TSTP Solution File: SYO242^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SYO242^5 : TPTP v7.5.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 0s
% DateTime : Tue Mar 29 00:50:59 EDT 2022
% Result : Timeout 286.22s 286.62s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.11 % Problem : SYO242^5 : TPTP v7.5.0. Released v4.0.0.
% 0.07/0.12 % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.12/0.33 % Computer : n002.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPUModel : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % RAMPerCPU : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % DateTime : Fri Mar 11 21:13:09 EST 2022
% 0.12/0.33 % CPUTime :
% 0.12/0.34 ModuleCmd_Load.c(213):ERROR:105: Unable to locate a modulefile for 'python/python27'
% 0.12/0.34 Python 2.7.5
% 3.21/3.45 Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 3.21/3.45 FOF formula (<kernel.Constant object at 0x1bbc368>, <kernel.Type object at 0x1bbc638>) of role type named a_type
% 3.21/3.45 Using role type
% 3.21/3.45 Declaring a:Type
% 3.21/3.45 FOF formula (((ex ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))) of role conjecture named cTHM539
% 3.21/3.45 Conjecture to prove = (((ex ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))):Prop
% 3.21/3.45 Parameter a_DUMMY:a.
% 3.21/3.45 We need to prove ['(((ex ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))']
% 3.21/3.45 Parameter a:Type.
% 3.21/3.45 Trying to prove (((ex ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X))))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 3.21/3.45 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (x:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((x Xx) Xy)) ((x Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x Xv) Xz))))))))))
% 3.21/3.45 Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 3.21/3.45 Found ((eta_expansion_dep0 (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 3.21/3.45 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (x:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((x Xx) Xy)) ((x Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x Xv) Xz))))))))))
% 4.78/4.97 Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found ((eta_expansion_dep0 (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 4.78/4.97 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 5.89/6.09 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x3:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b)
% 5.89/6.09 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) (fun (x:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 5.89/6.09 Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b)
% 5.89/6.09 Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b)
% 5.89/6.09 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b)
% 5.89/6.09 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b)
% 7.29/7.47 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b)
% 7.29/7.47 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 7.29/7.47 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.47 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.47 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.47 Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.47 Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 7.29/7.47 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 7.29/7.47 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.48 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.48 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 7.29/7.48 Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))
% 8.00/8.17 Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 8.00/8.17 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))):(((x2 Xy) Xx)->((x2 Xy) Xx))
% 8.00/8.17 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 8.00/8.17 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 8.00/8.17 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 8.00/8.17 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 8.00/8.17 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 8.00/8.17 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xx) Xy)->(((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 8.00/8.17 Found (and_rect00 (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 8.00/8.17 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 8.00/8.17 Found (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 8.00/8.17 Found (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 8.00/8.17 Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 8.00/8.17 Found ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 8.00/8.17 Found (fun (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 8.00/8.17 Found (fun (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy))
% 9.19/9.42 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 9.19/9.42 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 9.19/9.42 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))):(((x0 Xy) Xx)->((x0 Xy) Xx))
% 9.19/9.42 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 9.19/9.42 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 9.19/9.42 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 9.19/9.42 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 9.19/9.42 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 9.19/9.42 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 9.19/9.42 Found (and_rect00 (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 9.19/9.42 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 9.19/9.42 Found (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 9.19/9.42 Found (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 9.19/9.42 Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 9.19/9.42 Found ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 9.19/9.42 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 9.74/9.94 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 9.74/9.94 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 9.74/9.94 Found ex_ind00:=(ex_ind0 (((eq a) Xx) Xy)):((forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xx) Xy)))->(((eq a) Xx) Xy))
% 9.74/9.94 Instantiate: x0:=(fun (x4:a) (x30:a)=> (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) x30) x4)))):(a->(a->Prop))
% 9.74/9.94 Found (fun (x3:((x0 Xx) Xy))=> ex_ind00) as proof of (((x0 Xy) Xx)->(((eq a) Xx) Xy))
% 9.74/9.94 Found (fun (x3:((x0 Xx) Xy))=> ex_ind00) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->(((eq a) Xx) Xy)))
% 9.74/9.94 Found (and_rect00 (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found ((and_rect0 (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq a) Xx) Xy))
% 9.74/9.94 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xx) Xy)))
% 9.74/9.94 Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)))) as proof of (((eq a) Xx) Xy)
% 9.74/9.94 Found ((ex_ind0 (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> (ex_ind0 (((eq a) Xx) Xy)))))) as proof of (((eq a) Xx) Xy)
% 9.74/9.95 Found (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)))))) as proof of (((eq a) Xx) Xy)
% 9.74/9.95 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy))))))) as proof of (((eq a) Xx) Xy)
% 9.74/9.95 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy))))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 9.74/9.95 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy))))))) as proof of (forall (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 9.74/9.95 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x1) x)) (((eq a) Xx) Xy))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 22.46/22.65 Found x2:(S x1)
% 22.46/22.65 Instantiate: x3:=x1:a
