248_surj__def,axiom,
% 0.32/0.47 ! [F: c > c] :
% 0.32/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 = ( ! [Y: c] :
% 0.32/0.47 ? [X2: c] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_249_surj__def,axiom,
% 0.32/0.47 ! [F: c > a] :
% 0.32/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 = ( ! [Y: a] :
% 0.32/0.47 ? [X2: c] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_250_surj__def,axiom,
% 0.32/0.47 ! [F: c > b] :
% 0.32/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 = ( ! [Y: b] :
% 0.32/0.47 ? [X2: c] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_251_surj__def,axiom,
% 0.32/0.47 ! [F: a > c] :
% 0.32/0.47 ( ( ( image_a_c @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 = ( ! [Y: c] :
% 0.32/0.47 ? [X2: a] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_252_surj__def,axiom,
% 0.32/0.47 ! [F: a > a] :
% 0.32/0.47 ( ( ( image_a_a @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 = ( ! [Y: a] :
% 0.32/0.47 ? [X2: a] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_253_surj__def,axiom,
% 0.32/0.47 ! [F: a > b] :
% 0.32/0.47 ( ( ( image_a_b @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 = ( ! [Y: b] :
% 0.32/0.47 ? [X2: a] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_254_surj__def,axiom,
% 0.32/0.47 ! [F: b > c] :
% 0.32/0.47 ( ( ( image_b_c @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 = ( ! [Y: c] :
% 0.32/0.47 ? [X2: b] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_255_surj__def,axiom,
% 0.32/0.47 ! [F: b > a] :
% 0.32/0.47 ( ( ( image_b_a @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 = ( ! [Y: a] :
% 0.32/0.47 ? [X2: b] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_256_surj__def,axiom,
% 0.32/0.47 ! [F: b > b] :
% 0.32/0.47 ( ( ( image_b_b @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 = ( ! [Y: b] :
% 0.32/0.47 ? [X2: b] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_257_surj__def,axiom,
% 0.32/0.47 ! [F: c > c > a] :
% 0.32/0.47 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c_a )
% 0.32/0.47 = ( ! [Y: c > a] :
% 0.32/0.47 ? [X2: c] :
% 0.32/0.47 ( Y
% 0.32/0.47 = ( F @ X2 ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surj_def
% 0.32/0.47 thf(fact_258_surjI,axiom,
% 0.32/0.47 ! [G: c > c,F: c > c] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_c_c @ G @ top_top_set_c )
% 0.32/0.47 = top_top_set_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_259_surjI,axiom,
% 0.32/0.47 ! [G: c > a,F: a > c] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_c_a @ G @ top_top_set_c )
% 0.32/0.47 = top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_260_surjI,axiom,
% 0.32/0.47 ! [G: c > b,F: b > c] :
% 0.32/0.47 ( ! [X: b] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_c_b @ G @ top_top_set_c )
% 0.32/0.47 = top_top_set_b ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_261_surjI,axiom,
% 0.32/0.47 ! [G: a > c,F: c > a] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_a_c @ G @ top_top_set_a )
% 0.32/0.47 = top_top_set_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_262_surjI,axiom,
% 0.32/0.47 ! [G: a > a,F: a > a] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_a_a @ G @ top_top_set_a )
% 0.32/0.47 = top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_263_surjI,axiom,
% 0.32/0.47 ! [G: a > b,F: b > a] :
% 0.32/0.47 ( ! [X: b] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_a_b @ G @ top_top_set_a )
% 0.32/0.47 = top_top_set_b ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_264_surjI,axiom,
% 0.32/0.47 ! [G: b > c,F: c > b] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_b_c @ G @ top_top_set_b )
% 0.32/0.47 = top_top_set_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_265_surjI,axiom,
% 0.32/0.47 ! [G: b > a,F: a > b] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_b_a @ G @ top_top_set_b )
% 0.32/0.47 = top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_266_surjI,axiom,
% 0.32/0.47 ! [G: b > b,F: b > b] :
% 0.32/0.47 ( ! [X: b] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_b_b @ G @ top_top_set_b )
% 0.32/0.47 = top_top_set_b ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_267_surjI,axiom,
% 0.32/0.47 ! [G: c > c > a,F: ( c > a ) > c] :
% 0.32/0.47 ( ! [X: c > a] :
% 0.32/0.47 ( ( G @ ( F @ X ) )
% 0.32/0.47 = X )
% 0.32/0.47 => ( ( image_c_c_a @ G @ top_top_set_c )
% 0.32/0.47 = top_top_set_c_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjI
% 0.32/0.47 thf(fact_268_surjE,axiom,
% 0.32/0.47 ! [F: c > c,Y2: c] :
% 0.32/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ~ ! [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_269_surjE,axiom,
% 0.32/0.47 ! [F: c > a,Y2: a] :
% 0.32/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ~ ! [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_270_surjE,axiom,
% 0.32/0.47 ! [F: c > b,Y2: b] :
% 0.32/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ~ ! [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_271_surjE,axiom,
% 0.32/0.47 ! [F: a > c,Y2: c] :
% 0.32/0.47 ( ( ( image_a_c @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ~ ! [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_272_surjE,axiom,
% 0.32/0.47 ! [F: a > a,Y2: a] :
% 0.32/0.47 ( ( ( image_a_a @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ~ ! [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_273_surjE,axiom,
% 0.32/0.47 ! [F: a > b,Y2: b] :
% 0.32/0.47 ( ( ( image_a_b @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ~ ! [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_274_surjE,axiom,
% 0.32/0.47 ! [F: b > c,Y2: c] :
% 0.32/0.47 ( ( ( image_b_c @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ~ ! [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_275_surjE,axiom,
% 0.32/0.47 ! [F: b > a,Y2: a] :
% 0.32/0.47 ( ( ( image_b_a @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ~ ! [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_276_surjE,axiom,
% 0.32/0.47 ! [F: b > b,Y2: b] :
% 0.32/0.47 ( ( ( image_b_b @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ~ ! [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_277_surjE,axiom,
% 0.32/0.47 ! [F: c > c > a,Y2: c > a] :
% 0.32/0.47 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c_a )
% 0.32/0.47 => ~ ! [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 != ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjE
% 0.32/0.47 thf(fact_278_surjD,axiom,
% 0.32/0.47 ! [F: c > c,Y2: c] :
% 0.32/0.47 ( ( ( image_c_c @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ? [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_279_surjD,axiom,
% 0.32/0.47 ! [F: c > a,Y2: a] :
% 0.32/0.47 ( ( ( image_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ? [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_280_surjD,axiom,
% 0.32/0.47 ! [F: c > b,Y2: b] :
% 0.32/0.47 ( ( ( image_c_b @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ? [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_281_surjD,axiom,
% 0.32/0.47 ! [F: a > c,Y2: c] :
% 0.32/0.47 ( ( ( image_a_c @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ? [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_282_surjD,axiom,
% 0.32/0.47 ! [F: a > a,Y2: a] :
% 0.32/0.47 ( ( ( image_a_a @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ? [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_283_surjD,axiom,
% 0.32/0.47 ! [F: a > b,Y2: b] :
% 0.32/0.47 ( ( ( image_a_b @ F @ top_top_set_a )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ? [X: a] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_284_surjD,axiom,
% 0.32/0.47 ! [F: b > c,Y2: c] :
% 0.32/0.47 ( ( ( image_b_c @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_c )
% 0.32/0.47 => ? [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_285_surjD,axiom,
% 0.32/0.47 ! [F: b > a,Y2: a] :
% 0.32/0.47 ( ( ( image_b_a @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_a )
% 0.32/0.47 => ? [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_286_surjD,axiom,
% 0.32/0.47 ! [F: b > b,Y2: b] :
% 0.32/0.47 ( ( ( image_b_b @ F @ top_top_set_b )
% 0.32/0.47 = top_top_set_b )
% 0.32/0.47 => ? [X: b] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_287_surjD,axiom,
% 0.32/0.47 ! [F: c > c > a,Y2: c > a] :
% 0.32/0.47 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 0.32/0.47 = top_top_set_c_a )
% 0.32/0.47 => ? [X: c] :
% 0.32/0.47 ( Y2
% 0.32/0.47 = ( F @ X ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % surjD
% 0.32/0.47 thf(fact_288_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > c,A: set_c,G: a > c,B: set_a,H: c > b] :
% 0.32/0.47 ( ( ( image_c_c @ F @ A )
% 0.32/0.47 = ( image_a_c @ G @ B ) )
% 0.32/0.47 => ( ( image_c_b @ ( comp_c_b_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_289_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: a > c,A: set_a,G: c > c,B: set_c,H: c > b] :
% 0.32/0.47 ( ( ( image_a_c @ F @ A )
% 0.32/0.47 = ( image_c_c @ G @ B ) )
% 0.32/0.47 => ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_b @ ( comp_c_b_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_290_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: a > c,A: set_a,G: a > c,B: set_a,H: c > b] :
% 0.32/0.47 ( ( ( image_a_c @ F @ A )
% 0.32/0.47 = ( image_a_c @ G @ B ) )
% 0.32/0.47 => ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
% 0.32/0.47 = ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_291_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: a > a,A: set_a,G: a > a,B: set_a,H: a > b] :
% 0.32/0.47 ( ( ( image_a_a @ F @ A )
% 0.32/0.47 = ( image_a_a @ G @ B ) )
% 0.32/0.47 => ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
% 0.32/0.47 = ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_292_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: a > a,A: set_a,G: c > a,B: set_c,H: a > b] :
% 0.32/0.47 ( ( ( image_a_a @ F @ A )
% 0.32/0.47 = ( image_c_a @ G @ B ) )
% 0.32/0.47 => ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_293_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,G: a > a,B: set_a,H: a > b] :
% 0.32/0.47 ( ( ( image_c_a @ F @ A )
% 0.32/0.47 = ( image_a_a @ G @ B ) )
% 0.32/0.47 => ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_294_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > a] :
% 0.32/0.47 ( ( ( image_c_a @ F @ A )
% 0.32/0.47 = ( image_c_a @ G @ B ) )
% 0.32/0.47 => ( ( image_c_a @ ( comp_a_a_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_a @ ( comp_a_a_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_295_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > b] :
% 0.32/0.47 ( ( ( image_c_a @ F @ A )
% 0.32/0.47 = ( image_c_a @ G @ B ) )
% 0.32/0.47 => ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_296_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > a] :
% 0.32/0.47 ( ( ( image_c_b @ F @ A )
% 0.32/0.47 = ( image_c_b @ G @ B ) )
% 0.32/0.47 => ( ( image_c_a @ ( comp_b_a_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_a @ ( comp_b_a_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_297_image__eq__imp__comp,axiom,
% 0.32/0.47 ! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > b] :
% 0.32/0.47 ( ( ( image_c_b @ F @ A )
% 0.32/0.47 = ( image_c_b @ G @ B ) )
% 0.32/0.47 => ( ( image_c_b @ ( comp_b_b_c @ H @ F ) @ A )
% 0.32/0.47 = ( image_c_b @ ( comp_b_b_c @ H @ G ) @ B ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_eq_imp_comp
% 0.32/0.47 thf(fact_298_image__comp,axiom,
% 0.32/0.47 ! [F: a > a,G: c > a,R: set_c] :
% 0.32/0.47 ( ( image_a_a @ F @ ( image_c_a @ G @ R ) )
% 0.32/0.47 = ( image_c_a @ ( comp_a_a_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_299_image__comp,axiom,
% 0.32/0.47 ! [F: b > a,G: c > b,R: set_c] :
% 0.32/0.47 ( ( image_b_a @ F @ ( image_c_b @ G @ R ) )
% 0.32/0.47 = ( image_c_a @ ( comp_b_a_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_300_image__comp,axiom,
% 0.32/0.47 ! [F: b > b,G: c > b,R: set_c] :
% 0.32/0.47 ( ( image_b_b @ F @ ( image_c_b @ G @ R ) )
% 0.32/0.47 = ( image_c_b @ ( comp_b_b_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_301_image__comp,axiom,
% 0.32/0.47 ! [F: b > b,G: a > b,R: set_a] :
% 0.32/0.47 ( ( image_b_b @ F @ ( image_a_b @ G @ R ) )
% 0.32/0.47 = ( image_a_b @ ( comp_b_b_a @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_302_image__comp,axiom,
% 0.32/0.47 ! [F: c > a,G: c > c,R: set_c] :
% 0.32/0.47 ( ( image_c_a @ F @ ( image_c_c @ G @ R ) )
% 0.32/0.47 = ( image_c_a @ ( comp_c_a_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_303_image__comp,axiom,
% 0.32/0.47 ! [F: c > b,G: c > c,R: set_c] :
% 0.32/0.47 ( ( image_c_b @ F @ ( image_c_c @ G @ R ) )
% 0.32/0.47 = ( image_c_b @ ( comp_c_b_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_304_image__comp,axiom,
% 0.32/0.47 ! [F: c > b,G: a > c,R: set_a] :
% 0.32/0.47 ( ( image_c_b @ F @ ( image_a_c @ G @ R ) )
% 0.32/0.47 = ( image_a_b @ ( comp_c_b_a @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_305_image__comp,axiom,
% 0.32/0.47 ! [F: a > b,G: a > a,R: set_a] :
% 0.32/0.47 ( ( image_a_b @ F @ ( image_a_a @ G @ R ) )
% 0.32/0.47 = ( image_a_b @ ( comp_a_b_a @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_306_image__comp,axiom,
% 0.32/0.47 ! [F: a > b,G: c > a,R: set_c] :
% 0.32/0.47 ( ( image_a_b @ F @ ( image_c_a @ G @ R ) )
% 0.32/0.47 = ( image_c_b @ ( comp_a_b_c @ F @ G ) @ R ) ) ).
% 0.32/0.47
% 0.32/0.47 % image_comp
% 0.32/0.47 thf(fact_307_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_c,G: c > b,F: c > c] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ! [Xa: c] :
% 0.32/0.47 ( ( member_c @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_c_c @ F @ A )
% 0.32/0.47 => ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
% 0.32/0.47 = ( inj_on_c_b @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_308_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_c,G: c > a,F: c > c] :
% 0.32/0.47 ( ! [X: c] :
% 0.32/0.47 ( ( member_c @ X @ A )
% 0.32/0.47 => ! [Xa: c] :
% 0.32/0.47 ( ( member_c @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_c_c @ F @ A )
% 0.32/0.47 => ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
% 0.32/0.47 = ( inj_on_c_a @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_309_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_b,G: b > b,F: b > b] :
% 0.32/0.47 ( ! [X: b] :
% 0.32/0.47 ( ( member_b @ X @ A )
% 0.32/0.47 => ! [Xa: b] :
% 0.32/0.47 ( ( member_b @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_b_b @ F @ A )
% 0.32/0.47 => ( ( inj_on_b_b @ G @ ( image_b_b @ F @ A ) )
% 0.32/0.47 = ( inj_on_b_b @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_310_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_a,G: a > b,F: a > a] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ! [Xa: a] :
% 0.32/0.47 ( ( member_a @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) )
% 0.32/0.47 = ( inj_on_a_b @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_311_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_a,G: a > c,F: a > a] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ! [Xa: a] :
% 0.32/0.47 ( ( member_a @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( inj_on_a_c @ G @ ( image_a_a @ F @ A ) )
% 0.32/0.47 = ( inj_on_a_c @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_312_inj__on__image__iff,axiom,
% 0.32/0.47 ! [A: set_a,G: a > a,F: a > a] :
% 0.32/0.47 ( ! [X: a] :
% 0.32/0.47 ( ( member_a @ X @ A )
% 0.32/0.47 => ! [Xa: a] :
% 0.32/0.47 ( ( member_a @ Xa @ A )
% 0.32/0.47 => ( ( ( G @ ( F @ X ) )
% 0.32/0.47 = ( G @ ( F @ Xa ) ) )
% 0.32/0.47 = ( ( G @ X )
% 0.32/0.47 = ( G @ Xa ) ) ) ) )
% 0.32/0.47 => ( ( inj_on_a_a @ F @ A )
% 0.32/0.47 => ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A ) )
% 0.32/0.47 = ( inj_on_a_a @ G @ A ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_image_iff
% 0.32/0.47 thf(fact_313_inj__def,axiom,
% 0.32/0.47 ! [F: c > b] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.32/0.47 = ( ! [X2: c,Y: c] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_314_inj__def,axiom,
% 0.32/0.47 ! [F: c > a] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.32/0.47 = ( ! [X2: c,Y: c] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_315_inj__def,axiom,
% 0.32/0.47 ! [F: a > b] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.32/0.47 = ( ! [X2: a,Y: a] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_316_inj__def,axiom,
% 0.32/0.47 ! [F: a > c] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.32/0.47 = ( ! [X2: a,Y: a] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_317_inj__def,axiom,
% 0.32/0.47 ! [F: a > a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.32/0.47 = ( ! [X2: a,Y: a] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_318_inj__def,axiom,
% 0.32/0.47 ! [F: b > b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.32/0.47 = ( ! [X2: b,Y: b] :
% 0.32/0.47 ( ( ( F @ X2 )
% 0.32/0.47 = ( F @ Y ) )
% 0.32/0.47 => ( X2 = Y ) ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_def
% 0.32/0.47 thf(fact_319_inj__eq,axiom,
% 0.32/0.47 ! [F: c > b,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_320_inj__eq,axiom,
% 0.32/0.47 ! [F: c > a,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_321_inj__eq,axiom,
% 0.32/0.47 ! [F: a > b,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_322_inj__eq,axiom,
% 0.32/0.47 ! [F: a > c,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_323_inj__eq,axiom,
% 0.32/0.47 ! [F: a > a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_324_inj__eq,axiom,
% 0.32/0.47 ! [F: b > b,X3: b,Y2: b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 = ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_eq
% 0.32/0.47 thf(fact_325_injI,axiom,
% 0.32/0.47 ! [F: c > b] :
% 0.32/0.47 ( ! [X: c,Y3: c] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_c_b @ F @ top_top_set_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_326_injI,axiom,
% 0.32/0.47 ! [F: c > a] :
% 0.32/0.47 ( ! [X: c,Y3: c] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_c_a @ F @ top_top_set_c ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_327_injI,axiom,
% 0.32/0.47 ! [F: a > b] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_a_b @ F @ top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_328_injI,axiom,
% 0.32/0.47 ! [F: a > c] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_a_c @ F @ top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_329_injI,axiom,
% 0.32/0.47 ! [F: a > a] :
% 0.32/0.47 ( ! [X: a,Y3: a] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_330_injI,axiom,
% 0.32/0.47 ! [F: b > b] :
% 0.32/0.47 ( ! [X: b,Y3: b] :
% 0.32/0.47 ( ( ( F @ X )
% 0.32/0.47 = ( F @ Y3 ) )
% 0.32/0.47 => ( X = Y3 ) )
% 0.32/0.47 => ( inj_on_b_b @ F @ top_top_set_b ) ) ).
