%------------------------------------------------------------------------------
% File : Satallax---3.5
% Problem : ITP076^1 : TPTP v9.2.0. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% Computer : n028.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 : Thu Oct 9 01:46:54 PM UTC 2025
% Result : Theorem 205.41s 204.63s
% Output : Proof 205.41s
% Verified :
% SZS Type : Refutation
% Derivation depth : 3
% Number of leaves : 62
% Syntax : Number of formulae : 69 ( 12 unt; 0 typ; 3 def)
% Number of atoms : 203 ( 38 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 415 ( 51 ~; 33 |; 0 &; 257 @)
% ( 30 <=>; 44 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 1 ( 1 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 40 usr; 37 con; 0-2 aty)
% Number of variables : 57 ( 4 ^; 53 !; 0 ?; 57 :)
% Comments :
%------------------------------------------------------------------------------
thf(ty_hF_Mirabelle_hf,type,
hF_Mirabelle_hf: $tType ).
thf(ty_hF_Mir199975595nction,type,
hF_Mir199975595nction: hF_Mirabelle_hf > $o ).
thf(ty_hF_Mir1653039215strict,type,
hF_Mir1653039215strict: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(ty_eigen__2,type,
eigen__2: hF_Mirabelle_hf ).
thf(ty_eigen__1,type,
eigen__1: hF_Mirabelle_hf ).
thf(ty_eigen__0,type,
eigen__0: hF_Mirabelle_hf ).
thf(ty_r,type,
r: hF_Mirabelle_hf ).
thf(ty_hF_Mirabelle_hmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(ty_hF_Mirabelle_hpair,type,
hF_Mirabelle_hpair: hF_Mirabelle_hf > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(ty_x,type,
x: hF_Mirabelle_hf ).
thf(h0,assumption,
! [X1: hF_Mirabelle_hf > $o,X2: hF_Mirabelle_hf] :
( ( X1 @ X2 )
=> ( X1 @ ( eps__0 @ X1 ) ) ),
introduced(assumption,[]) ).
thf(eigendef_eigen__1,definition,
( eigen__1
= ( eps__0
@ ^ [X1: hF_Mirabelle_hf] :
~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( X1 = X2 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__1])]) ).
thf(eigendef_eigen__0,definition,
( eigen__0
= ( eps__0
@ ^ [X1: hF_Mirabelle_hf] :
~ ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( X2 = X3 ) ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__0])]) ).
thf(eigendef_eigen__2,definition,
( eigen__2
= ( eps__0
@ ^ [X1: hF_Mirabelle_hf] :
~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( eigen__1 = X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[eigen__2])]) ).
thf(sP1,plain,
( sP1
<=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( X2 = X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP1])]) ).
thf(sP2,plain,
( sP2
<=> ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ r )
=> ( eigen__1 = eigen__2 ) ) ),
introduced(definition,[new_symbols(definition,[sP2])]) ).
thf(sP3,plain,
( sP3
<=> ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
= ( ~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ r )
=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 )
= ( hF_Mirabelle_hpair @ X1 @ X2 ) )
=> ~ ( hF_Mirabelle_hmem @ X1 @ x ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP3])]) ).
thf(sP4,plain,
( sP4
<=> ( hF_Mir199975595nction @ r ) ),
introduced(definition,[new_symbols(definition,[sP4])]) ).
thf(sP5,plain,
( sP5
<=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X2 ) @ r )
=> ! [X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X1 @ X3 ) @ r )
=> ( X2 = X3 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP5])]) ).
thf(sP6,plain,
( sP6
<=> ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ ( hF_Mir1653039215strict @ r @ x ) ) ),
introduced(definition,[new_symbols(definition,[sP6])]) ).
thf(sP7,plain,
( sP7
<=> ( eigen__1 = eigen__2 ) ),
introduced(definition,[new_symbols(definition,[sP7])]) ).
thf(sP8,plain,
( sP8
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ r )
=> ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X2 ) @ r )
=> ( X1 = X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP8])]) ).
thf(sP9,plain,
( sP9
<=> ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ r )
=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ r )
=> ( eigen__1 = X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP9])]) ).
thf(sP10,plain,
( sP10
<=> ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ r )
=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 )
= ( hF_Mirabelle_hpair @ X1 @ X2 ) )
=> ~ ( hF_Mirabelle_hmem @ X1 @ x ) ) ) ),
introduced(definition,[new_symbols(definition,[sP10])]) ).
thf(sP11,plain,
( sP11
<=> ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ ( hF_Mir1653039215strict @ r @ x ) ) ),
introduced(definition,[new_symbols(definition,[sP11])]) ).
thf(sP12,plain,
( sP12
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ ( hF_Mir1653039215strict @ r @ X1 ) )
= ( ~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ r )
=> ! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 )
= ( hF_Mirabelle_hpair @ X2 @ X3 ) )
=> ~ ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP12])]) ).