% 22.46/22.65 Found x2 as proof of (S x3)
% 22.46/22.65 Found x2:(S x1)
% 22.46/22.65 Instantiate: x3:=x1:a
% 22.46/22.65 Found x2 as proof of (S x3)
% 22.46/22.65 Found x3:(S x2)
% 22.46/22.65 Instantiate: x1:=x2:a
% 22.46/22.65 Found x3 as proof of (S x1)
% 22.46/22.65 Found x3:(S x2)
% 22.46/22.65 Instantiate: x1:=x2:a
% 22.46/22.65 Found x3 as proof of (S x1)
% 22.46/22.65 Found x4:(P Xz)
% 22.46/22.65 Instantiate: x3:=Xz:a
% 22.46/22.65 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 22.46/22.65 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 22.46/22.65 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 22.46/22.65 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 22.46/22.65 Found x2:(P Xz)
% 22.46/22.65 Instantiate: x1:=Xz:a
% 22.46/22.65 Found (fun (x2:(P Xz))=> x2) as proof of (P x1)
% 22.46/22.65 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((P Xz)->(P x1))
% 22.46/22.65 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (((eq a) Xz) x1)
% 22.46/22.65 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 22.46/22.65 Found x4:(P Xz)
% 22.46/22.65 Instantiate: x1:=Xz:a
% 22.46/22.65 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 22.46/22.65 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 22.46/22.65 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 22.46/22.65 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x5:(S x4)
% 22.46/22.65 Instantiate: x3:=x4:a
% 22.46/22.65 Found x5 as proof of (S x3)
% 22.46/22.65 Found x3:(S x2)
% 22.46/22.65 Instantiate: x1:=x2:a
% 22.46/22.65 Found (fun (x3:(S x2))=> x3) as proof of (S x1)
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x5:(S x4)
% 22.46/22.65 Instantiate: x3:=x4:a
% 22.46/22.65 Found x5 as proof of (S x3)
% 22.46/22.65 Found x3:(S x2)
% 22.46/22.65 Instantiate: x1:=x2:a
% 22.46/22.65 Found (fun (x3:(S x2))=> x3) as proof of (S x1)
% 22.46/22.65 Found x5:(S x4)
% 22.46/22.65 Instantiate: x3:=x4:a
% 22.46/22.65 Found x5 as proof of (S x3)
% 22.46/22.65 Found x5:(S x4)
% 22.46/22.65 Instantiate: x3:=x4:a
% 22.46/22.65 Found x5 as proof of (S x3)
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x2:(S x1)
% 22.46/22.65 Instantiate: x5:=x1:a
% 22.46/22.65 Found x2 as proof of (S x5)
% 22.46/22.65 Found x4:(S x3)
% 22.46/22.65 Instantiate: x5:=x3:a
% 22.46/22.65 Found x4 as proof of (S x5)
% 22.46/22.65 Found x2:(S x1)
% 22.46/22.65 Instantiate: x5:=x1:a
% 22.46/22.65 Found x2 as proof of (S x5)
% 22.46/22.65 Found x2:(S x1)
% 22.46/22.65 Instantiate: x3:=x1:a
% 22.46/22.65 Found x2 as proof of (S x3)
% 22.46/22.65 Found x5:(S x4)
% 22.46/22.65 Instantiate: x3:=x4:a
% 22.46/22.65 Found x5 as proof of (S x3)
% 22.46/22.65 Found x4:(P Xz)
% 22.46/22.65 Instantiate: x3:=Xz:a
% 22.46/22.65 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 22.46/22.65 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 22.46/22.65 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 22.46/22.65 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x3:=x4:a
% 27.62/27.80 Found x5 as proof of (S x3)
% 27.62/27.80 Found x2:(S x1)
% 27.62/27.80 Instantiate: x3:=x1:a
% 27.62/27.80 Found x2 as proof of (S x3)
% 27.62/27.80 Found x3:(S x2)
% 27.62/27.80 Instantiate: x1:=x2:a
% 27.62/27.80 Found x3 as proof of (S x1)
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x1:=x4:a
% 27.62/27.80 Found x5 as proof of (S x1)
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x3:=x4:a
% 27.62/27.80 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 27.62/27.80 Found x4:(P Xz)
% 27.62/27.80 Instantiate: x1:=Xz:a
% 27.62/27.80 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 27.62/27.80 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 27.62/27.80 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 27.62/27.80 Found x3:(S x2)
% 27.62/27.80 Instantiate: x1:=x2:a
% 27.62/27.80 Found x3 as proof of (S x1)
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x1:=x4:a
% 27.62/27.80 Found x5 as proof of (S x1)
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x3:=x4:a
% 27.62/27.80 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 27.62/27.80 Found x6:(P Xz)
% 27.62/27.80 Instantiate: x5:=Xz:a
% 27.62/27.80 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 27.62/27.80 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 27.62/27.80 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 27.62/27.80 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 27.62/27.80 Found x6:(P Xz)
% 27.62/27.80 Instantiate: x5:=Xz:a
% 27.62/27.80 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 27.62/27.80 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 27.62/27.80 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 27.62/27.80 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x3:=x4:a
% 27.62/27.80 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 27.62/27.80 Found x5:(S x4)
% 27.62/27.80 Instantiate: x3:=x4:a
% 27.62/27.80 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 27.62/27.80 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 27.62/27.80 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (fun (x3:(S x2))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)) as proof of ((S x2)->(((eq a) Xz) x1))
% 27.62/27.80 Found (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x1)))
% 27.62/27.80 Found (ex_ind00 (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found ((ex_ind0 (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x1)
% 27.62/27.80 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 27.62/27.80 Found x6:(P Xz)
% 27.62/27.80 Instantiate: x3:=Xz:a
% 27.62/27.80 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 27.62/27.80 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 29.37/29.59 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 29.37/29.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 29.37/29.59 Found x6:(P Xz)
% 29.37/29.59 Instantiate: x3:=Xz:a
% 29.37/29.59 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 29.37/29.59 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 29.37/29.59 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 29.37/29.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 29.37/29.59 Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% 29.37/29.59 Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 29.37/29.59 Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% 29.37/29.59 Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 29.37/29.59 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 29.37/29.59 Found x4:(P Xz)
% 29.37/29.59 Instantiate: x1:=Xz:a
% 29.37/29.59 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 29.37/29.59 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 29.37/29.59 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 29.37/29.59 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 29.37/29.59 Found (fun (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((x0 x1) Xz)
% 29.37/29.59 Found (fun (x2:a) (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((S x2)->((x0 x1) Xz))
% 29.37/29.59 Found (fun (x2:a) (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x:a), ((S x)->((x0 x1) Xz)))
% 29.37/29.59 Found (ex_ind00 (fun (x2:a) (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 29.37/29.59 Found ((ex_ind0 ((x0 x1) Xz)) (fun (x2:a) (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 30.47/30.66 Found (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) ((x0 x1) Xz)) (fun (x2:a) (x3:(S x2)) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 30.47/30.66 Found x6:(P Xz)
% 30.47/30.66 Instantiate: x1:=Xz:a
% 30.47/30.66 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 30.47/30.66 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 30.47/30.66 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 30.47/30.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 30.47/30.66 Found x6:(P Xz)
