TSTP Solution File: SEV097^5 by cocATP---0.2.0
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%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV097^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n095.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 13:33:44 EDT 2014
% Result : Theorem 86.51s
% Output : Proof 86.51s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV097^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n095.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 08:02:56 CDT 2014
% % CPUTime : 86.51
% 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 0x113eb00>, <kernel.Type object at 0x113ebd8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x11ee878>, <kernel.Type object at 0x113e3f8>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x113e1b8>, <kernel.Constant object at 0x113eb00>) of role type named z
% Using role type
% Declaring z:a
% FOF formula (<kernel.Constant object at 0x113e320>, <kernel.DependentProduct object at 0x113d680>) of role type named cR
% Using role type
% Declaring cR:(a->(a->Prop))
% FOF formula (<kernel.Constant object at 0x113eb00>, <kernel.DependentProduct object at 0x113dcb0>) of role type named f
% Using role type
% Declaring f:(a->(b->Prop))
% FOF formula (<kernel.Constant object at 0x113e1b8>, <kernel.DependentProduct object at 0x113d5a8>) of role type named cS
% Using role type
% Declaring cS:(b->(b->Prop))
% FOF formula (((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))->(((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))) of role conjecture named cTHM552C_pme
% Conjecture to prove = (((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))->(((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))):Prop
% Parameter b_DUMMY:b.
% We need to prove ['(((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))->(((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter z:a.
% Parameter cR:(a->(a->Prop)).
% Parameter f:(a->(b->Prop)).
% Parameter cS:(b->(b->Prop)).
% Trying to prove (((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))->(((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x5)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x5)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x3)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x3)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x3)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x3)) ((and (((f Xw) x3)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x1)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (((f Xw) x1)->False)) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found (or_introl0 ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (((f Xw) x1)->False)) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x1)) ((and (((f Xw) x1)->False)) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (not ((f Xw) x5))) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found (or_introl0 ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found or_introl00:=(or_introl0 ((and (not ((f Xw) x5))) ((cR Xx) z))):(((f Xx0) x6)->((or ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found (or_introl0 ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found ((or_introl ((f Xx0) x6)) ((and (not ((f Xw) x5))) ((cR Xx) z))) as proof of (((f Xx0) x6)->((or ((f Xx) x5)) ((and (not ((f Xw) x5))) ((cR Xx) z))))
% Found x6:((f Xx0) x5)
% Instantiate: Xx0:=Xx:a;x7:=x5:b
% Found x6 as proof of ((f Xx) x7)
% Found (or_introl00 x6) as proof of ((or ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z)))
% Found ((or_introl0 ((and (((f Xw) x7)->False)) ((cR Xx) z))) x6) as proof of ((or ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z)))
% Found (((or_introl ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z))) x6) as proof of ((or ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z)))
% Found (fun (Xw:a)=> (((or_introl ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z))) x6)) as proof of ((or ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z)))
% Found (fun (Xw:a)=> (((or_introl ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z))) x6)) as proof of (forall (Xw:a), ((or ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z))))
% Found (ex_intro000 (fun (Xw:a)=> (((or_introl ((f Xx) x7)) ((and (((f Xw) x7)->False)) ((cR Xx) z))) x6))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found ((ex_intro00 x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (((ex_intro0 (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x5:b) (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))) as proof of (((f Xx0) x5)->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))
% Found (fun (x5:b) (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))) as proof of (forall (x:b), (((f Xx0) x)->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))
% Found (ex_ind00 (fun (x5:b) (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found ((ex_ind0 ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (((fun (P:Prop) (x5:(forall (x:b), (((f Xx0) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx0) Xy))) P) x5) x30)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx0) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))) as proof of ((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))
% Found (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))) as proof of ((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))
% Found (and_rect10 (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found ((and_rect1 ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))) as proof of ((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))
% Found (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))) as proof of (((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))
% Found (and_rect00 (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found ((and_rect0 ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))))) as proof of ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))
% Found (fun (x0:((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))) (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))))) as proof of (forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))
% Found (fun (x:((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))) (x0:((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))) (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))))) as proof of (((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))))
% Found (fun (x:((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))) (x0:((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))) (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6)))))))))) as proof of (((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))->(((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))->(forall (Xx:a), ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))))))
% Got proof (fun (x:((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))) (x0:((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))) (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))))))
% Time elapsed = 85.442089s
% node=9589 cost=1687.000000 depth=32
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (x:((and (forall (Xu:a) (Xv:a) (Xw:a), (((and ((cR Xu) Xv)) ((cR Xw) Xv))->((cR Xu) Xw)))) (forall (Xx:a), ((cR Xx) Xx)))) (x0:((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))) (Xx:a)=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))->((forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((f Xx) Xy))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((f Xx) Xy1)) ((f Xx) Xy2))->((cS Xy1) Xy2))))) (forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2)))) P) x1) x0)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x1:((and (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))) (x2:(forall (Xx1:a) (Xx2:a) (Xy:b), (((and ((f Xx1) Xy)) ((f Xx2) Xy))->((cR Xx1) Xx2))))=> (((fun (P:Type) (x3:((forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))->((forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))->P)))=> (((((and_rect (forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2)))) P) x3) x1)) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x3:(forall (Xx0:a), ((ex b) (fun (Xy:b)=> ((f Xx0) Xy))))) (x4:(forall (Xx0:a) (Xy1:b) (Xy2:b), (((and ((f Xx0) Xy1)) ((f Xx0) Xy2))->((cS Xy1) Xy2))))=> (((fun (P:Prop) (x5:(forall (x:b), (((f Xx) x)->P)))=> (((((ex_ind b) (fun (Xy:b)=> ((f Xx) Xy))) P) x5) (x3 Xx))) ((ex b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z))))))) (fun (x5:b) (x6:((f Xx) x5))=> ((((ex_intro b) (fun (Xy:b)=> (forall (Xw:a), ((or ((f Xx) Xy)) ((and (((f Xw) Xy)->False)) ((cR Xx) z)))))) x5) (fun (Xw:a)=> (((or_introl ((f Xx) x5)) ((and (((f Xw) x5)->False)) ((cR Xx) z))) x6))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------