% 0.32/0.47
% 0.32/0.47 % injI
% 0.32/0.47 thf(fact_331_injD,axiom,
% 0.32/0.47 ! [F: c > b,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_b @ F @ top_top_set_c )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_332_injD,axiom,
% 0.32/0.47 ! [F: c > a,X3: c,Y2: c] :
% 0.32/0.47 ( ( inj_on_c_a @ F @ top_top_set_c )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_333_injD,axiom,
% 0.32/0.47 ! [F: a > b,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_b @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_334_injD,axiom,
% 0.32/0.47 ! [F: a > c,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_c @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_335_injD,axiom,
% 0.32/0.47 ! [F: a > a,X3: a,Y2: a] :
% 0.32/0.47 ( ( inj_on_a_a @ F @ top_top_set_a )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_336_injD,axiom,
% 0.32/0.47 ! [F: b > b,X3: b,Y2: b] :
% 0.32/0.47 ( ( inj_on_b_b @ F @ top_top_set_b )
% 0.32/0.47 => ( ( ( F @ X3 )
% 0.32/0.47 = ( F @ Y2 ) )
% 0.32/0.47 => ( X3 = Y2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % injD
% 0.32/0.47 thf(fact_337_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: b > b,F: a > b,A: set_a] :
% 0.32/0.47 ( ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_a_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_338_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: c > b,F: a > c,A: set_a] :
% 0.32/0.47 ( ( inj_on_a_b @ ( comp_c_b_a @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_a_c @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_339_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: a > b,F: a > a,A: set_a] :
% 0.32/0.47 ( ( inj_on_a_b @ ( comp_a_b_a @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_a_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_340_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: b > b,F: c > b,A: set_c] :
% 0.32/0.47 ( ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_c_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_341_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: a > b,F: c > a,A: set_c] :
% 0.32/0.47 ( ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_c_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_342_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: b > a,F: c > b,A: set_c] :
% 0.32/0.47 ( ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_c_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_343_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: a > a,F: c > a,A: set_c] :
% 0.32/0.47 ( ( inj_on_c_a @ ( comp_a_a_c @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_c_a @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_344_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: b > b,F: b > b,A: set_b] :
% 0.32/0.47 ( ( inj_on_b_b @ ( comp_b_b_b @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_b_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_345_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: b > c,F: a > b,A: set_a] :
% 0.32/0.47 ( ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_a_b @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_346_inj__on__imageI2,axiom,
% 0.32/0.47 ! [F2: c > c,F: a > c,A: set_a] :
% 0.32/0.47 ( ( inj_on_a_c @ ( comp_c_c_a @ F2 @ F ) @ A )
% 0.32/0.47 => ( inj_on_a_c @ F @ A ) ) ).
% 0.32/0.47
% 0.32/0.47 % inj_on_imageI2
% 0.32/0.47 thf(fact_347_fun_Oinj__map__strong,axiom,
% 0.32/0.47 ! [X3: c > a,Xa2: c > a,F: a > b,Fa: a > b] :
% 0.32/0.47 ( ! [Z2: a,Za: a] :
% 0.32/0.47 ( ( member_a @ Z2 @ ( image_c_a @ X3 @ top_top_set_c ) )
% 0.32/0.47 => ( ( member_a @ Za @ ( image_c_a @ Xa2 @ top_top_set_c ) )
% 0.32/0.47 => ( ( ( F @ Z2 )
% 0.32/0.47 = ( Fa @ Za ) )
% 0.32/0.47 => ( Z2 = Za ) ) ) )
% 0.32/0.47 => ( ( ( comp_a_b_c @ F @ X3 )
% 0.32/0.47 = ( comp_a_b_c @ Fa @ Xa2 ) )
% 0.32/0.47 => ( X3 = Xa2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.inj_map_strong
% 0.32/0.47 thf(fact_348_fun_Oinj__map__strong,axiom,
% 0.32/0.47 ! [X3: a > c,Xa2: a > c,F: c > b,Fa: c > b] :
% 0.32/0.47 ( ! [Z2: c,Za: c] :
% 0.32/0.47 ( ( member_c @ Z2 @ ( image_a_c @ X3 @ top_top_set_a ) )
% 0.32/0.47 => ( ( member_c @ Za @ ( image_a_c @ Xa2 @ top_top_set_a ) )
% 0.32/0.47 => ( ( ( F @ Z2 )
% 0.32/0.47 = ( Fa @ Za ) )
% 0.32/0.47 => ( Z2 = Za ) ) ) )
% 0.32/0.47 => ( ( ( comp_c_b_a @ F @ X3 )
% 0.32/0.47 = ( comp_c_b_a @ Fa @ Xa2 ) )
% 0.32/0.47 => ( X3 = Xa2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.inj_map_strong
% 0.32/0.47 thf(fact_349_fun_Oinj__map__strong,axiom,
% 0.32/0.47 ! [X3: a > b,Xa2: a > b,F: b > b,Fa: b > b] :
% 0.32/0.47 ( ! [Z2: b,Za: b] :
% 0.32/0.47 ( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
% 0.32/0.47 => ( ( member_b @ Za @ ( image_a_b @ Xa2 @ top_top_set_a ) )
% 0.32/0.47 => ( ( ( F @ Z2 )
% 0.32/0.47 = ( Fa @ Za ) )
% 0.32/0.47 => ( Z2 = Za ) ) ) )
% 0.32/0.47 => ( ( ( comp_b_b_a @ F @ X3 )
% 0.32/0.47 = ( comp_b_b_a @ Fa @ Xa2 ) )
% 0.32/0.47 => ( X3 = Xa2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.inj_map_strong
% 0.32/0.47 thf(fact_350_fun_Oinj__map__strong,axiom,
% 0.32/0.47 ! [X3: a > a,Xa2: a > a,F: a > b,Fa: a > b] :
% 0.32/0.47 ( ! [Z2: a,Za: a] :
% 0.32/0.47 ( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
% 0.32/0.47 => ( ( member_a @ Za @ ( image_a_a @ Xa2 @ top_top_set_a ) )
% 0.32/0.47 => ( ( ( F @ Z2 )
% 0.32/0.47 = ( Fa @ Za ) )
% 0.32/0.47 => ( Z2 = Za ) ) ) )
% 0.32/0.47 => ( ( ( comp_a_b_a @ F @ X3 )
% 0.32/0.47 = ( comp_a_b_a @ Fa @ Xa2 ) )
% 0.32/0.47 => ( X3 = Xa2 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.inj_map_strong
% 0.32/0.47 thf(fact_351_fun_Omap__cong0,axiom,
% 0.32/0.47 ! [X3: a > b,F: b > b,G: b > b] :
% 0.32/0.47 ( ! [Z2: b] :
% 0.32/0.47 ( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
% 0.32/0.47 => ( ( F @ Z2 )
% 0.32/0.47 = ( G @ Z2 ) ) )
% 0.32/0.47 => ( ( comp_b_b_a @ F @ X3 )
% 0.32/0.47 = ( comp_b_b_a @ G @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.map_cong0
% 0.32/0.47 thf(fact_352_fun_Omap__cong0,axiom,
% 0.32/0.47 ! [X3: a > a,F: a > b,G: a > b] :
% 0.32/0.47 ( ! [Z2: a] :
% 0.32/0.47 ( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
% 0.32/0.47 => ( ( F @ Z2 )
% 0.32/0.47 = ( G @ Z2 ) ) )
% 0.32/0.47 => ( ( comp_a_b_a @ F @ X3 )
% 0.32/0.47 = ( comp_a_b_a @ G @ X3 ) ) ) ).
% 0.32/0.47
% 0.32/0.47 % fun.map_cong0
% 0.32/0.47
% 0.32/0.47 % Conjectures (3)
% 0.32/0.47 thf(conj_0,hypothesis,
% 0.32/0.47 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 ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(conj_1,hypothesis,
% 0.32/0.47 ( ( comp_a_b_c @ f @ g )
% 0.32/0.47 = ( comp_a_b_c @ f @ h ) ) ).
% 0.32/0.47
% 0.32/0.47 thf(conj_2,conjecture,
% 0.32/0.47 g = h ).
% 0.32/0.47
% 0.32/0.47 %------------------------------------------------------------------------------
% 0.32/0.47 ------------------------
% 2.61/1.45 ---- Embedded in THF ---
% 2.61/1.45 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.61/1.45 %%% Generated on Tue Feb 24 08:27:19 EST 2026
% 2.61/1.45 %%% using '$$dhol' embedding, version 1.3.0.
% 2.61/1.45 %%% Logic specification used:
% 2.61/1.45 %%% thf(spec, logic, $$dhol).
% 2.61/1.45
% 2.61/1.45 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2DTF24497.p
% See solution above
% 2.61/1.45 ------------------------
% 2.61/1.47 ---- Cleaned THF ---
% 2.61/1.48 thf(ty_n_t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 2.61/1.48 set_c_a: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
% 2.61/1.48 set_a_c: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
% 2.61/1.48 set_a_b: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
% 2.61/1.48 set_a_a: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_Itf__c_J,type,
% 2.61/1.48 set_c: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_Itf__b_J,type,
% 2.61/1.48 set_b: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 2.61/1.48 set_a: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_tf__c,type,
% 2.61/1.48 c: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_tf__b,type,
% 2.61/1.48 b: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(ty_n_tf__a,type,
% 2.61/1.48 a: $tType ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__a,type,
% 2.61/1.48 comp_a_a_a: ( a > a ) > ( a > a ) > a > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__a_001tf__c,type,
% 2.61/1.48 comp_a_a_c: ( a > a ) > ( c > a ) > c > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__a,type,
% 2.61/1.48 comp_a_b_a: ( a > b ) > ( a > a ) > a > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__b,type,
% 2.61/1.48 comp_a_b_b: ( a > b ) > ( b > a ) > b > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__b_001tf__c,type,
% 2.61/1.48 comp_a_b_c: ( a > b ) > ( c > a ) > c > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__a,type,
% 2.61/1.48 comp_a_c_a: ( a > c ) > ( a > a ) > a > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__a_001tf__c_001tf__c,type,
% 2.61/1.48 comp_a_c_c: ( a > c ) > ( c > a ) > c > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__a,type,
% 2.61/1.48 comp_b_a_a: ( b > a ) > ( a > b ) > a > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__a_001tf__c,type,
% 2.61/1.48 comp_b_a_c: ( b > a ) > ( c > b ) > c > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__a,type,
% 2.61/1.48 comp_b_b_a: ( b > b ) > ( a > b ) > a > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__b,type,
% 2.61/1.48 comp_b_b_b: ( b > b ) > ( b > b ) > b > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__b_001tf__c,type,
% 2.61/1.48 comp_b_b_c: ( b > b ) > ( c > b ) > c > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__a,type,
% 2.61/1.48 comp_b_c_a: ( b > c ) > ( a > b ) > a > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__b_001tf__c_001tf__c,type,
% 2.61/1.48 comp_b_c_c: ( b > c ) > ( c > b ) > c > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__a,type,
% 2.61/1.48 comp_c_a_a: ( c > a ) > ( a > c ) > a > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__b,type,
% 2.61/1.48 comp_c_a_b: ( c > a ) > ( b > c ) > b > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__a_001tf__c,type,
% 2.61/1.48 comp_c_a_c: ( c > a ) > ( c > c ) > c > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__a,type,
% 2.61/1.48 comp_c_b_a: ( c > b ) > ( a > c ) > a > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__b,type,
% 2.61/1.48 comp_c_b_b: ( c > b ) > ( b > c ) > b > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__b_001tf__c,type,
% 2.61/1.48 comp_c_b_c: ( c > b ) > ( c > c ) > c > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__a,type,
% 2.61/1.48 comp_c_c_a: ( c > c ) > ( a > c ) > a > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Ocomp_001tf__c_001tf__c_001tf__c,type,
% 2.61/1.48 comp_c_c_c: ( c > c ) > ( c > c ) > c > c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__a_J_001_062_Itf__a_Mtf__b_J,type,
% 2.61/1.48 inj_on_a_a_a_b: ( ( a > a ) > a > b ) > set_a_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__b_J_001_062_Itf__a_Mtf__b_J,type,
% 2.61/1.48 inj_on_a_b_a_b: ( ( a > b ) > a > b ) > set_a_b > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__a_Mtf__c_J_001_062_Itf__a_Mtf__b_J,type,
% 2.61/1.48 inj_on_a_c_a_b: ( ( a > c ) > a > b ) > set_a_c > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__a_J,type,
% 2.61/1.48 inj_on_c_a_c_a: ( ( c > a ) > c > a ) > set_c_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__b_J,type,
% 2.61/1.48 inj_on_c_a_c_b: ( ( c > a ) > c > b ) > set_c_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001_062_Itf__c_Mtf__c_J,type,
% 2.61/1.48 inj_on_c_a_c_c: ( ( c > a ) > c > c ) > set_c_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001_062_Itf__c_Mtf__a_J_001tf__a,type,
% 2.61/1.48 inj_on_c_a_a: ( ( c > a ) > a ) > set_c_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__a,type,
% 2.61/1.48 inj_on_a_a: ( a > a ) > set_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__b,type,
% 2.61/1.48 inj_on_a_b: ( a > b ) > set_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__a_001tf__c,type,
% 2.61/1.48 inj_on_a_c: ( a > c ) > set_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__a,type,
% 2.61/1.48 inj_on_b_a: ( b > a ) > set_b > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__b,type,
% 2.61/1.48 inj_on_b_b: ( b > b ) > set_b > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__b_001tf__c,type,
% 2.61/1.48 inj_on_b_c: ( b > c ) > set_b > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__a,type,
% 2.61/1.48 inj_on_c_a: ( c > a ) > set_c > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__b,type,
% 2.61/1.48 inj_on_c_b: ( c > b ) > set_c > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Fun_Oinj__on_001tf__c_001tf__c,type,
% 2.61/1.48 inj_on_c_c: ( c > c ) > set_c > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 2.61/1.48 sup_sup_set_c_a: set_c_a > set_c_a > set_c_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
% 2.61/1.48 sup_sup_set_a: set_a > set_a > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__b_J,type,
% 2.61/1.48 sup_sup_set_b: set_b > set_b > set_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__c_J,type,
% 2.61/1.48 sup_sup_set_c: set_c > set_c > set_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__a_J_J,type,
% 2.61/1.48 top_top_set_a_a: set_a_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__b_J_J,type,
% 2.61/1.48 top_top_set_a_b: set_a_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__a_Mtf__c_J_J,type,
% 2.61/1.48 top_top_set_a_c: set_a_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_062_Itf__c_Mtf__a_J_J,type,
% 2.61/1.48 top_top_set_c_a: set_c_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
% 2.61/1.48 top_top_set_a: set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__b_J,type,
% 2.61/1.48 top_top_set_b: set_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__c_J,type,
% 2.61/1.48 top_top_set_c: set_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_OCollect_001tf__a,type,
% 2.61/1.48 collect_a: ( a > $o ) > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001_062_Itf__c_Mtf__a_J_001tf__a,type,
% 2.61/1.48 image_c_a_a: ( ( c > a ) > a ) > set_c_a > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
% 2.61/1.48 image_a_a: ( a > a ) > set_a > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__a_001tf__b,type,
% 2.61/1.48 image_a_b: ( a > b ) > set_a > set_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__a_001tf__c,type,
% 2.61/1.48 image_a_c: ( a > c ) > set_a > set_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__b_001tf__a,type,
% 2.61/1.48 image_b_a: ( b > a ) > set_b > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__b_001tf__b,type,
% 2.61/1.48 image_b_b: ( b > b ) > set_b > set_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__b_001tf__c,type,
% 2.61/1.48 image_b_c: ( b > c ) > set_b > set_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__c_001_062_Itf__c_Mtf__a_J,type,
% 2.61/1.48 image_c_c_a: ( c > c > a ) > set_c > set_c_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__c_001tf__a,type,
% 2.61/1.48 image_c_a: ( c > a ) > set_c > set_a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__c_001tf__b,type,
% 2.61/1.48 image_c_b: ( c > b ) > set_c > set_b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_Set_Oimage_001tf__c_001tf__c,type,
% 2.61/1.48 image_c_c: ( c > c ) > set_c > set_c ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_member_001_062_Itf__c_Mtf__a_J,type,
% 2.61/1.48 member_c_a: ( c > a ) > set_c_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_member_001tf__a,type,
% 2.61/1.48 member_a: a > set_a > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_member_001tf__b,type,
% 2.61/1.48 member_b: b > set_b > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_c_member_001tf__c,type,
% 2.61/1.48 member_c: c > set_c > $o ).