% 30.47/30.66 Instantiate: x1:=Xz:a
% 30.47/30.66 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 30.47/30.66 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 30.47/30.66 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 30.47/30.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 30.47/30.66 Found x6:(P Xz)
% 30.47/30.66 Instantiate: x3:=Xz:a
% 30.47/30.66 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 30.47/30.66 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 30.47/30.66 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 30.47/30.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 30.47/30.66 Found (fun (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((x0 x3) Xz)
% 30.47/30.66 Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))->((x0 x3) Xz))
% 30.47/30.66 Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->((x0 x3) Xz)))
% 30.47/30.66 Found (ex_ind10 (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 30.47/30.66 Found ((ex_ind1 ((x0 x3) Xz)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 30.47/30.66 Found (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x4) x)) ((x0 x3) Xz)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 32.83/33.05 Found x3:=??:a
% 32.83/33.05 Found x3 as proof of a
% 32.83/33.05 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 32.83/33.05 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 32.83/33.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 32.83/33.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 32.83/33.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 32.83/33.05 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 32.83/33.05 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 32.83/33.05 Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))):(((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 32.83/33.05 Found (eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 32.83/33.05 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 32.83/33.05 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 32.83/33.05 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 32.83/33.05 Found x5:(S x4)
% 32.83/33.05 Instantiate: x3:=x4:a
% 32.83/33.05 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 32.83/33.05 Found x5:(S x4)
% 32.83/33.05 Instantiate: x3:=x4:a
% 32.83/33.05 Found (fun (x5:(S x4))=> x5) as proof of (S x3)
% 32.83/33.05 Found x3:=??:a
% 32.83/33.05 Found x3 as proof of a
% 32.83/33.05 Found x6:(P Xz)
% 32.83/33.05 Instantiate: x1:=Xz:a
% 32.83/33.05 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 32.83/33.05 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 32.83/33.05 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 32.83/33.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 32.83/33.05 Found (fun (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((x0 x1) Xz)
% 32.83/33.05 Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))->((x0 x1) Xz))
% 32.83/33.05 Found (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->((x0 x1) Xz)))
% 32.83/33.05 Found (ex_ind10 (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 34.98/35.24 Found ((ex_ind1 ((x0 x1) Xz)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 34.98/35.24 Found (((fun (P:Prop) (x4:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x4) x)) ((x0 x1) Xz)) (fun (x4:((a->Prop)->a)) (x5:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x4 X))))) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 34.98/35.24 Found x5:=??:a
% 34.98/35.24 Found x5 as proof of a
% 34.98/35.24 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 34.98/35.24 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 34.98/35.24 Found ((ex_intro11 x5) ((eq_ref a) Xz)) as proof of ((x2 x5) Xz)
% 34.98/35.24 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((x2 x5) Xz)
% 34.98/35.24 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 34.98/35.24 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 34.98/35.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 34.98/35.24 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 34.98/35.24 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 34.98/35.24 Found x5:(S x4)
% 34.98/35.24 Instantiate: x1:=x4:a
% 34.98/35.24 Found (fun (x5:(S x4))=> x5) as proof of (S x1)
% 34.98/35.24 Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% 34.98/35.24 Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 34.98/35.24 Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(a->Prop))->Prop)) a0) a0)
% 34.98/35.24 Found (eq_ref0 a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 34.98/35.24 Found ((eq_ref ((a->(a->Prop))->Prop)) a0) as proof of (((eq ((a->(a->Prop))->Prop)) a0) b)
% 34.98/35.24 Found x5:=x3:a
% 34.98/35.24 Found x5 as proof of a
% 34.98/35.24 Found x5:(S x4)
% 34.98/35.24 Instantiate: x1:=x4:a
% 34.98/35.24 Found (fun (x5:(S x4))=> x5) as proof of (S x1)
% 34.98/35.24 Found x4:(P Xz)
% 34.98/35.24 Instantiate: x1:=Xz:a
% 34.98/35.24 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 34.98/35.24 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 34.98/35.24 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 34.98/35.24 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 39.17/39.41 Found (fun (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((x0 x1) Xz)
% 39.17/39.41 Found (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))->((x0 x1) Xz))
% 39.17/39.41 Found (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->((x0 x1) Xz)))
% 39.17/39.41 Found (ex_ind00 (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 39.17/39.41 Found ((ex_ind0 ((x0 x1) Xz)) (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 39.17/39.41 Found (((fun (P:Prop) (x2:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P) x2) x)) ((x0 x1) Xz)) (fun (x2:((a->Prop)->a)) (x3:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x2 X))))) (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4)) as proof of ((x0 x1) Xz)
% 39.17/39.41 Found x5:=??:a
% 39.17/39.41 Found x5 as proof of a
% 39.17/39.41 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 39.17/39.41 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 39.17/39.41 Found ((ex_intro11 x5) ((eq_ref a) Xz)) as proof of ((x0 x5) Xz)
% 39.17/39.41 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((x0 x5) Xz)
% 39.17/39.41 Found x3:=??:a
% 39.17/39.41 Found x3 as proof of a
% 39.17/39.41 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 39.17/39.41 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 39.17/39.41 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 39.17/39.41 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 39.17/39.41 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 39.17/39.41 Found x5:=x3:a
% 39.17/39.41 Found x5 as proof of a
% 39.17/39.41 Found x3:=x4:a
% 39.17/39.41 Found x3 as proof of a
% 39.17/39.41 Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))):(((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 39.17/39.41 Found (eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) b)
% 39.17/39.41 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) b)
% 42.66/42.91 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) b)
% 42.66/42.91 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))) b)
