TSTP Solution File: SEU630^2 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU630^2 : TPTP v6.1.0. Released v3.7.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n185.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 13:32:39 EDT 2014
% Result : Theorem 2.37s
% Output : Proof 2.37s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU630^2 : TPTP v6.1.0. Released v3.7.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n185.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 10:57:36 CDT 2014
% % CPUTime : 2.37
% 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 0x16840e0>, <kernel.DependentProduct object at 0x16a3b90>) of role type named in_type
% Using role type
% Declaring in:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x16ffc68>, <kernel.Single object at 0x118d128>) of role type named emptyset_type
% Using role type
% Declaring emptyset:fofType
% FOF formula (<kernel.Constant object at 0x16840e0>, <kernel.DependentProduct object at 0x16a3dd0>) of role type named setadjoin_type
% Using role type
% Declaring setadjoin:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x118d128>, <kernel.DependentProduct object at 0x16a37a0>) of role type named powerset_type
% Using role type
% Declaring powerset:(fofType->fofType)
% FOF formula (<kernel.Constant object at 0x118d128>, <kernel.DependentProduct object at 0x16a35a8>) of role type named dsetconstr_type
% Using role type
% Declaring dsetconstr:(fofType->((fofType->Prop)->fofType))
% FOF formula (<kernel.Constant object at 0x16a3ab8>, <kernel.Sort object at 0x118c5a8>) of role type named dsetconstrI_type
% Using role type
% Declaring dsetconstrI:Prop
% FOF formula (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))) of role definition named dsetconstrI
% A new definition: (((eq Prop) dsetconstrI) (forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))))
% Defined: dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy)))))))
% FOF formula (<kernel.Constant object at 0x1191908>, <kernel.DependentProduct object at 0x1191e60>) of role type named binunion_type
% Using role type
% Declaring binunion:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0x1191050>, <kernel.DependentProduct object at 0x16a3dd0>) of role type named kpair_type
% Using role type
% Declaring kpair:(fofType->(fofType->fofType))
% FOF formula (((eq (fofType->(fofType->fofType))) kpair) (fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))) of role definition named kpair
% A new definition: (((eq (fofType->(fofType->fofType))) kpair) (fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))))
% Defined: kpair:=(fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset)))
% FOF formula (<kernel.Constant object at 0x1191e60>, <kernel.DependentProduct object at 0x16a37a0>) of role type named cartprod_type
% Using role type
% Declaring cartprod:(fofType->(fofType->fofType))
% FOF formula (((eq (fofType->(fofType->fofType))) cartprod) (fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz)))))))))))) of role definition named cartprod
% A new definition: (((eq (fofType->(fofType->fofType))) cartprod) (fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz))))))))))))
% Defined: cartprod:=(fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz)))))))))))
% FOF formula (<kernel.Constant object at 0x1191e60>, <kernel.Sort object at 0x118c5a8>) of role type named ubforcartprodlem3_type
% Using role type
% Declaring ubforcartprodlem3:Prop
% FOF formula (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))) of role definition named ubforcartprodlem3
% A new definition: (((eq Prop) ubforcartprodlem3) (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))))
% Defined: ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))))))
% FOF formula (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))) of role conjecture named cartprodpairin
% Conjecture to prove = (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))):Prop
% We need to prove ['(dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))))']
% Parameter fofType:Type.
% Parameter in:(fofType->(fofType->Prop)).
% Parameter emptyset:fofType.
% Parameter setadjoin:(fofType->(fofType->fofType)).
% Parameter powerset:(fofType->fofType).
% Parameter dsetconstr:(fofType->((fofType->Prop)->fofType)).
% Definition dsetconstrI:=(forall (A:fofType) (Xphi:(fofType->Prop)) (Xx:fofType), (((in Xx) A)->((Xphi Xx)->((in Xx) ((dsetconstr A) (fun (Xy:fofType)=> (Xphi Xy))))))):Prop.
% Parameter binunion:(fofType->(fofType->fofType)).
% Definition kpair:=(fun (Xx:fofType) (Xy:fofType)=> ((setadjoin ((setadjoin Xx) emptyset)) ((setadjoin ((setadjoin Xx) ((setadjoin Xy) emptyset))) emptyset))):(fofType->(fofType->fofType)).