% 2.61/1.48
% 2.61/1.48 thf(sy_v_f,type,
% 2.61/1.48 f: a > b ).
% 2.61/1.48
% 2.61/1.48 thf(sy_v_g,type,
% 2.61/1.48 g: c > a ).
% 2.61/1.48
% 2.61/1.48 thf(sy_v_h,type,
% 2.61/1.48 h: c > a ).
% 2.61/1.48
% 2.61/1.48 thf(fact_0_o__ext,axiom,
% 2.61/1.48 ! [H: a > c,F: c > b,G: c > b] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ ( image_a_c @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_c_b_a @ F @ H )
% 2.61/1.48 = ( comp_c_b_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_1_o__ext,axiom,
% 2.61/1.48 ! [H: a > b,F: b > b,G: b > b] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( member_b @ X @ ( image_a_b @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_b_b_a @ F @ H )
% 2.61/1.48 = ( comp_b_b_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_2_o__ext,axiom,
% 2.61/1.48 ! [H: a > a,F: a > b,G: a > b] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ ( image_a_a @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_a_b_a @ F @ H )
% 2.61/1.48 = ( comp_a_b_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_3_o__ext,axiom,
% 2.61/1.48 ! [H: c > a,F: a > b,G: a > b] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ ( image_c_a @ H @ top_top_set_c ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_a_b_c @ F @ H )
% 2.61/1.48 = ( comp_a_b_c @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_4_o__cong,axiom,
% 2.61/1.48 ! [H: a > c,I: a > c,F: c > b,G: c > b] :
% 2.61/1.48 ( ( H = I )
% 2.61/1.48 => ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ ( image_a_c @ I @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_c_b_a @ F @ H )
% 2.61/1.48 = ( comp_c_b_a @ F @ I ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_5_o__cong,axiom,
% 2.61/1.48 ! [H: a > b,I: a > b,F: b > b,G: b > b] :
% 2.61/1.48 ( ( H = I )
% 2.61/1.48 => ( ! [X: b] :
% 2.61/1.48 ( ( member_b @ X @ ( image_a_b @ I @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_b_b_a @ F @ H )
% 2.61/1.48 = ( comp_b_b_a @ F @ I ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_6_o__cong,axiom,
% 2.61/1.48 ! [H: a > a,I: a > a,F: a > b,G: a > b] :
% 2.61/1.48 ( ( H = I )
% 2.61/1.48 => ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ ( image_a_a @ I @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_a_b_a @ F @ H )
% 2.61/1.48 = ( comp_a_b_a @ F @ I ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_7_o__cong,axiom,
% 2.61/1.48 ! [H: c > a,I: c > a,F: a > b,G: a > b] :
% 2.61/1.48 ( ( H = I )
% 2.61/1.48 => ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ ( image_c_a @ I @ top_top_set_c ) )
% 2.61/1.48 => ( ( F @ X )
% 2.61/1.48 = ( G @ X ) ) )
% 2.61/1.48 => ( ( comp_a_b_c @ F @ H )
% 2.61/1.48 = ( comp_a_b_c @ F @ I ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_8_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > b,G: c > a,H: c > a] :
% 2.61/1.48 ( ( ( comp_a_b_c @ F @ G )
% 2.61/1.48 = ( comp_a_b_c @ F @ H ) )
% 2.61/1.48 => ( ( 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 ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_9_o__inj__on,axiom,
% 2.61/1.48 ! [F: b > b,G: c > b,H: c > b] :
% 2.61/1.48 ( ( ( comp_b_b_c @ F @ G )
% 2.61/1.48 = ( comp_b_b_c @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_c_b @ G @ top_top_set_c ) @ ( image_c_b @ H @ top_top_set_c ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_10_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > c,G: c > a,H: c > a] :
% 2.61/1.48 ( ( ( comp_a_c_c @ F @ G )
% 2.61/1.48 = ( comp_a_c_c @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_11_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > a,G: c > a,H: c > a] :
% 2.61/1.48 ( ( ( comp_a_a_c @ F @ G )
% 2.61/1.48 = ( comp_a_a_c @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_c_a @ G @ top_top_set_c ) @ ( image_c_a @ H @ top_top_set_c ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_12_o__inj__on,axiom,
% 2.61/1.48 ! [F: c > b,G: c > c,H: c > c] :
% 2.61/1.48 ( ( ( comp_c_b_c @ F @ G )
% 2.61/1.48 = ( comp_c_b_c @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_c_b @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_13_o__inj__on,axiom,
% 2.61/1.48 ! [F: c > a,G: c > c,H: c > c] :
% 2.61/1.48 ( ( ( comp_c_a_c @ F @ G )
% 2.61/1.48 = ( comp_c_a_c @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_c_a @ F @ ( sup_sup_set_c @ ( image_c_c @ G @ top_top_set_c ) @ ( image_c_c @ H @ top_top_set_c ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_14_o__inj__on,axiom,
% 2.61/1.48 ! [F: b > b,G: a > b,H: a > b] :
% 2.61/1.48 ( ( ( comp_b_b_a @ F @ G )
% 2.61/1.48 = ( comp_b_b_a @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ ( image_a_b @ G @ top_top_set_a ) @ ( image_a_b @ H @ top_top_set_a ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_15_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > b,G: a > a,H: a > a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ F @ G )
% 2.61/1.48 = ( comp_a_b_a @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_16_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > c,G: a > a,H: a > a] :
% 2.61/1.48 ( ( ( comp_a_c_a @ F @ G )
% 2.61/1.48 = ( comp_a_c_a @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_17_o__inj__on,axiom,
% 2.61/1.48 ! [F: a > a,G: a > a,H: a > a] :
% 2.61/1.48 ( ( ( comp_a_a_a @ F @ G )
% 2.61/1.48 = ( comp_a_a_a @ F @ H ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ ( image_a_a @ G @ top_top_set_a ) @ ( image_a_a @ H @ top_top_set_a ) ) )
% 2.61/1.48 => ( G = H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_18_ex__o__conv,axiom,
% 2.61/1.48 ! [G: a > b,F: c > b] :
% 2.61/1.48 ( ( ? [H2: a > c] :
% 2.61/1.48 ( G
% 2.61/1.48 = ( comp_c_b_a @ F @ H2 ) ) )
% 2.61/1.48 = ( ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 2.61/1.48 => ? [Y: c] :
% 2.61/1.48 ( X2
% 2.61/1.48 = ( F @ Y ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_19_ex__o__conv,axiom,
% 2.61/1.48 ! [G: a > b,F: a > b] :
% 2.61/1.48 ( ( ? [H2: a > a] :
% 2.61/1.48 ( G
% 2.61/1.48 = ( comp_a_b_a @ F @ H2 ) ) )
% 2.61/1.48 = ( ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 2.61/1.48 => ? [Y: a] :
% 2.61/1.48 ( X2
% 2.61/1.48 = ( F @ Y ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_20_ex__o__conv,axiom,
% 2.61/1.48 ! [G: a > b,F: b > b] :
% 2.61/1.48 ( ( ? [H2: a > b] :
% 2.61/1.48 ( G
% 2.61/1.48 = ( comp_b_b_a @ F @ H2 ) ) )
% 2.61/1.48 = ( ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ ( image_a_b @ G @ top_top_set_a ) )
% 2.61/1.48 => ? [Y: b] :
% 2.61/1.48 ( X2
% 2.61/1.48 = ( F @ Y ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_21_ex__o__conv,axiom,
% 2.61/1.48 ! [G: c > b,F: a > b] :
% 2.61/1.48 ( ( ? [H2: c > a] :
% 2.61/1.48 ( G
% 2.61/1.48 = ( comp_a_b_c @ F @ H2 ) ) )
% 2.61/1.48 = ( ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ ( image_c_b @ G @ top_top_set_c ) )
% 2.61/1.48 => ? [Y: a] :
% 2.61/1.48 ( X2
% 2.61/1.48 = ( F @ Y ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_22_o__eq__conv,axiom,
% 2.61/1.48 ! [F: c > b,H: a > c,G: c > b] :
% 2.61/1.48 ( ( ( comp_c_b_a @ F @ H )
% 2.61/1.48 = ( comp_c_b_a @ G @ H ) )
% 2.61/1.48 = ( ! [X2: c] :
% 2.61/1.48 ( ( member_c @ X2 @ ( image_a_c @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X2 )
% 2.61/1.48 = ( G @ X2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_23_o__eq__conv,axiom,
% 2.61/1.48 ! [F: a > b,H: a > a,G: a > b] :
% 2.61/1.48 ( ( ( comp_a_b_a @ F @ H )
% 2.61/1.48 = ( comp_a_b_a @ G @ H ) )
% 2.61/1.48 = ( ! [X2: a] :
% 2.61/1.48 ( ( member_a @ X2 @ ( image_a_a @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X2 )
% 2.61/1.48 = ( G @ X2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_24_o__eq__conv,axiom,
% 2.61/1.48 ! [F: b > b,H: a > b,G: b > b] :
% 2.61/1.48 ( ( ( comp_b_b_a @ F @ H )
% 2.61/1.48 = ( comp_b_b_a @ G @ H ) )
% 2.61/1.48 = ( ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ ( image_a_b @ H @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ X2 )
% 2.61/1.48 = ( G @ X2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_25_o__eq__conv,axiom,
% 2.61/1.48 ! [F: a > b,H: c > a,G: a > b] :
% 2.61/1.48 ( ( ( comp_a_b_c @ F @ H )
% 2.61/1.48 = ( comp_a_b_c @ G @ H ) )
% 2.61/1.48 = ( ! [X2: a] :
% 2.61/1.48 ( ( member_a @ X2 @ ( image_c_a @ H @ top_top_set_c ) )
% 2.61/1.48 => ( ( F @ X2 )
% 2.61/1.48 = ( G @ X2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_26_sup__top__left,axiom,
% 2.61/1.48 ! [X3: set_b] :
% 2.61/1.48 ( ( sup_sup_set_b @ top_top_set_b @ X3 )
% 2.61/1.48 = top_top_set_b ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_27_sup__top__left,axiom,
% 2.61/1.48 ! [X3: set_c_a] :
% 2.61/1.48 ( ( sup_sup_set_c_a @ top_top_set_c_a @ X3 )
% 2.61/1.48 = top_top_set_c_a ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_28_sup__top__left,axiom,
% 2.61/1.48 ! [X3: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ top_top_set_a @ X3 )
% 2.61/1.48 = top_top_set_a ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_29_sup__top__left,axiom,
% 2.61/1.48 ! [X3: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ top_top_set_c @ X3 )
% 2.61/1.48 = top_top_set_c ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_30_sup__top__right,axiom,
% 2.61/1.48 ! [X3: set_b] :
% 2.61/1.48 ( ( sup_sup_set_b @ X3 @ top_top_set_b )
% 2.61/1.48 = top_top_set_b ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_31_sup__top__right,axiom,
% 2.61/1.48 ! [X3: set_c_a] :
% 2.61/1.48 ( ( sup_sup_set_c_a @ X3 @ top_top_set_c_a )
% 2.61/1.48 = top_top_set_c_a ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_32_sup__top__right,axiom,
% 2.61/1.48 ! [X3: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ top_top_set_a )
% 2.61/1.48 = top_top_set_a ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_33_sup__top__right,axiom,
% 2.61/1.48 ! [X3: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ top_top_set_c )
% 2.61/1.48 = top_top_set_c ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_34_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: b > b,A: set_b,B: set_b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ ( sup_sup_set_b @ A @ B ) )
% 2.61/1.48 => ( ( ( image_b_b @ F @ A )
% 2.61/1.48 = ( image_b_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_35_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ ( sup_sup_set_a @ A @ B ) )
% 2.61/1.48 => ( ( ( image_a_c @ F @ A )
% 2.61/1.48 = ( image_a_c @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_36_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ ( sup_sup_set_a @ A @ B ) )
% 2.61/1.48 => ( ( ( image_a_a @ F @ A )
% 2.61/1.48 = ( image_a_a @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_37_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,B: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ ( sup_sup_set_c @ A @ B ) )
% 2.61/1.48 => ( ( ( image_c_b @ F @ A )
% 2.61/1.48 = ( image_c_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_38_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,B: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ ( sup_sup_set_c @ A @ B ) )
% 2.61/1.48 => ( ( ( image_c_a @ F @ A )
% 2.61/1.48 = ( image_c_a @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_39_inj__on__Un__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ ( sup_sup_set_a @ A @ B ) )
% 2.61/1.48 => ( ( ( image_a_b @ F @ A )
% 2.61/1.48 = ( image_a_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_40_inj__compose,axiom,
% 2.61/1.48 ! [F: b > b,G: a > b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( inj_on_a_b @ G @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_b @ ( comp_b_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_41_inj__compose,axiom,
% 2.61/1.48 ! [F: a > b,G: a > a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( inj_on_a_a @ G @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_b @ ( comp_a_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_42_inj__compose,axiom,
% 2.61/1.48 ! [F: a > b,G: c > a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( inj_on_c_a @ G @ top_top_set_c )
% 2.61/1.48 => ( inj_on_c_b @ ( comp_a_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_43_inj__compose,axiom,
% 2.61/1.48 ! [F: c > b,G: a > c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_b @ ( comp_c_b_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_44_inj__compose,axiom,
% 2.61/1.48 ! [F: c > b,G: c > c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_c_c @ G @ top_top_set_c )
% 2.61/1.48 => ( inj_on_c_b @ ( comp_c_b_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_45_inj__compose,axiom,
% 2.61/1.48 ! [F: c > a,G: c > c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_c_c @ G @ top_top_set_c )
% 2.61/1.48 => ( inj_on_c_a @ ( comp_c_a_c @ F @ G ) @ top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_46_inj__compose,axiom,
% 2.61/1.48 ! [F: c > c,G: a > c] :
% 2.61/1.48 ( ( inj_on_c_c @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_c @ ( comp_c_c_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_47_inj__compose,axiom,
% 2.61/1.48 ! [F: c > a,G: a > c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_a_c @ G @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_a @ ( comp_c_a_a @ F @ G ) @ top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_48_inj__compose,axiom,
% 2.61/1.48 ! [F: c > b,G: b > c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_b_c @ G @ top_top_set_b )
% 2.61/1.48 => ( inj_on_b_b @ ( comp_c_b_b @ F @ G ) @ top_top_set_b ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_49_inj__compose,axiom,
% 2.61/1.48 ! [F: c > a,G: b > c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( inj_on_b_c @ G @ top_top_set_b )
% 2.61/1.48 => ( inj_on_b_a @ ( comp_c_a_b @ F @ G ) @ top_top_set_b ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_50_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: c > b] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( inj_on_a_c_a_b @ ( comp_c_b_a @ F ) @ top_top_set_a_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_51_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: a > b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( inj_on_a_a_a_b @ ( comp_a_b_a @ F ) @ top_top_set_a_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_52_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: a > c] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( inj_on_c_a_c_c @ ( comp_a_c_c @ F ) @ top_top_set_c_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_53_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: a > a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( inj_on_c_a_c_a @ ( comp_a_a_c @ F ) @ top_top_set_c_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_54_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: b > b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( inj_on_a_b_a_b @ ( comp_b_b_a @ F ) @ top_top_set_a_b ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_55_fun_Oinj__map,axiom,
% 2.61/1.48 ! [F: a > b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( inj_on_c_a_c_b @ ( comp_a_b_c @ F ) @ top_top_set_c_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_56_comp__inj__on,axiom,
% 2.61/1.48 ! [F: b > a,A: set_b,G: a > b] :
% 2.61/1.48 ( ( inj_on_b_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_b @ G @ ( image_b_a @ F @ A ) )
% 2.61/1.48 => ( inj_on_b_b @ ( comp_a_b_b @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_57_comp__inj__on,axiom,
% 2.61/1.48 ! [F: c > c,A: set_c,G: c > b] :
% 2.61/1.48 ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
% 2.61/1.48 => ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_58_comp__inj__on,axiom,
% 2.61/1.48 ! [F: b > c,A: set_b,G: c > b] :
% 2.61/1.48 ( ( inj_on_b_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_b @ G @ ( image_b_c @ F @ A ) )
% 2.61/1.48 => ( inj_on_b_b @ ( comp_c_b_b @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_59_comp__inj__on,axiom,
% 2.61/1.48 ! [F: c > c,A: set_c,G: c > a] :
% 2.61/1.48 ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
% 2.61/1.48 => ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_60_comp__inj__on,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,G: b > c] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_c @ G @ ( image_a_b @ F @ A ) )
% 2.61/1.48 => ( inj_on_a_c @ ( comp_b_c_a @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_61_comp__inj__on,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,G: b > a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_a @ G @ ( image_a_b @ F @ A ) )
% 2.61/1.48 => ( inj_on_a_a @ ( comp_b_a_a @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_62_comp__inj__on,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,G: b > b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) )
% 2.61/1.48 => ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_63_comp__inj__on,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,G: b > a] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) )
% 2.61/1.48 => ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_64_comp__inj__on,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,G: b > b] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) )
% 2.61/1.48 => ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_65_comp__inj__on,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,G: a > b] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) )
% 2.61/1.48 => ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_66_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: a > c,F: c > a,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_c @ ( comp_a_c_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_c @ G @ ( image_c_a @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_67_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: a > b,F: a > a,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_a_b_a @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_68_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: c > b,F: a > c,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_c_b_a @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_b @ G @ ( image_a_c @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_69_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: b > b,F: a > b,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_b_b_a @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_b_b @ G @ ( image_a_b @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_70_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: a > b,F: c > a,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ ( comp_a_b_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_b @ G @ ( image_c_a @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_71_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: c > b,F: c > c,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ ( comp_c_b_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_72_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: b > b,F: c > b,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ ( comp_b_b_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_b_b @ G @ ( image_c_b @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_73_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: b > a,F: c > b,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ ( comp_b_a_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_b_a @ G @ ( image_c_b @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_74_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: c > a,F: c > c,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ ( comp_c_a_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_75_inj__on__imageI,axiom,
% 2.61/1.48 ! [G: a > a,F: c > a,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ ( comp_a_a_c @ G @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_a @ G @ ( image_c_a @ F @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_76_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: b > a,A: set_b,F2: a > b] :
% 2.61/1.48 ( ( inj_on_b_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_b @ F2 @ ( image_b_a @ F @ A ) )