% 42.66/42.91 Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% 42.66/42.91 Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% 42.66/42.91 Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 42.66/42.91 Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 42.66/42.91 Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 42.66/42.91 Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% 42.66/42.91 Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 42.66/42.91 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 42.66/42.91 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 42.66/42.91 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 42.66/42.91 Found x3:=??:a
% 42.66/42.91 Found x3 as proof of a
% 42.66/42.91 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 42.66/42.91 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 42.66/42.91 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 42.66/42.91 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 42.66/42.91 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 42.66/42.91 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 42.66/42.91 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 42.66/42.91 Found x3:=x4:a
% 42.66/42.91 Found x3 as proof of a
% 42.66/42.91 Found eq_ref00:=(eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))):(((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 42.66/42.91 Found (eq_ref0 (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 42.66/42.91 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 42.66/42.91 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 47.64/47.83 Found ((eq_ref Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) as proof of (((eq Prop) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))) b)
% 47.64/47.83 Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 47.64/47.83 Found (eq_ref0 (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) b)
% 47.64/47.83 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) b)
% 47.64/47.83 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) b)
% 47.64/47.83 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) b)
% 47.64/47.83 Found x5:(S x4)
% 47.64/47.83 Instantiate: x1:=x4:a
% 47.64/47.83 Found (fun (x5:(S x4))=> x5) as proof of (S x1)
% 47.64/47.83 Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 47.64/47.83 Found (eq_ref0 (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) b)
% 49.93/50.13 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) b)
% 49.93/50.13 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) b)
% 49.93/50.13 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) b)
% 49.93/50.13 Found x5:(S x4)
% 49.93/50.13 Instantiate: x1:=x4:a
% 49.93/50.13 Found (fun (x5:(S x4))=> x5) as proof of (S x1)
% 49.93/50.13 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 49.93/50.13 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 49.93/50.13 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 49.93/50.13 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 49.93/50.13 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 49.93/50.13 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 49.93/50.13 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 49.93/50.13 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 49.93/50.13 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 52.01/52.24 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 52.01/52.24 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 52.01/52.24 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 52.01/52.24 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 52.01/52.24 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found (fun (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of ((S x4)->(((eq a) Xz) x3))
% 52.01/52.24 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x3)))
% 52.01/52.24 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found ((ex_ind1 (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x3)
% 52.01/52.24 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 52.01/52.24 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 52.01/52.24 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.01/52.24 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.01/52.24 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.01/52.24 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.01/52.24 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 52.01/52.24 Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% 52.01/52.24 Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 52.86/53.11 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 52.86/53.11 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 52.86/53.11 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 52.86/53.11 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 52.86/53.11 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 52.86/53.11 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 52.86/53.11 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 52.86/53.11 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 53.03/53.26 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 53.03/53.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 53.03/53.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 53.03/53.26 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 53.03/53.26 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 53.03/53.26 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.03/53.26 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.15/53.37 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.15/53.37 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.15/53.37 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 53.15/53.37 Found x6:(P Xz)
% 53.15/53.37 Instantiate: x3:=Xz:a
% 53.15/53.37 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 53.15/53.37 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 53.15/53.37 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 53.15/53.37 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 53.15/53.37 Found (fun (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((x0 x3) Xz)
% 53.15/53.37 Found (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((S x4)->((x0 x3) Xz))
% 55.26/55.54 Found (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x:a), ((S x)->((x0 x3) Xz)))
% 55.26/55.54 Found (ex_ind10 (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 55.26/55.54 Found ((ex_ind1 ((x0 x3) Xz)) (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 55.26/55.54 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((x0 x3) Xz)) (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x3) Xz)
% 55.26/55.54 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 55.26/55.54 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (fun (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of ((S x4)->(((eq a) Xz) x1))
% 55.26/55.54 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x1)))
% 55.26/55.54 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found ((ex_ind1 (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 55.26/55.54 Found x6:(P Xz)
% 55.26/55.54 Instantiate: x1:=Xz:a
% 55.26/55.54 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 55.26/55.54 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 55.26/55.54 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 55.26/55.54 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 55.26/55.54 Found (fun (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((x0 x1) Xz)
% 55.26/55.54 Found (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((S x4)->((x0 x1) Xz))
% 55.26/55.54 Found (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x:a), ((S x)->((x0 x1) Xz)))
% 55.26/55.54 Found (ex_ind10 (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 55.26/55.54 Found ((ex_ind1 ((x0 x1) Xz)) (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 59.55/59.86 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((x0 x1) Xz)) (fun (x4:a) (x5:(S x4)) (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6)) as proof of ((x0 x1) Xz)
