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cocATP---0.2.0.THM-Prf.s

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%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU785^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n090.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:07 EDT 2014

% Result   : Theorem 5.35s
% Output   : Proof 5.35s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU785^2 : TPTP v6.1.0. Released v3.7.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:27:51 CDT 2014
% % CPUTime  : 5.35 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0xf0d488>, <kernel.DependentProduct object at 0xeee368>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xfbe290>, <kernel.DependentProduct object at 0xeee320>) of role type named binunion_type
% Using role type
% Declaring binunion:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xf0d7a0>, <kernel.DependentProduct object at 0xeee320>) of role type named kpair_type
% Using role type
% Declaring kpair:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xf0d440>, <kernel.DependentProduct object at 0xeee320>) of role type named breln1_type
% Using role type
% Declaring breln1:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0xf0d7a0>, <kernel.Sort object at 0xbf11b8>) of role type named breln1unionIL_type
% Using role type
% Declaring breln1unionIL:Prop
% FOF formula (((eq Prop) breln1unionIL) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))) of role definition named breln1unionIL
% A new definition: (((eq Prop) breln1unionIL) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Defined: breln1unionIL:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% FOF formula (<kernel.Constant object at 0xf0d7a0>, <kernel.Sort object at 0xbf11b8>) of role type named breln1unionIR_type
% Using role type
% Declaring breln1unionIR:Prop
% FOF formula (((eq Prop) breln1unionIR) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))) of role definition named breln1unionIR
% A new definition: (((eq Prop) breln1unionIR) (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Defined: breln1unionIR:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% FOF formula (breln1unionIL->(breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))) of role conjecture named breln1unionI
% Conjecture to prove = (breln1unionIL->(breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(breln1unionIL->(breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Parameter binunion:(fofType->(fofType->fofType)).
% Parameter kpair:(fofType->(fofType->fofType)).
% Parameter breln1:(fofType->(fofType->Prop)).
% Definition breln1unionIL:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))):Prop.
% Definition breln1unionIR:=(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))):Prop.
% Trying to prove (breln1unionIL->(breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))))
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x1:((breln1 A) R)
% Found x1 as proof of ((breln1 A) R)
% Found x1:((breln1 A) R)
% Found x1 as proof of ((breln1 A) R)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x1:((breln1 A) R)
% Found x1 as proof of ((breln1 A) R)
% Found x4:((in Xy) A)
% Found x4 as proof of ((in Xy) A)
% Found x3:((in Xx) A)
% Found x3 as proof of ((in Xx) A)
% Found x1:((breln1 A) R)
% Found x1 as proof of ((breln1 A) R)
% Found x2:((breln1 A) S)
% Found x2 as proof of ((breln1 A) S)
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x6:((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of ((in ((kpair Xx) Xy)) S)
% Found (fun (x6:((in ((kpair Xx) Xy)) S))=> x6) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) S))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x7:((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of ((in ((kpair Xx) Xy)) R)
% Found (fun (x7:((in ((kpair Xx) Xy)) R))=> x7) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) R))
% Found x600000000:=(x60000000 x4):(((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (x60000000 x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((x6000000 x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (((x600000 x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((x60000 x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (((((x6000 A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> (((((x600 A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> ((((x60 A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> (((x6 A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) R)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found x0000000000:=(x000000000 x4):(((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (x000000000 x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((x00000000 x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (((x0000000 x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((x000000 x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (((((x00000 A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> (((((x0000 A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> ((((x000 A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> (((x00 A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4) as proof of (((in ((kpair Xx) Xy)) S)->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found ((or_ind00 ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4)) as proof of ((in ((kpair Xx) Xy)) ((binunion R) S))
% Found (((or_ind0 ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4)) as proof of ((in ((kpair Xx) Xy)) ((binunion R) S))
% Found ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4)) as proof of ((in ((kpair Xx) Xy)) ((binunion R) S))
% Found (fun (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of ((in ((kpair Xx) Xy)) ((binunion R) S))
% Found (fun (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))
% Found (fun (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))
% Found (fun (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))
% Found (fun (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))
% Found (fun (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))
% Found (fun (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))
% Found (fun (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))
% Found (fun (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))
% Found (fun (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (forall (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% Found (fun (x0:breln1unionIR) (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))
% Found (fun (x:breln1unionIL) (x0:breln1unionIR) (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S))))))))))))
% Found (fun (x:breln1unionIL) (x0:breln1unionIR) (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4))) as proof of (breln1unionIL->(breln1unionIR->(forall (A:fofType) (R:fofType), (((breln1 A) R)->(forall (S:fofType), (((breln1 A) S)->(forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) A)->(((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S))->((in ((kpair Xx) Xy)) ((binunion R) S)))))))))))))
% Got proof (fun (x:breln1unionIL) (x0:breln1unionIR) (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4)))
% Time elapsed = 4.981328s
% node=586 cost=997.000000 depth=25
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:breln1unionIL) (x0:breln1unionIR) (A:fofType) (R:fofType) (x1:((breln1 A) R)) (S:fofType) (x2:((breln1 A) S)) (Xx:fofType) (x3:((in Xx) A)) (Xy:fofType) (x4:((in Xy) A)) (x5:((or ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)))=> ((((fun (P:Prop) (x6:(((in ((kpair Xx) Xy)) R)->P)) (x7:(((in ((kpair Xx) Xy)) S)->P))=> ((((((or_ind ((in ((kpair Xx) Xy)) R)) ((in ((kpair Xx) Xy)) S)) P) x6) x7) x5)) ((in ((kpair Xx) Xy)) ((binunion R) S))) ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S)) (x9:((in Xx) A0))=> ((((((fun (A0:fofType) (x7:((breln1 A0) R)) (x8:((breln1 A0) S))=> (((((fun (A0:fofType) (x7:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x A0) R)) A0) x7) S)) A0) x7) x8) Xx)) A0) x7) x8) x9) Xy)) A) x1) x2) x3) x4)) ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S)) (x8:((in Xx) A0))=> ((((((fun (A0:fofType) (x6:((breln1 A0) R)) (x7:((breln1 A0) S))=> (((((fun (A0:fofType) (x6:((breln1 A0) R))=> ((((fun (A0:fofType)=> ((x0 A0) R)) A0) x6) S)) A0) x6) x7) Xx)) A0) x6) x7) x8) Xy)) A) x1) x2) x3) x4)))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------