% 2.61/1.48 = ( inj_on_b_b @ ( comp_a_b_b @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_77_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: c > c,A: set_c,F2: c > b] :
% 2.61/1.48 ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_b @ F2 @ ( image_c_c @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_b @ ( comp_c_b_c @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_78_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: b > c,A: set_b,F2: c > b] :
% 2.61/1.48 ( ( inj_on_b_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_b @ F2 @ ( image_b_c @ F @ A ) )
% 2.61/1.48 = ( inj_on_b_b @ ( comp_c_b_b @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_79_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: c > c,A: set_c,F2: c > a] :
% 2.61/1.48 ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_a @ F2 @ ( image_c_c @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_a @ ( comp_c_a_c @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_80_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,F2: b > c] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_c @ F2 @ ( image_a_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_81_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,F2: b > a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_a @ F2 @ ( image_a_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_a @ ( comp_b_a_a @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_82_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,F2: b > b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_b @ F2 @ ( image_a_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_83_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,F2: b > a] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_a @ F2 @ ( image_c_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_84_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,F2: b > b] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_b @ F2 @ ( image_c_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_85_comp__inj__on__iff,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,F2: a > b] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_b @ F2 @ ( image_c_a @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_86_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: c > b,B2: b] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( member_b @ B2 @ ( image_c_b @ F @ top_top_set_c ) )
% 2.61/1.48 = ( ? [X2: c] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: c] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_87_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: c > a,B2: a] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( member_a @ B2 @ ( image_c_a @ F @ top_top_set_c ) )
% 2.61/1.48 = ( ? [X2: c] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: c] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_88_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: a > b,B2: b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_b @ B2 @ ( image_a_b @ F @ top_top_set_a ) )
% 2.61/1.48 = ( ? [X2: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_89_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: a > c,B2: c] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_c @ B2 @ ( image_a_c @ F @ top_top_set_a ) )
% 2.61/1.48 = ( ? [X2: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_90_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: a > a,B2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_a @ B2 @ ( image_a_a @ F @ top_top_set_a ) )
% 2.61/1.48 = ( ? [X2: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_91_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: b > b,B2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( member_b @ B2 @ ( image_b_b @ F @ top_top_set_b ) )
% 2.61/1.48 = ( ? [X2: b] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: b] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_92_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: b > a,B2: a] :
% 2.61/1.48 ( ( inj_on_b_a @ F @ top_top_set_b )
% 2.61/1.48 => ( ( member_a @ B2 @ ( image_b_a @ F @ top_top_set_b ) )
% 2.61/1.48 = ( ? [X2: b] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: b] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_93_range__ex1__eq,axiom,
% 2.61/1.48 ! [F: ( c > a ) > a,B2: a] :
% 2.61/1.48 ( ( inj_on_c_a_a @ F @ top_top_set_c_a )
% 2.61/1.48 => ( ( member_a @ B2 @ ( image_c_a_a @ F @ top_top_set_c_a ) )
% 2.61/1.48 = ( ? [X2: c > a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ X2 ) )
% 2.61/1.48 & ! [Y: c > a] :
% 2.61/1.48 ( ( B2
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( Y = X2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_94_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,B: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_b @ F @ A )
% 2.61/1.48 = ( image_c_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_95_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,B: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_a @ F @ A )
% 2.61/1.48 = ( image_c_a @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_96_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_b @ F @ A )
% 2.61/1.48 = ( image_a_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_97_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_c @ F @ A )
% 2.61/1.48 = ( image_a_c @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_98_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,B: set_a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_a @ F @ A )
% 2.61/1.48 = ( image_a_a @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_99_inj__image__eq__iff,axiom,
% 2.61/1.48 ! [F: b > b,A: set_b,B: set_b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( ( image_b_b @ F @ A )
% 2.61/1.48 = ( image_b_b @ F @ B ) )
% 2.61/1.48 = ( A = B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_100_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: c > b,A2: c,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( member_b @ ( F @ A2 ) @ ( image_c_b @ F @ A ) )
% 2.61/1.48 = ( member_c @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_101_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: c > a,A2: c,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( member_a @ ( F @ A2 ) @ ( image_c_a @ F @ A ) )
% 2.61/1.48 = ( member_c @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_102_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: a > b,A2: a,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_b @ ( F @ A2 ) @ ( image_a_b @ F @ A ) )
% 2.61/1.48 = ( member_a @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_103_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: a > c,A2: a,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_c @ ( F @ A2 ) @ ( image_a_c @ F @ A ) )
% 2.61/1.48 = ( member_a @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_104_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: a > a,A2: a,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( ( member_a @ ( F @ A2 ) @ ( image_a_a @ F @ A ) )
% 2.61/1.48 = ( member_a @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_105_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: b > b,A2: b,A: set_b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( member_b @ ( F @ A2 ) @ ( image_b_b @ F @ A ) )
% 2.61/1.48 = ( member_b @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_106_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: b > a,A2: b,A: set_b] :
% 2.61/1.48 ( ( inj_on_b_a @ F @ top_top_set_b )
% 2.61/1.48 => ( ( member_a @ ( F @ A2 ) @ ( image_b_a @ F @ A ) )
% 2.61/1.48 = ( member_b @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_107_inj__image__mem__iff,axiom,
% 2.61/1.48 ! [F: ( c > a ) > a,A2: c > a,A: set_c_a] :
% 2.61/1.48 ( ( inj_on_c_a_a @ F @ top_top_set_c_a )
% 2.61/1.48 => ( ( member_a @ ( F @ A2 ) @ ( image_c_a_a @ F @ A ) )
% 2.61/1.48 = ( member_c_a @ A2 @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_108_comp__surj,axiom,
% 2.61/1.48 ! [F: c > c,G: c > c] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_c @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( image_c_c @ ( comp_c_c_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_109_comp__surj,axiom,
% 2.61/1.48 ! [F: c > c,G: c > a] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_a @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( image_c_a @ ( comp_c_a_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_110_comp__surj,axiom,
% 2.61/1.48 ! [F: c > c,G: c > b] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_b @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( image_c_b @ ( comp_c_b_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_b ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_111_comp__surj,axiom,
% 2.61/1.48 ! [F: c > a,G: a > c] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_c @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( image_c_c @ ( comp_a_c_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_112_comp__surj,axiom,
% 2.61/1.48 ! [F: c > a,G: a > a] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_a @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( image_c_a @ ( comp_a_a_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_113_comp__surj,axiom,
% 2.61/1.48 ! [F: c > a,G: a > b] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( ( image_a_b @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( image_c_b @ ( comp_a_b_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_b ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_114_comp__surj,axiom,
% 2.61/1.48 ! [F: c > b,G: b > c] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( ( image_b_c @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( image_c_c @ ( comp_b_c_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_115_comp__surj,axiom,
% 2.61/1.48 ! [F: c > b,G: b > a] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( ( image_b_a @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ( ( image_c_a @ ( comp_b_a_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_a ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_116_comp__surj,axiom,
% 2.61/1.48 ! [F: c > b,G: b > b] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( ( image_b_b @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ( ( image_c_b @ ( comp_b_b_c @ G @ F ) @ top_top_set_c )
% 2.61/1.48 = top_top_set_b ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_117_comp__surj,axiom,
% 2.61/1.48 ! [F: a > c,G: c > c] :
% 2.61/1.48 ( ( ( image_a_c @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( ( image_c_c @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ( ( image_a_c @ ( comp_c_c_a @ G @ F ) @ top_top_set_a )
% 2.61/1.48 = top_top_set_c ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_118_sup_Oright__idem,axiom,
% 2.61/1.48 ! [A2: set_a,B2: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
% 2.61/1.48 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_119_sup_Oright__idem,axiom,
% 2.61/1.48 ! [A2: set_c,B2: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ B2 )
% 2.61/1.48 = ( sup_sup_set_c @ A2 @ B2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_120_sup__left__idem,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 2.61/1.48 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_121_sup__left__idem,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
% 2.61/1.48 = ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_122_sup_Oleft__idem,axiom,
% 2.61/1.48 ! [A2: set_a,B2: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
% 2.61/1.48 = ( sup_sup_set_a @ A2 @ B2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_123_sup_Oleft__idem,axiom,
% 2.61/1.48 ! [A2: set_c,B2: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ A2 @ B2 ) )
% 2.61/1.48 = ( sup_sup_set_c @ A2 @ B2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_124_sup__idem,axiom,
% 2.61/1.48 ! [X3: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ X3 )
% 2.61/1.48 = X3 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_125_sup__idem,axiom,
% 2.61/1.48 ! [X3: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ X3 )
% 2.61/1.48 = X3 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_126_sup_Oidem,axiom,
% 2.61/1.48 ! [A2: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ A2 @ A2 )
% 2.61/1.48 = A2 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_127_sup_Oidem,axiom,
% 2.61/1.48 ! [A2: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ A2 @ A2 )
% 2.61/1.48 = A2 ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_128_comp__apply,axiom,
% 2.61/1.48 ( comp_a_b_c
% 2.61/1.48 = ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_129_comp__apply,axiom,
% 2.61/1.48 ( comp_c_b_a
% 2.61/1.48 = ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_130_comp__apply,axiom,
% 2.61/1.48 ( comp_a_b_a
% 2.61/1.48 = ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_131_comp__apply,axiom,
% 2.61/1.48 ( comp_b_b_a
% 2.61/1.48 = ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_132_sup__left__commute,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
% 2.61/1.48 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_133_sup__left__commute,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
% 2.61/1.48 = ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_134_sup_Oleft__commute,axiom,
% 2.61/1.48 ! [B2: set_a,A2: set_a,C: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
% 2.61/1.48 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_135_sup_Oleft__commute,axiom,
% 2.61/1.48 ! [B2: set_c,A2: set_c,C: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ B2 @ ( sup_sup_set_c @ A2 @ C ) )
% 2.61/1.48 = ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_136_sup__commute,axiom,
% 2.61/1.48 ( sup_sup_set_a
% 2.61/1.48 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_137_sup__commute,axiom,
% 2.61/1.48 ( sup_sup_set_c
% 2.61/1.48 = ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_138_sup_Ocommute,axiom,
% 2.61/1.48 ( sup_sup_set_a
% 2.61/1.48 = ( ^ [A3: set_a,B3: set_a] : ( sup_sup_set_a @ B3 @ A3 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_139_sup_Ocommute,axiom,
% 2.61/1.48 ( sup_sup_set_c
% 2.61/1.48 = ( ^ [A3: set_c,B3: set_c] : ( sup_sup_set_c @ B3 @ A3 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_140_sup__assoc,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
% 2.61/1.48 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_141_sup__assoc,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
% 2.61/1.48 = ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_142_sup_Oassoc,axiom,
% 2.61/1.48 ! [A2: set_a,B2: set_a,C: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
% 2.61/1.48 = ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_143_sup_Oassoc,axiom,
% 2.61/1.48 ! [A2: set_c,B2: set_c,C: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ ( sup_sup_set_c @ A2 @ B2 ) @ C )
% 2.61/1.48 = ( sup_sup_set_c @ A2 @ ( sup_sup_set_c @ B2 @ C ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_144_boolean__algebra__cancel_Osup2,axiom,
% 2.61/1.48 ! [B: set_a,K: set_a,B2: set_a,A2: set_a] :
% 2.61/1.48 ( ( B
% 2.61/1.48 = ( sup_sup_set_a @ K @ B2 ) )
% 2.61/1.48 => ( ( sup_sup_set_a @ A2 @ B )
% 2.61/1.48 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_145_boolean__algebra__cancel_Osup2,axiom,
% 2.61/1.48 ! [B: set_c,K: set_c,B2: set_c,A2: set_c] :
% 2.61/1.48 ( ( B
% 2.61/1.48 = ( sup_sup_set_c @ K @ B2 ) )
% 2.61/1.48 => ( ( sup_sup_set_c @ A2 @ B )
% 2.61/1.48 = ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_146_boolean__algebra__cancel_Osup1,axiom,
% 2.61/1.48 ! [A: set_a,K: set_a,A2: set_a,B2: set_a] :
% 2.61/1.48 ( ( A
% 2.61/1.48 = ( sup_sup_set_a @ K @ A2 ) )
% 2.61/1.48 => ( ( sup_sup_set_a @ A @ B2 )
% 2.61/1.48 = ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_147_boolean__algebra__cancel_Osup1,axiom,
% 2.61/1.48 ! [A: set_c,K: set_c,A2: set_c,B2: set_c] :
% 2.61/1.48 ( ( A
% 2.61/1.48 = ( sup_sup_set_c @ K @ A2 ) )
% 2.61/1.48 => ( ( sup_sup_set_c @ A @ B2 )
% 2.61/1.48 = ( sup_sup_set_c @ K @ ( sup_sup_set_c @ A2 @ B2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_148_inf__sup__aci_I5_J,axiom,
% 2.61/1.48 ( sup_sup_set_a
% 2.61/1.48 = ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_149_inf__sup__aci_I5_J,axiom,
% 2.61/1.48 ( sup_sup_set_c
% 2.61/1.48 = ( ^ [X2: set_c,Y: set_c] : ( sup_sup_set_c @ Y @ X2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_150_inf__sup__aci_I6_J,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z )
% 2.61/1.48 = ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_151_inf__sup__aci_I6_J,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ ( sup_sup_set_c @ X3 @ Y2 ) @ Z )
% 2.61/1.48 = ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_152_inf__sup__aci_I7_J,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a,Z: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z ) )
% 2.61/1.48 = ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_153_inf__sup__aci_I7_J,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c,Z: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ Y2 @ Z ) )
% 2.61/1.48 = ( sup_sup_set_c @ Y2 @ ( sup_sup_set_c @ X3 @ Z ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_154_inf__sup__aci_I8_J,axiom,
% 2.61/1.48 ! [X3: set_a,Y2: set_a] :
% 2.61/1.48 ( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
% 2.61/1.48 = ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_155_inf__sup__aci_I8_J,axiom,
% 2.61/1.48 ! [X3: set_c,Y2: set_c] :
% 2.61/1.48 ( ( sup_sup_set_c @ X3 @ ( sup_sup_set_c @ X3 @ Y2 ) )
% 2.61/1.48 = ( sup_sup_set_c @ X3 @ Y2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_156_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: c > b,F: a > c,V: c > a] :
% 2.61/1.48 ( ( comp_c_b_c @ G @ ( comp_a_c_c @ F @ V ) )
% 2.61/1.48 = ( comp_a_b_c @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_157_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: b > b,F: a > b,V: c > a] :
% 2.61/1.48 ( ( comp_b_b_c @ G @ ( comp_a_b_c @ F @ V ) )
% 2.61/1.48 = ( comp_a_b_c @ ( comp_b_b_a @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_158_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: a > b,F: c > a,V: c > c] :
% 2.61/1.48 ( ( comp_a_b_c @ G @ ( comp_c_a_c @ F @ V ) )
% 2.61/1.48 = ( comp_c_b_c @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_159_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: a > b,F: a > a,V: c > a] :
% 2.61/1.48 ( ( comp_a_b_c @ G @ ( comp_a_a_c @ F @ V ) )
% 2.61/1.48 = ( comp_a_b_c @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_160_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: c > b,F: c > c,V: a > c] :
% 2.61/1.48 ( ( comp_c_b_a @ G @ ( comp_c_c_a @ F @ V ) )
% 2.61/1.48 = ( comp_c_b_a @ ( comp_c_b_c @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_161_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: c > b,F: a > c,V: a > a] :
% 2.61/1.48 ( ( comp_c_b_a @ G @ ( comp_a_c_a @ F @ V ) )
% 2.61/1.48 = ( comp_a_b_a @ ( comp_c_b_a @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_162_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: c > b,F: b > c,V: a > b] :
% 2.61/1.48 ( ( comp_c_b_a @ G @ ( comp_b_c_a @ F @ V ) )
% 2.61/1.48 = ( comp_b_b_a @ ( comp_c_b_b @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_163_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: a > b,F: c > a,V: a > c] :
% 2.61/1.48 ( ( comp_a_b_a @ G @ ( comp_c_a_a @ F @ V ) )
% 2.61/1.48 = ( comp_c_b_a @ ( comp_a_b_c @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_164_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: a > b,F: a > a,V: a > a] :
% 2.61/1.48 ( ( comp_a_b_a @ G @ ( comp_a_a_a @ F @ V ) )
% 2.61/1.48 = ( comp_a_b_a @ ( comp_a_b_a @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_165_fun_Omap__comp,axiom,
% 2.61/1.48 ! [G: a > b,F: b > a,V: a > b] :
% 2.61/1.48 ( ( comp_a_b_a @ G @ ( comp_b_a_a @ F @ V ) )
% 2.61/1.48 = ( comp_b_b_a @ ( comp_a_b_b @ G @ F ) @ V ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_166_comp__eq__dest__lhs,axiom,
% 2.61/1.48 ! [A2: a > b,B2: c > a,C: c > b,V: c] :