% 59.55/59.86 Found x10:(P0 (f x0))
% 59.55/59.86 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 59.55/59.86 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 59.55/59.86 Found x10:(P0 (f x0))
% 59.55/59.86 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 59.55/59.86 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 59.55/59.86 Found x4:(P Xz)
% 59.55/59.86 Instantiate: x3:=Xz:a
% 59.55/59.86 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 59.55/59.86 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 59.55/59.86 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 59.55/59.86 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 59.55/59.86 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 59.55/59.86 Found x3:=??:a
% 59.55/59.86 Found x3 as proof of a
% 59.55/59.86 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 59.55/59.86 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 59.55/59.86 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 59.55/59.86 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 59.55/59.86 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 59.55/59.86 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found (fun (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((S x4)->((x2 x3) Xz))
% 59.55/59.86 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of (forall (x:a), ((S x)->((x2 x3) Xz)))
% 59.55/59.86 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found ((ex_ind1 ((x2 x3) Xz)) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((x2 x3) Xz)) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x2 x3) Xz)
% 59.55/59.86 Found x2:(P Xz)
% 59.55/59.86 Instantiate: x1:=Xz:a
% 59.55/59.86 Found (fun (x2:(P Xz))=> x2) as proof of (P x1)
% 59.55/59.86 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((P Xz)->(P x1))
% 59.55/59.86 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 59.55/59.86 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (((eq a) Xz) x1)
% 59.55/59.86 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 59.55/59.86 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 59.55/59.86 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.55/59.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.55/59.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.55/59.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.55/59.86 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 59.55/59.86 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.55/59.86 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.65/59.88 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 59.84/60.05 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.84/60.05 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.84/60.05 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.84/60.05 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 59.84/60.05 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 59.84/60.05 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 59.84/60.05 Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 61.32/61.55 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 61.32/61.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 61.32/61.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 61.32/61.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 61.32/61.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 61.32/61.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 61.32/61.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 61.32/61.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 61.32/61.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 61.32/61.55 Found x4:(P Xz)
% 61.32/61.55 Instantiate: x1:=Xz:a
% 61.32/61.55 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 61.32/61.55 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 61.32/61.55 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 61.32/61.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 61.32/61.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 61.32/61.55 Found x3:=??:a
% 61.32/61.55 Found x3 as proof of a
% 61.32/61.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 61.32/61.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 61.32/61.55 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 61.32/61.55 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((x0 x3) Xz)
% 61.32/61.55 Found (fun (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((x0 x3) Xz)
% 61.32/61.55 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((S x4)->((x0 x3) Xz))
% 62.95/63.25 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of (forall (x:a), ((S x)->((x0 x3) Xz)))
% 62.95/63.25 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x0 x3) Xz)
% 62.95/63.25 Found ((ex_ind1 ((x0 x3) Xz)) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x0 x3) Xz)
% 62.95/63.25 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((x0 x3) Xz)) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((x0 x3) Xz)
% 62.95/63.25 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 62.95/63.25 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 62.95/63.25 Found x4:(P Xz)
% 62.95/63.25 Instantiate: x3:=Xz:a
% 62.95/63.25 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 62.95/63.25 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 62.95/63.25 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 62.95/63.25 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 62.95/63.25 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 62.95/63.25 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 62.95/63.25 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 62.95/63.25 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 62.95/63.25 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 62.95/63.25 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 62.95/63.25 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 62.95/63.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 62.95/63.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 62.95/63.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 62.95/63.25 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 62.95/63.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 69.74/70.00 Found x2:(P Xz)
% 69.74/70.00 Instantiate: x1:=Xz:a
% 69.74/70.00 Found (fun (x2:(P Xz))=> x2) as proof of (P x1)
% 69.74/70.00 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (((eq a) Xz) x1)
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 69.74/70.00 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 69.74/70.00 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 69.74/70.00 Found x4:(P Xz)
% 69.74/70.00 Instantiate: x1:=Xz:a
% 69.74/70.00 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 69.74/70.00 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 69.74/70.00 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 69.74/70.00 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 69.74/70.00 Found x30:(P0 (f x2))
% 69.74/70.00 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 69.74/70.00 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 69.74/70.00 Found x30:(P0 (f x2))