thf(sP13,plain,
( sP13
<=> ( sP11
=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( eigen__1 = X1 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP13])]) ).
thf(sP14,plain,
( sP14
<=> ( sP11
= ( ~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ r )
=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 )
= ( hF_Mirabelle_hpair @ X1 @ X2 ) )
=> ~ ( hF_Mirabelle_hmem @ X1 @ x ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP14])]) ).
thf(sP15,plain,
( sP15
<=> ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ r ) ),
introduced(definition,[new_symbols(definition,[sP15])]) ).
thf(sP16,plain,
( sP16
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ ( hF_Mir1653039215strict @ r @ X1 ) )
= ( ~ ( sP15
=> ! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 )
= ( hF_Mirabelle_hpair @ X2 @ X3 ) )
=> ~ ( hF_Mirabelle_hmem @ X2 @ X1 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP16])]) ).
thf(sP17,plain,
( sP17
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mir199975595nction @ X1 )
= ( ! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X3 ) @ X1 )
=> ! [X4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X4 ) @ X1 )
=> ( X3 = X4 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP17])]) ).
thf(sP18,plain,
( sP18
<=> ( sP6
=> sP7 ) ),
introduced(definition,[new_symbols(definition,[sP18])]) ).
thf(sP19,plain,
( sP19
<=> ( hF_Mir199975595nction
= ( ^ [X1: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X3 ) @ X1 )
=> ! [X4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ X2 @ X4 ) @ X1 )
=> ( X3 = X4 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP19])]) ).
thf(sP20,plain,
( sP20
<=> ( sP4 = sP5 ) ),
introduced(definition,[new_symbols(definition,[sP20])]) ).
thf(sP21,plain,
( sP21
<=> ( hF_Mir199975595nction @ ( hF_Mir1653039215strict @ r @ x ) ) ),
introduced(definition,[new_symbols(definition,[sP21])]) ).
thf(sP22,plain,
( sP22
<=> ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ r ) ),
introduced(definition,[new_symbols(definition,[sP22])]) ).
thf(sP23,plain,
( sP23
<=> ( sP22
=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 )
= ( hF_Mirabelle_hpair @ X1 @ X2 ) )
=> ~ ( hF_Mirabelle_hmem @ X1 @ x ) ) ) ),
introduced(definition,[new_symbols(definition,[sP23])]) ).
thf(sP24,plain,
( sP24
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X2 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( X1 = X2 ) ) ) ),
introduced(definition,[new_symbols(definition,[sP24])]) ).
thf(sP25,plain,
( sP25
<=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ ( hF_Mir1653039215strict @ X1 @ X2 ) )
= ( ~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 ) @ X1 )
=> ! [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__2 )
= ( hF_Mirabelle_hpair @ X3 @ X4 ) )
=> ~ ( hF_Mirabelle_hmem @ X3 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP25])]) ).
thf(sP26,plain,
( sP26
<=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ ( hF_Mir1653039215strict @ X1 @ X2 ) )
= ( ~ ( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 ) @ X1 )
=> ! [X3: hF_Mirabelle_hf,X4: hF_Mirabelle_hf] :
( ( ( hF_Mirabelle_hpair @ eigen__0 @ eigen__1 )
= ( hF_Mirabelle_hpair @ X3 @ X4 ) )
=> ~ ( hF_Mirabelle_hmem @ X3 @ X2 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP26])]) ).
thf(sP27,plain,
( sP27
<=> ( sP21 = sP1 ) ),
introduced(definition,[new_symbols(definition,[sP27])]) ).
thf(sP28,plain,
( sP28
<=> ! [X1: hF_Mirabelle_hf,X2: hF_Mirabelle_hf,X3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X1 @ ( hF_Mir1653039215strict @ X2 @ X3 ) )
= ( ~ ( ( hF_Mirabelle_hmem @ X1 @ X2 )
=> ! [X4: hF_Mirabelle_hf,X5: hF_Mirabelle_hf] :
( ( X1
= ( hF_Mirabelle_hpair @ X4 @ X5 ) )
=> ~ ( hF_Mirabelle_hmem @ X4 @ X3 ) ) ) ) ) ),
introduced(definition,[new_symbols(definition,[sP28])]) ).
thf(sP29,plain,
( sP29
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ r )
=> ( eigen__1 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP29])]) ).
thf(sP30,plain,
( sP30
<=> ! [X1: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ ( hF_Mirabelle_hpair @ eigen__0 @ X1 ) @ ( hF_Mir1653039215strict @ r @ x ) )
=> ( eigen__1 = X1 ) ) ),
introduced(definition,[new_symbols(definition,[sP30])]) ).
thf(conj_1,conjecture,
sP21 ).