% 2.61/1.48 ( ( ( comp_a_b_c @ A2 @ B2 )
% 2.61/1.48 = C )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ V ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_167_comp__eq__dest__lhs,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = C )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ V ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_168_comp__eq__dest__lhs,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = C )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ V ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_169_comp__eq__dest__lhs,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = C )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ V ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_170_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: a > b,B2: c > a,C: a > b,D: c > a] :
% 2.61/1.48 ( ( ( comp_a_b_c @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_c @ C @ D ) )
% 2.61/1.48 => ! [V2: c] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_171_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: c > b,D: a > c] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_172_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: a > b,D: a > a] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_173_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: b > b,D: a > b] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_174_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: c > b,D: a > c] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_175_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: a > b,D: a > a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_176_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: b > b,D: a > b] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_177_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: c > b,D: a > c] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_178_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: a > b,D: a > a] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_179_comp__eq__elim,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: b > b,D: a > b] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ! [V2: a] :
% 2.61/1.48 ( ( A2 @ ( B2 @ V2 ) )
% 2.61/1.48 = ( C @ ( D @ V2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_180_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: a > b,B2: c > a,C: a > b,D: c > a,V: c] :
% 2.61/1.48 ( ( ( comp_a_b_c @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_c @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_181_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: c > b,D: a > c,V: a] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_182_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: a > b,D: a > a,V: a] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_183_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: c > b,B2: a > c,C: b > b,D: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_c_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_184_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: c > b,D: a > c,V: a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_185_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: a > b,D: a > a,V: a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_186_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: a > b,B2: a > a,C: b > b,D: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_a_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_187_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: c > b,D: a > c,V: a] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_c_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_188_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: a > b,D: a > a,V: a] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_a_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_189_comp__eq__dest,axiom,
% 2.61/1.48 ! [A2: b > b,B2: a > b,C: b > b,D: a > b,V: a] :
% 2.61/1.48 ( ( ( comp_b_b_a @ A2 @ B2 )
% 2.61/1.48 = ( comp_b_b_a @ C @ D ) )
% 2.61/1.48 => ( ( A2 @ ( B2 @ V ) )
% 2.61/1.48 = ( C @ ( D @ V ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_190_comp__assoc,axiom,
% 2.61/1.48 ! [F: a > b,G: c > a,H: c > c] :
% 2.61/1.48 ( ( comp_c_b_c @ ( comp_a_b_c @ F @ G ) @ H )
% 2.61/1.48 = ( comp_a_b_c @ F @ ( comp_c_a_c @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_191_comp__assoc,axiom,
% 2.61/1.48 ! [F: c > b,G: a > c,H: c > a] :
% 2.61/1.48 ( ( comp_a_b_c @ ( comp_c_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_c_b_c @ F @ ( comp_a_c_c @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_192_comp__assoc,axiom,
% 2.61/1.48 ! [F: a > b,G: a > a,H: c > a] :
% 2.61/1.48 ( ( comp_a_b_c @ ( comp_a_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_a_b_c @ F @ ( comp_a_a_c @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_193_comp__assoc,axiom,
% 2.61/1.48 ! [F: b > b,G: a > b,H: c > a] :
% 2.61/1.48 ( ( comp_a_b_c @ ( comp_b_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_b_b_c @ F @ ( comp_a_b_c @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_194_comp__assoc,axiom,
% 2.61/1.48 ! [F: c > b,G: c > c,H: a > c] :
% 2.61/1.48 ( ( comp_c_b_a @ ( comp_c_b_c @ F @ G ) @ H )
% 2.61/1.48 = ( comp_c_b_a @ F @ ( comp_c_c_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_195_comp__assoc,axiom,
% 2.61/1.48 ! [F: b > b,G: c > b,H: a > c] :
% 2.61/1.48 ( ( comp_c_b_a @ ( comp_b_b_c @ F @ G ) @ H )
% 2.61/1.48 = ( comp_b_b_a @ F @ ( comp_c_b_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_196_comp__assoc,axiom,
% 2.61/1.48 ! [F: a > b,G: c > a,H: a > c] :
% 2.61/1.48 ( ( comp_c_b_a @ ( comp_a_b_c @ F @ G ) @ H )
% 2.61/1.48 = ( comp_a_b_a @ F @ ( comp_c_a_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_197_comp__assoc,axiom,
% 2.61/1.48 ! [F: c > b,G: a > c,H: a > a] :
% 2.61/1.48 ( ( comp_a_b_a @ ( comp_c_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_c_b_a @ F @ ( comp_a_c_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_198_comp__assoc,axiom,
% 2.61/1.48 ! [F: a > b,G: a > a,H: a > a] :
% 2.61/1.48 ( ( comp_a_b_a @ ( comp_a_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_a_b_a @ F @ ( comp_a_a_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_199_comp__assoc,axiom,
% 2.61/1.48 ! [F: b > b,G: a > b,H: a > a] :
% 2.61/1.48 ( ( comp_a_b_a @ ( comp_b_b_a @ F @ G ) @ H )
% 2.61/1.48 = ( comp_b_b_a @ F @ ( comp_a_b_a @ G @ H ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_200_comp__def,axiom,
% 2.61/1.48 ( comp_a_b_c
% 2.61/1.48 = ( ^ [F3: a > b,G2: c > a,X2: c] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_201_comp__def,axiom,
% 2.61/1.48 ( comp_c_b_a
% 2.61/1.48 = ( ^ [F3: c > b,G2: a > c,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_202_comp__def,axiom,
% 2.61/1.48 ( comp_a_b_a
% 2.61/1.48 = ( ^ [F3: a > b,G2: a > a,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_203_comp__def,axiom,
% 2.61/1.48 ( comp_b_b_a
% 2.61/1.48 = ( ^ [F3: b > b,G2: a > b,X2: a] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_204_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_c,G: b > c,F: c > b] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_c_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_205_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_c,G: a > c,F: c > a] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_c_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_206_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_b,G: b > b,F: b > b] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( member_b @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_b_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_207_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_a,G: b > a,F: a > b] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_a_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_208_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_a,G: c > a,F: a > c] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_a_c @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_209_inj__on__inverseI,axiom,
% 2.61/1.48 ! [A: set_a,G: a > a,F: a > a] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( G @ ( F @ X ) )
% 2.61/1.48 = X ) )
% 2.61/1.48 => ( inj_on_a_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_210_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_211_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_212_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_b @ X3 @ A )
% 2.61/1.48 => ( ( member_b @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_213_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_214_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_215_inj__on__contraD,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( X3 != Y2 )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( F @ X3 )
% 2.61/1.48 != ( F @ Y2 ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_216_mem__Collect__eq,axiom,
% 2.61/1.48 ! [A2: a,P: a > $o] :
% 2.61/1.48 ( ( member_a @ A2 @ ( collect_a @ P ) )
% 2.61/1.48 = ( P @ A2 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_217_Collect__mem__eq,axiom,
% 2.61/1.48 ! [A: set_a] :
% 2.61/1.48 ( ( collect_a
% 2.61/1.48 @ ^ [X2: a] : ( member_a @ X2 @ A ) )
% 2.61/1.48 = A ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_218_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_219_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_220_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ A )
% 2.61/1.48 => ( ( member_b @ X3 @ A )
% 2.61/1.48 => ( ( member_b @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_221_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_222_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ A )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_223_inj__on__eq__iff,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_224_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_c,F: c > b,G: c > b] :
% 2.61/1.48 ( ! [A4: c] :
% 2.61/1.48 ( ( member_c @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 = ( inj_on_c_b @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_225_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_c,F: c > a,G: c > a] :
% 2.61/1.48 ( ! [A4: c] :
% 2.61/1.48 ( ( member_c @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 = ( inj_on_c_a @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_226_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_b,F: b > b,G: b > b] :
% 2.61/1.48 ( ! [A4: b] :
% 2.61/1.48 ( ( member_b @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_b_b @ F @ A )
% 2.61/1.48 = ( inj_on_b_b @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_227_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_a,F: a > b,G: a > b] :
% 2.61/1.48 ( ! [A4: a] :
% 2.61/1.48 ( ( member_a @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 = ( inj_on_a_b @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_228_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_a,F: a > c,G: a > c] :
% 2.61/1.48 ( ! [A4: a] :
% 2.61/1.48 ( ( member_a @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_a_c @ F @ A )
% 2.61/1.48 = ( inj_on_a_c @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_229_inj__on__cong,axiom,
% 2.61/1.48 ! [A: set_a,F: a > a,G: a > a] :
% 2.61/1.48 ( ! [A4: a] :
% 2.61/1.48 ( ( member_a @ A4 @ A )
% 2.61/1.48 => ( ( F @ A4 )
% 2.61/1.48 = ( G @ A4 ) ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 = ( inj_on_a_a @ G @ A ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_230_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_a_b
% 2.61/1.48 = ( ^ [F3: a > b,A5: set_a] :
% 2.61/1.48 ! [X2: a] :
% 2.61/1.48 ( ( member_a @ X2 @ A5 )
% 2.61/1.48 => ! [Y: a] :
% 2.61/1.48 ( ( member_a @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_231_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_c_b
% 2.61/1.48 = ( ^ [F3: c > b,A5: set_c] :
% 2.61/1.48 ! [X2: c] :
% 2.61/1.48 ( ( member_c @ X2 @ A5 )
% 2.61/1.48 => ! [Y: c] :
% 2.61/1.48 ( ( member_c @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_232_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_c_a
% 2.61/1.48 = ( ^ [F3: c > a,A5: set_c] :
% 2.61/1.48 ! [X2: c] :
% 2.61/1.48 ( ( member_c @ X2 @ A5 )
% 2.61/1.48 => ! [Y: c] :
% 2.61/1.48 ( ( member_c @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_233_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_b_b
% 2.61/1.48 = ( ^ [F3: b > b,A5: set_b] :
% 2.61/1.48 ! [X2: b] :
% 2.61/1.48 ( ( member_b @ X2 @ A5 )
% 2.61/1.48 => ! [Y: b] :
% 2.61/1.48 ( ( member_b @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_234_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_a_c
% 2.61/1.48 = ( ^ [F3: a > c,A5: set_a] :
% 2.61/1.48 ! [X2: a] :
% 2.61/1.48 ( ( member_a @ X2 @ A5 )
% 2.61/1.48 => ! [Y: a] :
% 2.61/1.48 ( ( member_a @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_235_inj__on__def,axiom,
% 2.61/1.48 ( inj_on_a_a
% 2.61/1.48 = ( ^ [F3: a > a,A5: set_a] :
% 2.61/1.48 ! [X2: a] :
% 2.61/1.48 ( ( member_a @ X2 @ A5 )
% 2.61/1.48 => ! [Y: a] :
% 2.61/1.48 ( ( member_a @ Y @ A5 )
% 2.61/1.48 => ( ( ( F3 @ X2 )
% 2.61/1.48 = ( F3 @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_236_inj__onI,axiom,
% 2.61/1.48 ! [A: set_c,F: c > b] :
% 2.61/1.48 ( ! [X: c,Y3: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ( ( member_c @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_c_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_237_inj__onI,axiom,
% 2.61/1.48 ! [A: set_c,F: c > a] :
% 2.61/1.48 ( ! [X: c,Y3: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ( ( member_c @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_c_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_238_inj__onI,axiom,
% 2.61/1.48 ! [A: set_b,F: b > b] :
% 2.61/1.48 ( ! [X: b,Y3: b] :
% 2.61/1.48 ( ( member_b @ X @ A )
% 2.61/1.48 => ( ( member_b @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_b_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_239_inj__onI,axiom,
% 2.61/1.48 ! [A: set_a,F: a > b] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( member_a @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_a_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_240_inj__onI,axiom,
% 2.61/1.48 ! [A: set_a,F: a > c] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( member_a @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_a_c @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_241_inj__onI,axiom,
% 2.61/1.48 ! [A: set_a,F: a > a] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ( ( member_a @ Y3 @ A )
% 2.61/1.48 => ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) ) ) )
% 2.61/1.48 => ( inj_on_a_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_242_inj__onD,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_243_inj__onD,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_c @ X3 @ A )
% 2.61/1.48 => ( ( member_c @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_244_inj__onD,axiom,
% 2.61/1.48 ! [F: b > b,A: set_b,X3: b,Y2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_b @ X3 @ A )
% 2.61/1.48 => ( ( member_b @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_245_inj__onD,axiom,
% 2.61/1.48 ! [F: a > b,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_246_inj__onD,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_247_inj__onD,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( ( member_a @ X3 @ A )
% 2.61/1.48 => ( ( member_a @ Y2 @ A )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_248_surj__def,axiom,
% 2.61/1.48 ! [F: c > c] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 = ( ! [Y: c] :
% 2.61/1.48 ? [X2: c] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_249_surj__def,axiom,
% 2.61/1.48 ! [F: c > a] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 = ( ! [Y: a] :
% 2.61/1.48 ? [X2: c] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_250_surj__def,axiom,
% 2.61/1.48 ! [F: c > b] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 = ( ! [Y: b] :
% 2.61/1.48 ? [X2: c] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_251_surj__def,axiom,
% 2.61/1.48 ! [F: a > c] :
% 2.61/1.48 ( ( ( image_a_c @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 = ( ! [Y: c] :
% 2.61/1.48 ? [X2: a] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_252_surj__def,axiom,
% 2.61/1.48 ! [F: a > a] :
% 2.61/1.48 ( ( ( image_a_a @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 = ( ! [Y: a] :
% 2.61/1.48 ? [X2: a] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_253_surj__def,axiom,
% 2.61/1.48 ! [F: a > b] :
% 2.61/1.48 ( ( ( image_a_b @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 = ( ! [Y: b] :
% 2.61/1.48 ? [X2: a] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_254_surj__def,axiom,
% 2.61/1.48 ! [F: b > c] :
% 2.61/1.48 ( ( ( image_b_c @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 = ( ! [Y: c] :
% 2.61/1.48 ? [X2: b] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_255_surj__def,axiom,
% 2.61/1.48 ! [F: b > a] :
% 2.61/1.48 ( ( ( image_b_a @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 = ( ! [Y: a] :
% 2.61/1.48 ? [X2: b] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_256_surj__def,axiom,
% 2.61/1.48 ! [F: b > b] :
% 2.61/1.48 ( ( ( image_b_b @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 = ( ! [Y: b] :
% 2.61/1.48 ? [X2: b] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_257_surj__def,axiom,
% 2.61/1.48 ! [F: c > c > a] :
% 2.61/1.48 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c_a )
% 2.61/1.48 = ( ! [Y: c > a] :
% 2.61/1.48 ? [X2: c] :
% 2.61/1.48 ( Y
% 2.61/1.48 = ( F @ X2 ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_258_surjI,axiom,
% 2.61/1.48 ! [G: c > c,F: c > c] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_c_c @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_259_surjI,axiom,
% 2.61/1.48 ! [G: c > a,F: a > c] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_c_a @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_260_surjI,axiom,
% 2.61/1.48 ! [G: c > b,F: b > c] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_c_b @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_b ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_261_surjI,axiom,
% 2.61/1.48 ! [G: a > c,F: c > a] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_a_c @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_262_surjI,axiom,
% 2.61/1.48 ! [G: a > a,F: a > a] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_a_a @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_263_surjI,axiom,
% 2.61/1.48 ! [G: a > b,F: b > a] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_a_b @ G @ top_top_set_a )
% 2.61/1.48 = top_top_set_b ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_264_surjI,axiom,
% 2.61/1.48 ! [G: b > c,F: c > b] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_b_c @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_265_surjI,axiom,
% 2.61/1.48 ! [G: b > a,F: a > b] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_b_a @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_266_surjI,axiom,
% 2.61/1.48 ! [G: b > b,F: b > b] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_b_b @ G @ top_top_set_b )
% 2.61/1.48 = top_top_set_b ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_267_surjI,axiom,
% 2.61/1.48 ! [G: c > c > a,F: ( c > a ) > c] :
% 2.61/1.48 ( ! [X: c > a] :
% 2.61/1.48 ( ( G @ ( F @ X ) )
% 2.61/1.48 = X )
% 2.61/1.48 => ( ( image_c_c_a @ G @ top_top_set_c )
% 2.61/1.48 = top_top_set_c_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_268_surjE,axiom,
% 2.61/1.48 ! [F: c > c,Y2: c] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ~ ! [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_269_surjE,axiom,
% 2.61/1.48 ! [F: c > a,Y2: a] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ~ ! [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_270_surjE,axiom,
% 2.61/1.48 ! [F: c > b,Y2: b] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ~ ! [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_271_surjE,axiom,