% 69.74/70.00 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 69.74/70.00 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 69.74/70.00 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 69.74/70.00 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 69.94/70.22 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 69.94/70.22 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 69.94/70.22 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 69.94/70.22 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 69.94/70.22 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 69.94/70.22 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 69.94/70.22 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 74.31/74.59 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 74.31/74.59 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 74.31/74.59 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 74.31/74.59 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 74.31/74.59 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.31/74.59 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.31/74.59 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.31/74.59 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.31/74.59 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 74.31/74.59 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.31/74.59 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.37/74.60 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.37/74.60 Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.37/74.60 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 74.37/74.60 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 74.37/74.61 Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.37/74.61 Found ((ex_ind0 (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.37/74.61 Found (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 74.46/74.71 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.46/74.71 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.46/74.71 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.46/74.71 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 74.46/74.71 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 74.46/74.71 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.46/74.71 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.50/74.72 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.50/74.72 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.50/74.72 Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.50/74.72 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 74.51/74.74 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 74.51/74.74 Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.51/74.74 Found ((ex_ind0 (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 74.51/74.74 Found (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 78.21/78.51 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 78.21/78.51 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 78.21/78.51 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 78.21/78.51 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 78.21/78.51 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 78.21/78.51 Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 78.21/78.51 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 78.21/78.51 Found x4:(P Xz)
% 78.21/78.51 Instantiate: x3:=Xz:a
% 78.21/78.51 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 78.21/78.51 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 78.21/78.51 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 78.21/78.51 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 78.21/78.51 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 78.21/78.51 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 78.21/78.51 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 78.21/78.51 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 78.21/78.51 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 78.21/78.51 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 78.21/78.51 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 78.21/78.51 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 78.21/78.51 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 78.21/78.51 Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 80.33/80.55 Found x3:=??:a
% 80.33/80.55 Found x3 as proof of a
% 80.33/80.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 80.33/80.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 80.33/80.55 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 80.33/80.55 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 80.33/80.55 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 80.33/80.55 Found x4:(P Xz)
% 80.33/80.55 Instantiate: x1:=Xz:a
% 80.33/80.55 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 80.33/80.55 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 80.33/80.55 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 80.33/80.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 80.33/80.55 Found x5:=??:a
% 80.33/80.55 Found x5 as proof of a
% 80.33/80.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 80.33/80.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 80.33/80.55 Found ((ex_intro11 x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 80.33/80.55 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 80.33/80.55 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 80.33/80.55 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 80.33/80.55 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 80.33/80.55 Found x6:(P Xz)
% 80.33/80.55 Instantiate: x5:=Xz:a
% 80.33/80.55 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 80.33/80.55 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 80.33/80.55 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 80.33/80.55 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 80.33/80.55 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 81.11/81.35 Found x6:(P Xz)
% 81.11/81.35 Instantiate: x5:=Xz:a
% 81.11/81.35 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 81.11/81.35 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 81.11/81.35 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 81.11/81.35 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 81.11/81.35 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 81.11/81.35 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 81.11/81.35 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.11/81.35 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.11/81.35 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.11/81.35 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.11/81.35 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 81.11/81.35 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 81.11/81.35 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 81.11/81.35 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 81.77/82.02 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.77/82.02 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.77/82.02 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.77/82.02 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 81.77/82.02 Found x3:=??:a
% 81.77/82.02 Found x3 as proof of a
% 81.77/82.02 Found x4:(P Xz)
% 81.77/82.02 Instantiate: x3:=Xz:a
% 81.77/82.02 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 81.77/82.02 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 81.77/82.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 81.77/82.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 81.77/82.02 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 81.77/82.02 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 81.77/82.02 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 81.77/82.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 83.82/84.14 Found x5:=x3:a
% 83.82/84.14 Found x5 as proof of a