thf(h1,negated_conjecture,
~ sP21,
inference(assume_negation,[status(cth)],[conj_1]) ).
thf(1,plain,
( ~ sP8
| sP9 ),
inference(all_rule,[status(thm)],[]) ).
thf(2,plain,
( ~ sP9
| ~ sP22
| sP29 ),
inference(prop_rule,[status(thm)],[]) ).
thf(3,plain,
( ~ sP29
| sP2 ),
inference(all_rule,[status(thm)],[]) ).
thf(4,plain,
( ~ sP2
| ~ sP15
| sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(5,plain,
( sP10
| sP15 ),
inference(prop_rule,[status(thm)],[]) ).
thf(6,plain,
( ~ sP3
| ~ sP6
| ~ sP10 ),
inference(prop_rule,[status(thm)],[]) ).
thf(7,plain,
( sP18
| ~ sP7 ),
inference(prop_rule,[status(thm)],[]) ).
thf(8,plain,
( sP18
| sP6 ),
inference(prop_rule,[status(thm)],[]) ).
thf(9,plain,
( ~ sP28
| sP25 ),
inference(all_rule,[status(thm)],[]) ).
thf(10,plain,
( ~ sP25
| sP16 ),
inference(all_rule,[status(thm)],[]) ).
thf(11,plain,
( ~ sP16
| sP3 ),
inference(all_rule,[status(thm)],[]) ).
thf(12,plain,
( ~ sP5
| sP8 ),
inference(all_rule,[status(thm)],[]) ).
thf(13,plain,
( sP23
| sP22 ),
inference(prop_rule,[status(thm)],[]) ).
thf(14,plain,
( ~ sP14
| ~ sP11
| ~ sP23 ),
inference(prop_rule,[status(thm)],[]) ).
thf(15,plain,
( sP30
| ~ sP18 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__2]) ).
thf(16,plain,
( sP13
| ~ sP30 ),
inference(prop_rule,[status(thm)],[]) ).
thf(17,plain,
( sP13
| sP11 ),
inference(prop_rule,[status(thm)],[]) ).
thf(18,plain,
( ~ sP28
| sP26 ),
inference(all_rule,[status(thm)],[]) ).
thf(19,plain,
( ~ sP26
| sP12 ),
inference(all_rule,[status(thm)],[]) ).
thf(20,plain,
( ~ sP12
| sP14 ),
inference(all_rule,[status(thm)],[]) ).
thf(21,plain,
( sP24
| ~ sP13 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__1]) ).
thf(22,plain,
( ~ sP20
| ~ sP4
| sP5 ),
inference(prop_rule,[status(thm)],[]) ).
thf(23,plain,
( sP1
| ~ sP24 ),
inference(eigen_choice_rule,[status(thm),assumptions([h0])],[h0,eigendef_eigen__0]) ).
thf(24,plain,
( ~ sP27
| sP21
| ~ sP1 ),
inference(prop_rule,[status(thm)],[]) ).
thf(25,plain,
( ~ sP17
| sP27 ),
inference(all_rule,[status(thm)],[]) ).
thf(26,plain,
( ~ sP17
| sP20 ),
inference(all_rule,[status(thm)],[]) ).
thf(27,plain,
( ~ sP19
| sP17 ),
inference(prop_rule,[status(thm)],[]) ).
thf(conj_0,axiom,
sP4 ).
thf(fact_2_hrestrict__iff,axiom,
sP28 ).
thf(fact_1_hfunction__def,axiom,
sP19 ).
thf(28,plain,
$false,
inference(prop_unsat,[status(thm),assumptions([h1,h0])],[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,conj_0,fact_2_hrestrict__iff,fact_1_hfunction__def,h1]) ).
thf(29,plain,
$false,
inference(eigenvar_choice,[status(thm),assumptions([h1]),eigenvar_choice(discharge,[h0])],[28,h0]) ).
thf(0,theorem,
sP21,
inference(contra,[status(thm),contra(discharge,[h1])],[28,h1]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : ITP076^1 : TPTP v9.2.0. Released v7.5.0.
% 0.03/0.13 % Command : satallax -E eprover-ho -P picomus -M modes -p tstp -t %d %s
% 0.13/0.34 % Computer : n028.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Wed Oct 8 18:47:23 EDT 2025
% 0.13/0.34 % CPUTime :
% 205.41/204.63 % SZS status Theorem
% 205.41/204.63 % Mode: mode506:USE_SINE=true:SINE_TOLERANCE=5.0:SINE_GENERALITY_THRESHOLD=4:SINE_RANK_LIMIT=1.:SINE_DEPTH=0
% 205.41/204.63 % Inferences: 20680
% 205.41/204.63 % SZS output start Proof
% See solution above
%------------------------------------------------------------------------------