% 2.61/1.48 ! [F: a > c,Y2: c] :
% 2.61/1.48 ( ( ( image_a_c @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ~ ! [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_272_surjE,axiom,
% 2.61/1.48 ! [F: a > a,Y2: a] :
% 2.61/1.48 ( ( ( image_a_a @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ~ ! [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_273_surjE,axiom,
% 2.61/1.48 ! [F: a > b,Y2: b] :
% 2.61/1.48 ( ( ( image_a_b @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ~ ! [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_274_surjE,axiom,
% 2.61/1.48 ! [F: b > c,Y2: c] :
% 2.61/1.48 ( ( ( image_b_c @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ~ ! [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_275_surjE,axiom,
% 2.61/1.48 ! [F: b > a,Y2: a] :
% 2.61/1.48 ( ( ( image_b_a @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ~ ! [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_276_surjE,axiom,
% 2.61/1.48 ! [F: b > b,Y2: b] :
% 2.61/1.48 ( ( ( image_b_b @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ~ ! [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_277_surjE,axiom,
% 2.61/1.48 ! [F: c > c > a,Y2: c > a] :
% 2.61/1.48 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c_a )
% 2.61/1.48 => ~ ! [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 != ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_278_surjD,axiom,
% 2.61/1.48 ! [F: c > c,Y2: c] :
% 2.61/1.48 ( ( ( image_c_c @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ? [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_279_surjD,axiom,
% 2.61/1.48 ! [F: c > a,Y2: a] :
% 2.61/1.48 ( ( ( image_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ? [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_280_surjD,axiom,
% 2.61/1.48 ! [F: c > b,Y2: b] :
% 2.61/1.48 ( ( ( image_c_b @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ? [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_281_surjD,axiom,
% 2.61/1.48 ! [F: a > c,Y2: c] :
% 2.61/1.48 ( ( ( image_a_c @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ? [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_282_surjD,axiom,
% 2.61/1.48 ! [F: a > a,Y2: a] :
% 2.61/1.48 ( ( ( image_a_a @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ? [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_283_surjD,axiom,
% 2.61/1.48 ! [F: a > b,Y2: b] :
% 2.61/1.48 ( ( ( image_a_b @ F @ top_top_set_a )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ? [X: a] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_284_surjD,axiom,
% 2.61/1.48 ! [F: b > c,Y2: c] :
% 2.61/1.48 ( ( ( image_b_c @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_c )
% 2.61/1.48 => ? [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_285_surjD,axiom,
% 2.61/1.48 ! [F: b > a,Y2: a] :
% 2.61/1.48 ( ( ( image_b_a @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_a )
% 2.61/1.48 => ? [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_286_surjD,axiom,
% 2.61/1.48 ! [F: b > b,Y2: b] :
% 2.61/1.48 ( ( ( image_b_b @ F @ top_top_set_b )
% 2.61/1.48 = top_top_set_b )
% 2.61/1.48 => ? [X: b] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_287_surjD,axiom,
% 2.61/1.48 ! [F: c > c > a,Y2: c > a] :
% 2.61/1.48 ( ( ( image_c_c_a @ F @ top_top_set_c )
% 2.61/1.48 = top_top_set_c_a )
% 2.61/1.48 => ? [X: c] :
% 2.61/1.48 ( Y2
% 2.61/1.48 = ( F @ X ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_288_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > c,A: set_c,G: a > c,B: set_a,H: c > b] :
% 2.61/1.48 ( ( ( image_c_c @ F @ A )
% 2.61/1.48 = ( image_a_c @ G @ B ) )
% 2.61/1.48 => ( ( image_c_b @ ( comp_c_b_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_289_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,G: c > c,B: set_c,H: c > b] :
% 2.61/1.48 ( ( ( image_a_c @ F @ A )
% 2.61/1.48 = ( image_c_c @ G @ B ) )
% 2.61/1.48 => ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_b @ ( comp_c_b_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_290_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: a > c,A: set_a,G: a > c,B: set_a,H: c > b] :
% 2.61/1.48 ( ( ( image_a_c @ F @ A )
% 2.61/1.48 = ( image_a_c @ G @ B ) )
% 2.61/1.48 => ( ( image_a_b @ ( comp_c_b_a @ H @ F ) @ A )
% 2.61/1.48 = ( image_a_b @ ( comp_c_b_a @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_291_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,G: a > a,B: set_a,H: a > b] :
% 2.61/1.48 ( ( ( image_a_a @ F @ A )
% 2.61/1.48 = ( image_a_a @ G @ B ) )
% 2.61/1.48 => ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
% 2.61/1.48 = ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_292_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: a > a,A: set_a,G: c > a,B: set_c,H: a > b] :
% 2.61/1.48 ( ( ( image_a_a @ F @ A )
% 2.61/1.48 = ( image_c_a @ G @ B ) )
% 2.61/1.48 => ( ( image_a_b @ ( comp_a_b_a @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_293_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,G: a > a,B: set_a,H: a > b] :
% 2.61/1.48 ( ( ( image_c_a @ F @ A )
% 2.61/1.48 = ( image_a_a @ G @ B ) )
% 2.61/1.48 => ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_a_b @ ( comp_a_b_a @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_294_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > a] :
% 2.61/1.48 ( ( ( image_c_a @ F @ A )
% 2.61/1.48 = ( image_c_a @ G @ B ) )
% 2.61/1.48 => ( ( image_c_a @ ( comp_a_a_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_a @ ( comp_a_a_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_295_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > a,A: set_c,G: c > a,B: set_c,H: a > b] :
% 2.61/1.48 ( ( ( image_c_a @ F @ A )
% 2.61/1.48 = ( image_c_a @ G @ B ) )
% 2.61/1.48 => ( ( image_c_b @ ( comp_a_b_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_b @ ( comp_a_b_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_296_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > a] :
% 2.61/1.48 ( ( ( image_c_b @ F @ A )
% 2.61/1.48 = ( image_c_b @ G @ B ) )
% 2.61/1.48 => ( ( image_c_a @ ( comp_b_a_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_a @ ( comp_b_a_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_297_image__eq__imp__comp,axiom,
% 2.61/1.48 ! [F: c > b,A: set_c,G: c > b,B: set_c,H: b > b] :
% 2.61/1.48 ( ( ( image_c_b @ F @ A )
% 2.61/1.48 = ( image_c_b @ G @ B ) )
% 2.61/1.48 => ( ( image_c_b @ ( comp_b_b_c @ H @ F ) @ A )
% 2.61/1.48 = ( image_c_b @ ( comp_b_b_c @ H @ G ) @ B ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_298_image__comp,axiom,
% 2.61/1.48 ! [F: a > a,G: c > a,R: set_c] :
% 2.61/1.48 ( ( image_a_a @ F @ ( image_c_a @ G @ R ) )
% 2.61/1.48 = ( image_c_a @ ( comp_a_a_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_299_image__comp,axiom,
% 2.61/1.48 ! [F: b > a,G: c > b,R: set_c] :
% 2.61/1.48 ( ( image_b_a @ F @ ( image_c_b @ G @ R ) )
% 2.61/1.48 = ( image_c_a @ ( comp_b_a_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_300_image__comp,axiom,
% 2.61/1.48 ! [F: b > b,G: c > b,R: set_c] :
% 2.61/1.48 ( ( image_b_b @ F @ ( image_c_b @ G @ R ) )
% 2.61/1.48 = ( image_c_b @ ( comp_b_b_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_301_image__comp,axiom,
% 2.61/1.48 ! [F: b > b,G: a > b,R: set_a] :
% 2.61/1.48 ( ( image_b_b @ F @ ( image_a_b @ G @ R ) )
% 2.61/1.48 = ( image_a_b @ ( comp_b_b_a @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_302_image__comp,axiom,
% 2.61/1.48 ! [F: c > a,G: c > c,R: set_c] :
% 2.61/1.48 ( ( image_c_a @ F @ ( image_c_c @ G @ R ) )
% 2.61/1.48 = ( image_c_a @ ( comp_c_a_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_303_image__comp,axiom,
% 2.61/1.48 ! [F: c > b,G: c > c,R: set_c] :
% 2.61/1.48 ( ( image_c_b @ F @ ( image_c_c @ G @ R ) )
% 2.61/1.48 = ( image_c_b @ ( comp_c_b_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_304_image__comp,axiom,
% 2.61/1.48 ! [F: c > b,G: a > c,R: set_a] :
% 2.61/1.48 ( ( image_c_b @ F @ ( image_a_c @ G @ R ) )
% 2.61/1.48 = ( image_a_b @ ( comp_c_b_a @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_305_image__comp,axiom,
% 2.61/1.48 ! [F: a > b,G: a > a,R: set_a] :
% 2.61/1.48 ( ( image_a_b @ F @ ( image_a_a @ G @ R ) )
% 2.61/1.48 = ( image_a_b @ ( comp_a_b_a @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_306_image__comp,axiom,
% 2.61/1.48 ! [F: a > b,G: c > a,R: set_c] :
% 2.61/1.48 ( ( image_a_b @ F @ ( image_c_a @ G @ R ) )
% 2.61/1.48 = ( image_c_b @ ( comp_a_b_c @ F @ G ) @ R ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_307_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_c,G: c > b,F: c > c] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ! [Xa: c] :
% 2.61/1.48 ( ( member_c @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_b @ G @ ( image_c_c @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_b @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_308_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_c,G: c > a,F: c > c] :
% 2.61/1.48 ( ! [X: c] :
% 2.61/1.48 ( ( member_c @ X @ A )
% 2.61/1.48 => ! [Xa: c] :
% 2.61/1.48 ( ( member_c @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_c_c @ F @ A )
% 2.61/1.48 => ( ( inj_on_c_a @ G @ ( image_c_c @ F @ A ) )
% 2.61/1.48 = ( inj_on_c_a @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_309_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_b,G: b > b,F: b > b] :
% 2.61/1.48 ( ! [X: b] :
% 2.61/1.48 ( ( member_b @ X @ A )
% 2.61/1.48 => ! [Xa: b] :
% 2.61/1.48 ( ( member_b @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_b_b @ F @ A )
% 2.61/1.48 => ( ( inj_on_b_b @ G @ ( image_b_b @ F @ A ) )
% 2.61/1.48 = ( inj_on_b_b @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_310_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_a,G: a > b,F: a > a] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ! [Xa: a] :
% 2.61/1.48 ( ( member_a @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_b @ G @ ( image_a_a @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_b @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_311_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_a,G: a > c,F: a > a] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ! [Xa: a] :
% 2.61/1.48 ( ( member_a @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_c @ G @ ( image_a_a @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_c @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_312_inj__on__image__iff,axiom,
% 2.61/1.48 ! [A: set_a,G: a > a,F: a > a] :
% 2.61/1.48 ( ! [X: a] :
% 2.61/1.48 ( ( member_a @ X @ A )
% 2.61/1.48 => ! [Xa: a] :
% 2.61/1.48 ( ( member_a @ Xa @ A )
% 2.61/1.48 => ( ( ( G @ ( F @ X ) )
% 2.61/1.48 = ( G @ ( F @ Xa ) ) )
% 2.61/1.48 = ( ( G @ X )
% 2.61/1.48 = ( G @ Xa ) ) ) ) )
% 2.61/1.48 => ( ( inj_on_a_a @ F @ A )
% 2.61/1.48 => ( ( inj_on_a_a @ G @ ( image_a_a @ F @ A ) )
% 2.61/1.48 = ( inj_on_a_a @ G @ A ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_313_inj__def,axiom,
% 2.61/1.48 ! [F: c > b] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 = ( ! [X2: c,Y: c] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_314_inj__def,axiom,
% 2.61/1.48 ! [F: c > a] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 = ( ! [X2: c,Y: c] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_315_inj__def,axiom,
% 2.61/1.48 ! [F: a > b] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 = ( ! [X2: a,Y: a] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_316_inj__def,axiom,
% 2.61/1.48 ! [F: a > c] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 = ( ! [X2: a,Y: a] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_317_inj__def,axiom,
% 2.61/1.48 ! [F: a > a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 = ( ! [X2: a,Y: a] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_318_inj__def,axiom,
% 2.61/1.48 ! [F: b > b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 = ( ! [X2: b,Y: b] :
% 2.61/1.48 ( ( ( F @ X2 )
% 2.61/1.48 = ( F @ Y ) )
% 2.61/1.48 => ( X2 = Y ) ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_319_inj__eq,axiom,
% 2.61/1.48 ! [F: c > b,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_320_inj__eq,axiom,
% 2.61/1.48 ! [F: c > a,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_321_inj__eq,axiom,
% 2.61/1.48 ! [F: a > b,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_322_inj__eq,axiom,
% 2.61/1.48 ! [F: a > c,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_323_inj__eq,axiom,
% 2.61/1.48 ! [F: a > a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_324_inj__eq,axiom,
% 2.61/1.48 ! [F: b > b,X3: b,Y2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 = ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_325_injI,axiom,
% 2.61/1.48 ! [F: c > b] :
% 2.61/1.48 ( ! [X: c,Y3: c] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_c_b @ F @ top_top_set_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_326_injI,axiom,
% 2.61/1.48 ! [F: c > a] :
% 2.61/1.48 ( ! [X: c,Y3: c] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_c_a @ F @ top_top_set_c ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_327_injI,axiom,
% 2.61/1.48 ! [F: a > b] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_a_b @ F @ top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_328_injI,axiom,
% 2.61/1.48 ! [F: a > c] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_a_c @ F @ top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_329_injI,axiom,
% 2.61/1.48 ! [F: a > a] :
% 2.61/1.48 ( ! [X: a,Y3: a] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_a_a @ F @ top_top_set_a ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_330_injI,axiom,
% 2.61/1.48 ! [F: b > b] :
% 2.61/1.48 ( ! [X: b,Y3: b] :
% 2.61/1.48 ( ( ( F @ X )
% 2.61/1.48 = ( F @ Y3 ) )
% 2.61/1.48 => ( X = Y3 ) )
% 2.61/1.48 => ( inj_on_b_b @ F @ top_top_set_b ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_331_injD,axiom,
% 2.61/1.48 ! [F: c > b,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_b @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_332_injD,axiom,
% 2.61/1.48 ! [F: c > a,X3: c,Y2: c] :
% 2.61/1.48 ( ( inj_on_c_a @ F @ top_top_set_c )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_333_injD,axiom,
% 2.61/1.48 ! [F: a > b,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_b @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_334_injD,axiom,
% 2.61/1.48 ! [F: a > c,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_c @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_335_injD,axiom,
% 2.61/1.48 ! [F: a > a,X3: a,Y2: a] :
% 2.61/1.48 ( ( inj_on_a_a @ F @ top_top_set_a )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_336_injD,axiom,
% 2.61/1.48 ! [F: b > b,X3: b,Y2: b] :
% 2.61/1.48 ( ( inj_on_b_b @ F @ top_top_set_b )
% 2.61/1.48 => ( ( ( F @ X3 )
% 2.61/1.48 = ( F @ Y2 ) )
% 2.61/1.48 => ( X3 = Y2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_337_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: b > b,F: a > b,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_b_b_a @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_338_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: c > b,F: a > c,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_c_b_a @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_c @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_339_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: a > b,F: a > a,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_b @ ( comp_a_b_a @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_340_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: b > b,F: c > b,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ ( comp_b_b_c @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_341_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: a > b,F: c > a,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_b @ ( comp_a_b_c @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_342_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: b > a,F: c > b,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ ( comp_b_a_c @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_343_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: a > a,F: c > a,A: set_c] :
% 2.61/1.48 ( ( inj_on_c_a @ ( comp_a_a_c @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_c_a @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_344_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: b > b,F: b > b,A: set_b] :
% 2.61/1.48 ( ( inj_on_b_b @ ( comp_b_b_b @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_b_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_345_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: b > c,F: a > b,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_c @ ( comp_b_c_a @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_b @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_346_inj__on__imageI2,axiom,
% 2.61/1.48 ! [F2: c > c,F: a > c,A: set_a] :
% 2.61/1.48 ( ( inj_on_a_c @ ( comp_c_c_a @ F2 @ F ) @ A )
% 2.61/1.48 => ( inj_on_a_c @ F @ A ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_347_fun_Oinj__map__strong,axiom,
% 2.61/1.48 ! [X3: c > a,Xa2: c > a,F: a > b,Fa: a > b] :
% 2.61/1.48 ( ! [Z2: a,Za: a] :
% 2.61/1.48 ( ( member_a @ Z2 @ ( image_c_a @ X3 @ top_top_set_c ) )
% 2.61/1.48 => ( ( member_a @ Za @ ( image_c_a @ Xa2 @ top_top_set_c ) )
% 2.61/1.48 => ( ( ( F @ Z2 )
% 2.61/1.48 = ( Fa @ Za ) )
% 2.61/1.48 => ( Z2 = Za ) ) ) )
% 2.61/1.48 => ( ( ( comp_a_b_c @ F @ X3 )
% 2.61/1.48 = ( comp_a_b_c @ Fa @ Xa2 ) )
% 2.61/1.48 => ( X3 = Xa2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_348_fun_Oinj__map__strong,axiom,
% 2.61/1.48 ! [X3: a > c,Xa2: a > c,F: c > b,Fa: c > b] :
% 2.61/1.48 ( ! [Z2: c,Za: c] :
% 2.61/1.48 ( ( member_c @ Z2 @ ( image_a_c @ X3 @ top_top_set_a ) )
% 2.61/1.48 => ( ( member_c @ Za @ ( image_a_c @ Xa2 @ top_top_set_a ) )
% 2.61/1.48 => ( ( ( F @ Z2 )
% 2.61/1.48 = ( Fa @ Za ) )
% 2.61/1.48 => ( Z2 = Za ) ) ) )
% 2.61/1.48 => ( ( ( comp_c_b_a @ F @ X3 )
% 2.61/1.48 = ( comp_c_b_a @ Fa @ Xa2 ) )
% 2.61/1.48 => ( X3 = Xa2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_349_fun_Oinj__map__strong,axiom,
% 2.61/1.48 ! [X3: a > b,Xa2: a > b,F: b > b,Fa: b > b] :
% 2.61/1.48 ( ! [Z2: b,Za: b] :
% 2.61/1.48 ( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
% 2.61/1.48 => ( ( member_b @ Za @ ( image_a_b @ Xa2 @ top_top_set_a ) )
% 2.61/1.48 => ( ( ( F @ Z2 )
% 2.61/1.48 = ( Fa @ Za ) )
% 2.61/1.48 => ( Z2 = Za ) ) ) )
% 2.61/1.48 => ( ( ( comp_b_b_a @ F @ X3 )
% 2.61/1.48 = ( comp_b_b_a @ Fa @ Xa2 ) )
% 2.61/1.48 => ( X3 = Xa2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_350_fun_Oinj__map__strong,axiom,
% 2.61/1.48 ! [X3: a > a,Xa2: a > a,F: a > b,Fa: a > b] :
% 2.61/1.48 ( ! [Z2: a,Za: a] :
% 2.61/1.48 ( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
% 2.61/1.48 => ( ( member_a @ Za @ ( image_a_a @ Xa2 @ top_top_set_a ) )
% 2.61/1.48 => ( ( ( F @ Z2 )
% 2.61/1.48 = ( Fa @ Za ) )
% 2.61/1.48 => ( Z2 = Za ) ) ) )
% 2.61/1.48 => ( ( ( comp_a_b_a @ F @ X3 )
% 2.61/1.48 = ( comp_a_b_a @ Fa @ Xa2 ) )
% 2.61/1.48 => ( X3 = Xa2 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_351_fun_Omap__cong0,axiom,
% 2.61/1.48 ! [X3: a > b,F: b > b,G: b > b] :
% 2.61/1.48 ( ! [Z2: b] :
% 2.61/1.48 ( ( member_b @ Z2 @ ( image_a_b @ X3 @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ Z2 )
% 2.61/1.48 = ( G @ Z2 ) ) )
% 2.61/1.48 => ( ( comp_b_b_a @ F @ X3 )
% 2.61/1.48 = ( comp_b_b_a @ G @ X3 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(fact_352_fun_Omap__cong0,axiom,
% 2.61/1.48 ! [X3: a > a,F: a > b,G: a > b] :
% 2.61/1.48 ( ! [Z2: a] :
% 2.61/1.48 ( ( member_a @ Z2 @ ( image_a_a @ X3 @ top_top_set_a ) )
% 2.61/1.48 => ( ( F @ Z2 )
% 2.61/1.48 = ( G @ Z2 ) ) )
% 2.61/1.48 => ( ( comp_a_b_a @ F @ X3 )
% 2.61/1.48 = ( comp_a_b_a @ G @ X3 ) ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(conj_0,hypothesis,
% 2.61/1.48 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 ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(conj_1,hypothesis,
% 2.61/1.48 ( ( comp_a_b_c @ f @ g )
% 2.61/1.48 = ( comp_a_b_c @ f @ h ) ) ).