% 83.82/84.14 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 83.82/84.14 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 83.82/84.14 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 83.82/84.14 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 83.82/84.14 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 83.82/84.14 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 83.82/84.14 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 83.82/84.14 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 83.82/84.14 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 83.82/84.14 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 83.82/84.14 Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 83.82/84.14 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 83.82/84.14 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 83.82/84.14 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 83.82/84.14 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 83.82/84.14 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 83.82/84.14 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 83.82/84.14 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 83.82/84.14 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 83.82/84.14 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 83.82/84.14 Found x5:=??:a
% 83.82/84.14 Found x5 as proof of a
% 83.82/84.14 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 83.82/84.14 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x5)
% 83.82/84.14 Found ((ex_intro11 x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 84.66/84.89 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 84.66/84.89 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x5))) x5) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x5)))
% 84.66/84.89 Found x3:=??:a
% 84.66/84.89 Found x3 as proof of a
% 84.66/84.89 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 84.66/84.89 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 84.66/84.89 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 84.66/84.89 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 84.66/84.89 Found x6:(P Xz)
% 84.66/84.89 Instantiate: x3:=Xz:a
% 84.66/84.89 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 84.66/84.89 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 84.66/84.89 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 84.66/84.89 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 84.66/84.89 Found x6:(P Xz)
% 84.66/84.89 Instantiate: x3:=Xz:a
% 84.66/84.89 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 84.66/84.89 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 84.66/84.89 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 84.66/84.89 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 84.66/84.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 84.66/84.89 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 84.66/84.89 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.66/84.89 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.66/84.89 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.66/84.89 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.66/84.89 Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.66/84.89 Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.66/84.89 Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.66/84.89 Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.66/84.89 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.89 Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.89 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.90 Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.90 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.90 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.66/84.91 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.66/84.91 Found ((eq_sym00 (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.66/84.91 Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.71/84.94 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) ((((eq_sym Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.71/84.94 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 84.71/84.94 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.71/84.94 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.71/84.94 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.71/84.94 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 84.71/84.94 Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.71/84.94 Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.71/84.94 Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.71/84.94 Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% 84.71/84.94 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.94 Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.94 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.96 Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.96 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.96 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0))
% 84.71/84.96 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.71/84.96 Found ((eq_sym00 (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.71/84.96 Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (((eq_sym0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 84.71/84.96 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) (f x0)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) ((((eq_sym Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 86.38/86.64 Found x4:(P Xz)
% 86.38/86.64 Instantiate: x1:=Xz:a
% 86.38/86.64 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 86.38/86.64 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 86.38/86.64 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 86.38/86.64 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 86.38/86.64 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 86.38/86.64 Found x5:=x3:a
% 86.38/86.64 Found x5 as proof of a
% 86.38/86.64 Found x3:=x4:a
% 86.38/86.64 Found x3 as proof of a
% 86.38/86.64 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 86.38/86.64 Found (eq_ref00 P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x5)
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 86.38/86.64 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 86.38/86.64 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 86.38/86.64 Found (eq_ref00 P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 86.38/86.64 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 87.12/87.38 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x5))
% 87.12/87.38 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x5)
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 87.12/87.38 Found x6:(P Xz)
% 87.12/87.38 Instantiate: x5:=Xz:a
% 87.12/87.38 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 87.12/87.38 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 87.12/87.38 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 87.12/87.38 Found x6:(P Xz)
% 87.12/87.38 Instantiate: x5:=Xz:a
% 87.12/87.38 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 87.12/87.38 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 87.12/87.38 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 87.12/87.38 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 87.12/87.38 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 87.12/87.38 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 87.12/87.38 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 87.12/87.38 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 87.12/87.38 Found x3:=??:a
% 89.78/90.05 Found x3 as proof of a
% 89.78/90.05 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 89.78/90.05 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 89.78/90.05 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 89.78/90.05 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 89.78/90.05 Found x6:(P Xz)
% 89.78/90.05 Instantiate: x1:=Xz:a