% Definition cartprod:=(fun (A:fofType) (B:fofType)=> ((dsetconstr (powerset (powerset ((binunion A) B)))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((and ((in Xy) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) Xx) ((kpair Xy) Xz))))))))))):(fofType->(fofType->fofType)).
% Definition ubforcartprodlem3:=(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B)))))))):Prop.
% Trying to prove (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))))
% Found x0000000:=(x000000 x2):((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found (x000000 x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found ((x00000 Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found (((x0000 x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found ((((x000 Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found (((((x00 B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found ((((((x0 A) B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found ((((((x0 A) B) Xx) x1) Xy) x2) as proof of ((in ((kpair Xx) Xy)) (powerset (powerset ((binunion A) B))))
% Found x1:((in Xx) A)
% Instantiate: x4:=Xx:fofType
% Found x1 as proof of ((in x4) A)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))):(((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) (fun (x:fofType)=> ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz))))))))
% Found (eta_expansion_dep00 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz))))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))))
% Found (fun (x4:fofType)=> ((eq_ref Prop) (f x4))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and ((in x) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x) Xz))))))))
% Found x2:((in Xy) B)
% Instantiate: x5:=Xy:fofType
% Found x2 as proof of ((in x5) B)
% Found eq_ref00:=(eq_ref0 ((kpair Xx) Xy)):(((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))
% Found (eq_ref0 ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))
% Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))
% Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))
% Found ((eq_ref fofType) ((kpair Xx) Xy)) as proof of (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))
% Found ((conj10 x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)))
% Found (((conj1 (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)))
% Found ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)))
% Found ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy))) as proof of ((and ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5)))
% Found (ex_intro010 ((((conj ((in x5) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) x5))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))
% Found ((ex_intro01 Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))
% Found (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))
% Found (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))) as proof of ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))
% Found ((conj00 x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))
% Found (((conj0 ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))
% Found ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))
% Found ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))) as proof of ((and ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))))
% Found (ex_intro000 ((((conj ((in x4) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair x4) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz))))))))
% Found ((ex_intro00 Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz))))))))
% Found (((ex_intro0 (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) (((ex_intro0 (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz))))))))
% Found ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz))))))))
% Found ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))) as proof of ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz))))))))
% Found ((x300 ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B))
% Found (((x30 ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B))
% Found ((((x3 (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B))
% Found (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B))
% Found (fun (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of ((in ((kpair Xx) Xy)) ((cartprod A) B))
% Found (fun (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))
% Found (fun (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))
% Found (fun (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))
% Found (fun (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))
% Found (fun (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))
% Found (fun (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))
% Found (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B)))))))
% Found (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy)))))))) as proof of (dsetconstrI->(ubforcartprodlem3->(forall (A:fofType) (B:fofType) (Xx:fofType), (((in Xx) A)->(forall (Xy:fofType), (((in Xy) B)->((in ((kpair Xx) Xy)) ((cartprod A) B))))))))
% Got proof (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))))
% Time elapsed = 2.022636s
% node=291 cost=1508.000000 depth=33
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:dsetconstrI) (x0:ubforcartprodlem3) (A:fofType) (B:fofType) (Xx:fofType) (x1:((in Xx) A)) (Xy:fofType) (x2:((in Xy) B))=> (((((x (powerset (powerset ((binunion A) B)))) (fun (x7:fofType)=> ((ex fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) x7) ((kpair Xy0) Xz)))))))))) ((kpair Xx) Xy)) ((((((x0 A) B) Xx) x1) Xy) x2)) ((((ex_intro fofType) (fun (Xy0:fofType)=> ((and ((in Xy0) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xy0) Xz)))))))) Xx) ((((conj ((in Xx) A)) ((ex fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz)))))) x1) ((((ex_intro fofType) (fun (Xz:fofType)=> ((and ((in Xz) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xz))))) Xy) ((((conj ((in Xy) B)) (((eq fofType) ((kpair Xx) Xy)) ((kpair Xx) Xy))) x2) ((eq_ref fofType) ((kpair Xx) Xy))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------