% 2.61/1.48
% 2.61/1.48 thf(conj_2,conjecture,
% 2.61/1.48 g = h ).
% 2.61/1.48 ------------------------
% 2.78/1.50 % (24607)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_24497 for (2999ds/2Mi)
% 2.78/1.50 % (24603)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/183Mi)
% 2.78/1.50 % (24609)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/18Mi)
% 2.78/1.50 % (24607)Instruction limit reached!
% 2.78/1.50 % (24607)------------------------------
% 2.78/1.50 % (24607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.50 % (24607)Termination reason: Unknown
% 2.78/1.50 % (24607)Termination phase: shuffling
% 2.78/1.50
% 2.78/1.50 % (24607)Memory used [KB]: 1407
% 2.78/1.50 % (24607)Time elapsed: 0.002 s
% 2.78/1.50 % (24607)Instructions burned: 2 (million)
% 2.78/1.50 % (24607)------------------------------
% 2.78/1.50 % (24607)------------------------------
% 2.78/1.50 % (24605)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_24497 for (2999ds/27Mi)
% 2.78/1.51 % (24609)Instruction limit reached!
% 2.78/1.51 % (24609)------------------------------
% 2.78/1.51 % (24609)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.51 % (24609)Termination reason: Unknown
% 2.78/1.51 % (24609)Termination phase: shuffling
% 2.78/1.51
% 2.78/1.51 % (24609)Memory used [KB]: 1663
% 2.78/1.51 % (24609)Time elapsed: 0.007 s
% 2.78/1.51 % (24609)Instructions burned: 18 (million)
% 2.78/1.51 % (24609)------------------------------
% 2.78/1.51 % (24609)------------------------------
% 2.78/1.51 % (24604)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_24497 for (2999ds/4Mi)
% 2.78/1.51 % (24606)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/2Mi)
% 2.78/1.51 % (24608)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_24497 for (2999ds/275Mi)
% 2.78/1.51 % (24606)Instruction limit reached!
% 2.78/1.51 % (24606)------------------------------
% 2.78/1.51 % (24606)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.51 % (24606)Termination reason: Unknown
% 2.78/1.51 % (24606)Termination phase: shuffling
% 2.78/1.51
% 2.78/1.51 % (24606)Memory used [KB]: 1407
% 2.78/1.51 % (24606)Time elapsed: 0.004 s
% 2.78/1.51 % (24606)Instructions burned: 3 (million)
% 2.78/1.51 % (24606)------------------------------
% 2.78/1.51 % (24606)------------------------------
% 2.78/1.51 % (24610)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/3Mi)
% 2.78/1.51 % (24604)Instruction limit reached!
% 2.78/1.51 % (24604)------------------------------
% 2.78/1.51 % (24604)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.51 % (24605)Instruction limit reached!
% 2.78/1.51 % (24605)------------------------------
% 2.78/1.51 % (24605)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.51 % (24605)Termination reason: Unknown
% 2.78/1.51 % (24605)Termination phase: Property scanning
% 2.78/1.51
% 2.78/1.51 % (24605)Memory used [KB]: 1791
% 2.78/1.51 % (24605)Time elapsed: 0.011 s
% 2.78/1.51 % (24605)Instructions burned: 29 (million)
% 2.78/1.51 % (24605)------------------------------
% 2.78/1.51 % (24605)------------------------------
% 2.78/1.51 % (24604)Termination reason: Unknown
% 2.78/1.51 % (24604)Termination phase: shuffling
% 2.78/1.51
% 2.78/1.51 % (24604)Memory used [KB]: 1407
% 2.78/1.51 % (24604)Time elapsed: 0.005 s
% 2.78/1.51 % (24604)Instructions burned: 5 (million)
% 2.78/1.51 % (24604)------------------------------
% 2.78/1.51 % (24604)------------------------------
% 2.78/1.51 % (24610)Instruction limit reached!
% 2.78/1.51 % (24610)------------------------------
% 2.78/1.51 % (24610)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.51 % (24610)Termination reason: Unknown
% 2.78/1.51 % (24610)Termination phase: shuffling
% 2.78/1.51
% 2.78/1.51 % (24610)Memory used [KB]: 1407
% 2.78/1.51 % (24610)Time elapsed: 0.002 s
% 2.78/1.51 % (24610)Instructions burned: 3 (million)
% 2.78/1.51 % (24610)------------------------------
% 2.78/1.51 % (24610)------------------------------
% 2.78/1.51 % (24611)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_24497 for (2999ds/37Mi)
% 2.78/1.52 % (24615)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/7Mi)
% 2.78/1.52 % (24613)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_24497 for (2999ds/3Mi)
% 2.78/1.52 % (24613)Instruction limit reached!
% 2.78/1.52 % (24613)------------------------------
% 2.78/1.52 % (24613)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.52 % (24613)Termination reason: Unknown
% 2.78/1.52 % (24613)Termination phase: shuffling
% 2.78/1.52
% 2.78/1.52 % (24613)Memory used [KB]: 1407
% 2.78/1.52 % (24613)Time elapsed: 0.003 s
% 2.78/1.52 % (24613)Instructions burned: 3 (million)
% 2.78/1.52 % (24613)------------------------------
% 2.78/1.52 % (24613)------------------------------
% 2.78/1.52 % (24615)Instruction limit reached!
% 2.78/1.52 % (24615)------------------------------
% 2.78/1.52 % (24615)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.52 % (24615)Termination reason: Unknown
% 2.78/1.52 % (24615)Termination phase: shuffling
% 2.78/1.52
% 2.78/1.52 % (24615)Memory used [KB]: 1535
% 2.78/1.52 % (24615)Time elapsed: 0.004 s
% 2.78/1.52 % (24615)Instructions burned: 10 (million)
% 2.78/1.52 % (24615)------------------------------
% 2.78/1.52 % (24615)------------------------------
% 2.78/1.52 % (24611)Instruction limit reached!
% 2.78/1.52 % (24611)------------------------------
% 2.78/1.52 % (24611)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.52 % (24611)Termination reason: Unknown
% 2.78/1.52 % (24611)Termination phase: shuffling
% 2.78/1.52
% 2.78/1.52 % (24611)Memory used [KB]: 2174
% 2.78/1.52 % (24611)Time elapsed: 0.011 s
% 2.78/1.52 % (24611)Instructions burned: 38 (million)
% 2.78/1.52 % (24611)------------------------------
% 2.78/1.52 % (24611)------------------------------
% 2.78/1.53 % (24612)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_24497 for (2999ds/15Mi)
% 2.78/1.53 % (24614)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_24497 for (2999ds/1041Mi)
% 2.78/1.53 % (24616)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/16Mi)
% 2.78/1.53 % (24617)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_24497 for (2999ds/3Mi)
% 2.78/1.53 % (24617)Instruction limit reached!
% 2.78/1.53 % (24617)------------------------------
% 2.78/1.53 % (24617)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.53 % (24617)Termination reason: Unknown
% 2.78/1.53 % (24617)Termination phase: shuffling
% 2.78/1.53
% 2.78/1.53 % (24617)Memory used [KB]: 1407
% 2.78/1.53 % (24617)Time elapsed: 0.002 s
% 2.78/1.53 % (24617)Instructions burned: 3 (million)
% 2.78/1.53 % (24617)------------------------------
% 2.78/1.53 % (24617)------------------------------
% 2.78/1.53 % (24618)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_24497 for (2999ds/3Mi)
% 2.78/1.54 % (24618)Instruction limit reached!
% 2.78/1.54 % (24618)------------------------------
% 2.78/1.54 % (24618)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.54 % (24618)Termination reason: Unknown
% 2.78/1.54 % (24618)Termination phase: shuffling
% 2.78/1.54
% 2.78/1.54 % (24618)Memory used [KB]: 1407
% 2.78/1.54 % (24618)Time elapsed: 0.002 s
% 2.78/1.54 % (24618)Instructions burned: 3 (million)
% 2.78/1.54 % (24618)------------------------------
% 2.78/1.54 % (24618)------------------------------
% 2.78/1.54 % (24612)Instruction limit reached!
% 2.78/1.54 % (24612)------------------------------
% 2.78/1.54 % (24612)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.54 % (24612)Termination reason: Unknown
% 2.78/1.54 % (24612)Termination phase: shuffling
% 2.78/1.54
% 2.78/1.54 % (24612)Memory used [KB]: 1663
% 2.78/1.54 % (24612)Time elapsed: 0.011 s
% 2.78/1.54 % (24612)Instructions burned: 15 (million)
% 2.78/1.54 % (24612)------------------------------
% 2.78/1.54 % (24612)------------------------------
% 2.78/1.54 % (24616)Instruction limit reached!
% 2.78/1.54 % (24616)------------------------------
% 2.78/1.54 % (24616)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.54 % (24616)Termination reason: Unknown
% 2.78/1.54 % (24616)Termination phase: shuffling
% 2.78/1.54
% 2.78/1.54 % (24616)Memory used [KB]: 1663
% 2.78/1.54 % (24616)Time elapsed: 0.008 s
% 2.78/1.54 % (24616)Instructions burned: 18 (million)
% 2.78/1.54 % (24616)------------------------------
% 2.78/1.54 % (24616)------------------------------
% 2.78/1.54 % (24619)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/7Mi)
% 2.78/1.54 % (24620)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/3Mi)
% 2.78/1.54 % (24619)Instruction limit reached!
% 2.78/1.54 % (24619)------------------------------
% 2.78/1.54 % (24619)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.54 % (24619)Termination reason: Unknown
% 2.78/1.54 % (24619)Termination phase: shuffling
% 2.78/1.54
% 2.78/1.54 % (24619)Memory used [KB]: 1535
% 2.78/1.54 % (24619)Time elapsed: 0.003 s
% 2.78/1.54 % (24619)Instructions burned: 7 (million)
% 2.78/1.54 % (24619)------------------------------
% 2.78/1.54 % (24619)------------------------------
% 2.78/1.55 % (24620)Instruction limit reached!
% 2.78/1.55 % (24620)------------------------------
% 2.78/1.55 % (24620)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.55 % (24620)Termination reason: Unknown
% 2.78/1.55 % (24620)Termination phase: shuffling
% 2.78/1.55
% 2.78/1.55 % (24620)Memory used [KB]: 1407
% 2.78/1.55 % (24620)Time elapsed: 0.002 s
% 2.78/1.55 % (24620)Instructions burned: 4 (million)
% 2.78/1.55 % (24620)------------------------------
% 2.78/1.55 % (24620)------------------------------
% 2.78/1.55 % (24608)First to succeed.
% 2.78/1.55 % (24603)Instruction limit reached!
% 2.78/1.55 % (24603)------------------------------
% 2.78/1.55 % (24603)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.55 % (24603)Termination reason: Unknown
% 2.78/1.55 % (24603)Termination phase: Saturation
% 2.78/1.55
% 2.78/1.55 % (24622)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_24497 for (2999ds/18Mi)
% 2.78/1.55 % (24603)Memory used [KB]: 7164
% 2.78/1.55 % (24603)Time elapsed: 0.050 s
% 2.78/1.55 % (24603)Instructions burned: 184 (million)
% 2.78/1.55 % (24603)------------------------------
% 2.78/1.55 % (24603)------------------------------
% 2.78/1.55 % (24608)Refutation found. Thanks to Tanya!
% 2.78/1.55 % SZS status Theorem for DTF2THF_24497
% 2.78/1.55 % SZS output start Proof for DTF2THF_24497
% 2.78/1.55 thf(type_def_5, type, a: $tType).
% 2.78/1.55 thf(type_def_7, type, c: $tType).
% 2.78/1.55 thf(type_def_8, type, b: $tType).
% 2.78/1.55 thf(type_def_9, type, set_a_a: $tType).
% 2.78/1.55 thf(type_def_10, type, set_a_b: $tType).
% 2.78/1.55 thf(type_def_11, type, set_a_c: $tType).
% 2.78/1.55 thf(type_def_12, type, set_c_a: $tType).
% 2.78/1.55 thf(type_def_13, type, set_a: $tType).
% 2.78/1.55 thf(type_def_14, type, set_b: $tType).
% 2.78/1.55 thf(type_def_15, type, set_c: $tType).
% 2.78/1.55 thf(func_def_0, type, set_c_a: $tType).
% 2.78/1.55 thf(func_def_1, type, set_a_c: $tType).
% 2.78/1.55 thf(func_def_2, type, set_a_b: $tType).
% 2.78/1.55 thf(func_def_3, type, set_a_a: $tType).
% 2.78/1.55 thf(func_def_4, type, set_c: $tType).
% 2.78/1.55 thf(func_def_5, type, set_b: $tType).
% 2.78/1.55 thf(func_def_6, type, set_a: $tType).
% 2.78/1.55 thf(func_def_7, type, c: $tType).
% 2.78/1.55 thf(func_def_8, type, b: $tType).
% 2.78/1.55 thf(func_def_9, type, a: $tType).
% 2.78/1.55 thf(func_def_10, type, comp_a_a_a: (a > a) > (a > a) > a > a).
% 2.78/1.55 thf(func_def_11, type, comp_a_a_c: (a > a) > (c > a) > c > a).
% 2.78/1.55 thf(func_def_12, type, comp_a_b_a: (a > b) > (a > a) > a > b).
% 2.78/1.55 thf(func_def_13, type, comp_a_b_b: (a > b) > (b > a) > b > b).
% 2.78/1.55 thf(func_def_14, type, comp_a_b_c: (a > b) > (c > a) > c > b).
% 2.78/1.55 thf(func_def_15, type, comp_a_c_a: (a > c) > (a > a) > a > c).
% 2.78/1.55 thf(func_def_16, type, comp_a_c_c: (a > c) > (c > a) > c > c).
% 2.78/1.55 thf(func_def_17, type, comp_b_a_a: (b > a) > (a > b) > a > a).
% 2.78/1.55 thf(func_def_18, type, comp_b_a_c: (b > a) > (c > b) > c > a).
% 2.78/1.55 thf(func_def_19, type, comp_b_b_a: (b > b) > (a > b) > a > b).
% 2.78/1.55 thf(func_def_20, type, comp_b_b_b: (b > b) > (b > b) > b > b).
% 2.78/1.55 thf(func_def_21, type, comp_b_b_c: (b > b) > (c > b) > c > b).
% 2.78/1.55 thf(func_def_22, type, comp_b_c_a: (b > c) > (a > b) > a > c).
% 2.78/1.55 thf(func_def_23, type, comp_b_c_c: (b > c) > (c > b) > c > c).
% 2.78/1.55 thf(func_def_24, type, comp_c_a_a: (c > a) > (a > c) > a > a).
% 2.78/1.55 thf(func_def_25, type, comp_c_a_b: (c > a) > (b > c) > b > a).
% 2.78/1.55 thf(func_def_26, type, comp_c_a_c: (c > a) > (c > c) > c > a).
% 2.78/1.55 thf(func_def_27, type, comp_c_b_a: (c > b) > (a > c) > a > b).
% 2.78/1.55 thf(func_def_28, type, comp_c_b_b: (c > b) > (b > c) > b > b).
% 2.78/1.55 thf(func_def_29, type, comp_c_b_c: (c > b) > (c > c) > c > b).
% 2.78/1.55 thf(func_def_30, type, comp_c_c_a: (c > c) > (a > c) > a > c).
% 2.78/1.55 thf(func_def_31, type, comp_c_c_c: (c > c) > (c > c) > c > c).
% 2.78/1.55 thf(func_def_32, type, inj_on_a_a_a_b: ((a > a) > a > b) > set_a_a > $o).
% 2.78/1.55 thf(func_def_33, type, inj_on_a_b_a_b: ((a > b) > a > b) > set_a_b > $o).
% 2.78/1.55 thf(func_def_34, type, inj_on_a_c_a_b: ((a > c) > a > b) > set_a_c > $o).
% 2.78/1.55 thf(func_def_35, type, inj_on_c_a_c_a: ((c > a) > c > a) > set_c_a > $o).
% 2.78/1.55 thf(func_def_36, type, inj_on_c_a_c_b: ((c > a) > c > b) > set_c_a > $o).
% 2.78/1.55 thf(func_def_37, type, inj_on_c_a_c_c: ((c > a) > c > c) > set_c_a > $o).
% 2.78/1.55 thf(func_def_38, type, inj_on_c_a_a: ((c > a) > a) > set_c_a > $o).
% 2.78/1.55 thf(func_def_39, type, inj_on_a_a: (a > a) > set_a > $o).
% 2.78/1.55 thf(func_def_40, type, inj_on_a_b: (a > b) > set_a > $o).
% 2.78/1.55 thf(func_def_41, type, inj_on_a_c: (a > c) > set_a > $o).
% 2.78/1.55 thf(func_def_42, type, inj_on_b_a: (b > a) > set_b > $o).
% 2.78/1.55 thf(func_def_43, type, inj_on_b_b: (b > b) > set_b > $o).
% 2.78/1.55 thf(func_def_44, type, inj_on_b_c: (b > c) > set_b > $o).
% 2.78/1.55 thf(func_def_45, type, inj_on_c_a: (c > a) > set_c > $o).
% 2.78/1.55 thf(func_def_46, type, inj_on_c_b: (c > b) > set_c > $o).
% 2.78/1.55 thf(func_def_47, type, inj_on_c_c: (c > c) > set_c > $o).
% 2.78/1.55 thf(func_def_48, type, sup_sup_set_c_a: set_c_a > set_c_a > set_c_a).
% 2.78/1.55 thf(func_def_49, type, sup_sup_set_a: set_a > set_a > set_a).
% 2.78/1.55 thf(func_def_50, type, sup_sup_set_b: set_b > set_b > set_b).
% 2.78/1.55 thf(func_def_51, type, sup_sup_set_c: set_c > set_c > set_c).
% 2.78/1.55 thf(func_def_52, type, top_top_set_a_a: set_a_a).
% 2.78/1.55 thf(func_def_53, type, top_top_set_a_b: set_a_b).
% 2.78/1.55 thf(func_def_54, type, top_top_set_a_c: set_a_c).
% 2.78/1.55 thf(func_def_55, type, top_top_set_c_a: set_c_a).
% 2.78/1.55 thf(func_def_56, type, top_top_set_a: set_a).
% 2.78/1.55 thf(func_def_57, type, top_top_set_b: set_b).
% 2.78/1.55 thf(func_def_58, type, top_top_set_c: set_c).
% 2.78/1.55 thf(func_def_59, type, collect_a: (a > $o) > set_a).
% 2.78/1.55 thf(func_def_60, type, image_c_a_a: ((c > a) > a) > set_c_a > set_a).
% 2.78/1.55 thf(func_def_61, type, image_a_a: (a > a) > set_a > set_a).
% 2.78/1.55 thf(func_def_62, type, image_a_b: (a > b) > set_a > set_b).
% 2.78/1.55 thf(func_def_63, type, image_a_c: (a > c) > set_a > set_c).
% 2.78/1.55 thf(func_def_64, type, image_b_a: (b > a) > set_b > set_a).
% 2.78/1.55 thf(func_def_65, type, image_b_b: (b > b) > set_b > set_b).
% 2.78/1.55 thf(func_def_66, type, image_b_c: (b > c) > set_b > set_c).
% 2.78/1.55 thf(func_def_67, type, image_c_c_a: (c > c > a) > set_c > set_c_a).
% 2.78/1.55 thf(func_def_68, type, image_c_a: (c > a) > set_c > set_a).
% 2.78/1.55 thf(func_def_69, type, image_c_b: (c > b) > set_c > set_b).
% 2.78/1.55 thf(func_def_70, type, image_c_c: (c > c) > set_c > set_c).
% 2.78/1.55 thf(func_def_71, type, member_c_a: (c > a) > set_c_a > $o).
% 2.78/1.55 thf(func_def_72, type, member_a: a > set_a > $o).
% 2.78/1.55 thf(func_def_73, type, member_b: b > set_b > $o).
% 2.78/1.55 thf(func_def_74, type, member_c: c > set_c > $o).
% 2.78/1.55 thf(func_def_75, type, f: a > b).
% 2.78/1.55 thf(func_def_76, type, g: c > a).
% 2.78/1.55 thf(func_def_77, type, h: c > a).
% 2.78/1.55 thf(func_def_89, type, sK0: (a > b) > set_a > (b > a) > a).
% 2.78/1.55 thf(func_def_90, type, sK1: (c > a) > c).
% 2.78/1.55 thf(func_def_91, type, sK2: (c > a) > c).
% 2.78/1.55 thf(func_def_92, type, sK3: (a > b) > (a > b) > (c > a) > a).
% 2.78/1.55 thf(func_def_93, type, sK4: (b > a) > (a > b) > b).
% 2.78/1.55 thf(func_def_94, type, sK5: (c > a) > a > c).
% 2.78/1.55 thf(func_def_95, type, sK6: a > (c > a) > c).
% 2.78/1.55 thf(func_def_96, type, sK7: (c > a) > (a > b) > (a > b) > (c > a) > a).
% 2.78/1.55 thf(func_def_97, type, sK8: (c > a) > (a > b) > (a > b) > (c > a) > a).
% 2.78/1.55 thf(func_def_98, type, sK9: set_a > (a > b) > a).
% 2.78/1.55 thf(func_def_99, type, sK10: set_a > (a > b) > a).
% 2.78/1.55 thf(func_def_100, type, sK11: (c > b) > c).
% 2.78/1.55 thf(func_def_101, type, sK12: (c > b) > c).
% 2.78/1.55 thf(func_def_102, type, sK13: (a > b) > (a > b) > (c > a) > a).
% 2.78/1.55 thf(func_def_103, type, sK14: b > (a > b) > a).
% 2.78/1.55 thf(func_def_104, type, sK15: (a > b) > a).
% 2.78/1.55 thf(func_def_105, type, sK16: (a > b) > a).
% 2.78/1.55 thf(func_def_106, type, sK17: (a > b) > b).
% 2.78/1.55 thf(func_def_107, type, sK18: b > (a > b) > a).
% 2.78/1.55 thf(func_def_108, type, sK19: (a > b) > (a > b) > set_a > a).
% 2.78/1.55 thf(func_def_109, type, sK20: (c > b) > b > c).
% 2.78/1.55 thf(func_def_110, type, sK21: (a > b) > (c > a) > (a > b) > a).
% 2.78/1.55 thf(func_def_111, type, sK22: (c > b) > c).
% 2.78/1.55 thf(func_def_112, type, sK23: (c > b) > c).
% 2.78/1.55 thf(func_def_113, type, sK24: (a > c) > (c > a) > a).
% 2.78/1.55 thf(func_def_114, type, sK25: b > (c > b) > c).
% 2.78/1.55 thf(func_def_115, type, sK26: (c > b) > b).
% 2.78/1.55 thf(func_def_116, type, sK27: (c > b) > (b > c) > b).
% 2.78/1.55 thf(func_def_117, type, sK28: (c > a) > a > c).
% 2.78/1.55 thf(func_def_118, type, sK29: c > (c > a) > a > c).
% 2.78/1.55 thf(func_def_119, type, sK30: (c > a) > a).
% 2.78/1.55 thf(func_def_120, type, sK31: a > (c > a) > c).
% 2.78/1.55 thf(func_def_121, type, sK32: b > (c > b) > c).
% 2.78/1.55 thf(func_def_122, type, sK33: (c > a) > c).
% 2.78/1.55 thf(func_def_123, type, sK34: (c > a) > c).
% 2.78/1.55 thf(func_def_124, type, sK35: b > (a > b) > a).
% 2.78/1.55 thf(func_def_125, type, sK36: (a > b) > a).
% 2.78/1.55 thf(func_def_126, type, sK37: (a > b) > a).
% 2.78/1.55 thf(func_def_128, type, ph39: !>[X0: $tType]:(X0)).
% 2.78/1.55 thf(func_def_129, type, sK40: c).
% 2.78/1.55 thf(f1153,plain,(
% 2.78/1.55 $false),
% 2.78/1.55 inference(subsumption_resolution,[status(thm)],[f1152,f966])).
% 2.78/1.55 thf(f966,plain,(
% 2.78/1.55 (g != h)),
% 2.78/1.55 inference(cnf_transformation,[status(thm)],[f787])).
% 2.78/1.55 thf(f787,plain,(
% 2.78/1.55 (g != h)),
% 2.78/1.55 inference(flattening,[status(thm)],[f357])).
% 2.78/1.55 thf(f357,negated_conjecture,(
% 2.78/1.55 ~(g = h)),
% 2.78/1.55 inference(negated_conjecture,[status(cth)],[f356])).
% 2.78/1.55 thf(f356,conjecture,(
% 2.78/1.55 (g = h)),
% 2.78/1.55 file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_2)).
% 2.78/1.55 thf(f1152,plain,(
% 2.78/1.55 (g = h)),
% 2.78/1.55 inference(subsumption_resolution,[status(thm)],[f1151,f1029])).
% 2.78/1.55 thf(f1029,plain,(
% 2.78/1.55 ((comp_a_b_c @ f @ g) = (comp_a_b_c @ f @ h))),
% 2.78/1.55 inference(cnf_transformation,[status(thm)],[f355])).
% 2.78/1.55 thf(f355,axiom,(
% 2.78/1.55 ((comp_a_b_c @ f @ g) = (comp_a_b_c @ f @ h))),
% 2.78/1.55 file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_1)).
% 2.78/1.55 thf(f1151,plain,(
% 2.78/1.55 ((comp_a_b_c @ f @ g) != (comp_a_b_c @ f @ h)) | (g = h)),
% 2.78/1.55 inference(trivial_inequality_removal,[status(thm)],[f1142])).
% 2.78/1.55 thf(f1142,plain,(
% 2.78/1.55 ($true != $true) | ((comp_a_b_c @ f @ g) != (comp_a_b_c @ f @ h)) | (g = h)),
% 2.78/1.55 inference(superposition,[status(thm)],[f1059,f986])).
% 2.78/1.55 thf(f986,plain,(
% 2.78/1.55 ((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)),
% 2.78/1.55 inference(cnf_transformation,[status(thm)],[f576])).
% 2.78/1.55 thf(f576,plain,(
% 2.78/1.55 ((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))) = $true)),
% 2.78/1.55 inference(fool_elimination,[status(thm)],[f575])).
% 2.78/1.55 thf(f575,plain,(
% 2.78/1.55 (inj_on_a_b @ f @ (sup_sup_set_a @ (image_c_a @ g @ top_top_set_c) @ (image_c_a @ h @ top_top_set_c)))),
% 2.78/1.55 inference(rectify,[status(thm)],[f354])).
% 2.78/1.55 thf(f354,axiom,(
% 2.78/1.55 (inj_on_a_b @ f @ (sup_sup_set_a @ (image_c_a @ g @ top_top_set_c) @ (image_c_a @ h @ top_top_set_c)))),
% 2.78/1.55 file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',conj_0)).
% 2.78/1.55 thf(f1059,plain,(
% 2.78/1.55 ( ! [X2 : c > a,X0 : c > a,X1 : a > b] : (($true != (inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X0 @ top_top_set_c) @ (image_c_a @ X2 @ top_top_set_c)))) | ((comp_a_b_c @ X1 @ X0) != (comp_a_b_c @ X1 @ X2)) | (X0 = X2)) )),
% 2.78/1.55 inference(cnf_transformation,[status(thm)],[f880])).
% 2.78/1.55 thf(f880,plain,(
% 2.78/1.55 ! [X0 : c > a,X1 : a > b,X2 : c > a] : (($true != (inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X0 @ top_top_set_c) @ (image_c_a @ X2 @ top_top_set_c)))) | ((comp_a_b_c @ X1 @ X0) != (comp_a_b_c @ X1 @ X2)) | (X0 = X2))),
% 2.78/1.55 inference(flattening,[status(thm)],[f879])).
% 2.78/1.55 thf(f879,plain,(
% 2.78/1.55 ! [X0 : c > a,X1 : a > b,X2 : c > a] : (((X0 = X2) | ($true != (inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X0 @ top_top_set_c) @ (image_c_a @ X2 @ top_top_set_c))))) | ((comp_a_b_c @ X1 @ X0) != (comp_a_b_c @ X1 @ X2)))),
% 2.78/1.55 inference(ennf_transformation,[status(thm)],[f544])).
% 2.78/1.55 thf(f544,plain,(
% 2.78/1.55 ! [X0 : c > a,X1 : a > b,X2 : c > a] : (((comp_a_b_c @ X1 @ X0) = (comp_a_b_c @ X1 @ X2)) => (($true = (inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X0 @ top_top_set_c) @ (image_c_a @ X2 @ top_top_set_c)))) => (X0 = X2)))),
% 2.78/1.55 inference(fool_elimination,[status(thm)],[f543])).
% 2.78/1.55 thf(f543,plain,(
% 2.78/1.55 ! [X0 : c > a,X1 : a > b,X2 : c > a] : (((comp_a_b_c @ X1 @ X0) = (comp_a_b_c @ X1 @ X2)) => ((inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X0 @ top_top_set_c) @ (image_c_a @ X2 @ top_top_set_c))) => (X0 = X2)))),
% 2.78/1.55 inference(rectify,[status(thm)],[f9])).
% 2.78/1.55 thf(f9,axiom,(
% 2.78/1.55 ! [X2 : c > a,X1 : a > b,X0 : c > a] : (((comp_a_b_c @ X1 @ X0) = (comp_a_b_c @ X1 @ X2)) => ((inj_on_a_b @ X1 @ (sup_sup_set_a @ (image_c_a @ X2 @ top_top_set_c) @ (image_c_a @ X0 @ top_top_set_c))) => (X0 = X2)))),
% 2.78/1.55 file('/export/starexec/sandbox/tmp/tmp.H7rtIyjQOT/DTF2THF_24497.p',fact_8_o__inj__on)).
% 2.78/1.55 % SZS output end Proof for DTF2THF_24497
% 2.78/1.55 % (24608)------------------------------
% 2.78/1.55 % (24608)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.78/1.55 % (24608)Termination reason: Refutation
% 2.78/1.55
% 2.78/1.55 % (24608)Memory used [KB]: 6652
% 2.78/1.55 % (24608)Time elapsed: 0.042 s
% 2.78/1.55 % (24608)Instructions burned: 82 (million)
% 2.78/1.55 % (24608)------------------------------
% 2.78/1.55 % (24608)------------------------------
% 2.78/1.55 % (24602)Success in time 0.058 s
%------------------------------------------------------------------------------