% 89.78/90.05 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 89.78/90.05 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 89.78/90.05 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 89.78/90.05 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 89.78/90.05 Found x6:(P Xz)
% 89.78/90.05 Instantiate: x1:=Xz:a
% 89.78/90.05 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 89.78/90.05 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 89.78/90.05 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 89.78/90.05 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 89.78/90.05 Found x3:=x4:a
% 89.78/90.05 Found x3 as proof of a
% 89.78/90.05 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 89.78/90.05 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 89.78/90.05 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 89.78/90.05 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 89.78/90.05 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 89.78/90.05 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 89.78/90.05 Found x6:(P Xz)
% 89.78/90.05 Instantiate: x3:=Xz:a
% 89.78/90.05 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 91.19/91.42 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 91.19/91.42 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 91.19/91.42 Found x6:(P Xz)
% 91.19/91.42 Instantiate: x3:=Xz:a
% 91.19/91.42 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 91.19/91.42 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 91.19/91.42 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 91.19/91.42 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 91.19/91.42 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 91.19/91.42 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 91.19/91.42 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 91.19/91.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 91.19/91.42 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 91.19/91.42 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x3:(S x2))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)) as proof of ((S x2)->(((eq a) Xz) x1))
% 91.19/91.42 Found (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x1)))
% 91.19/91.42 Found (ex_ind00 (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found ((ex_ind0 (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 91.19/91.42 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x1)
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x2:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x2) x00)) (((eq a) Xz) x1)) (fun (x2:a) (x3:(S x2))=> ((eq_ref a) Xz)))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 93.71/94.02 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 93.71/94.02 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 93.71/94.02 Found x6:(P Xz)
% 93.71/94.02 Instantiate: x1:=Xz:a
% 93.71/94.02 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 93.71/94.02 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 93.71/94.02 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 93.71/94.02 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 93.71/94.02 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 93.71/94.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 93.71/94.02 Found x6:(P Xz)
% 93.71/94.02 Instantiate: x1:=Xz:a
% 93.71/94.02 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 93.71/94.02 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 100.67/100.92 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 100.67/100.92 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 100.67/100.92 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 100.67/100.92 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 100.67/100.92 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 100.67/100.92 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 100.67/100.92 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 100.67/100.92 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 100.67/100.92 Found (eq_sym010 ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 100.67/100.92 Found ((eq_sym01 b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 100.67/100.92 Found (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 100.67/100.92 Found (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 100.67/100.92 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.92 Found (((eq_trans000 (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.92 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.92 Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.94 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.94 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 100.67/100.94 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 100.67/100.94 Found ((eq_sym00 (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 107.92/108.23 Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 107.92/108.23 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) ((((eq_sym Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 107.92/108.23 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 112.65/112.89 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found ((eq_sym00 Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (((eq_sym0 x5) Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 112.65/112.89 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 112.65/112.89 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 112.65/112.89 Found (eq_sym000 ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found ((eq_sym00 Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (((eq_sym0 x5) Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 112.65/112.89 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 112.65/112.89 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 112.65/112.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 112.65/112.89 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 112.65/112.89 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 112.65/112.89 Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 112.65/112.89 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 116.64/116.91 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 116.64/116.91 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 116.64/116.91 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 116.64/116.91 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 116.64/116.91 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 116.64/116.91 Found (eq_sym000 ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 116.64/116.91 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 116.64/116.91 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 116.64/116.91 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 116.64/116.91 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 116.64/116.91 Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 116.64/116.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 120.13/120.41 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 120.13/120.41 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 120.13/120.41 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 120.13/120.41 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 120.13/120.41 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 120.13/120.41 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 120.13/120.41 Found (eq_sym000 ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 120.13/120.41 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 120.13/120.41 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 120.13/120.41 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (((eq a) Xz) x1)
% 120.13/120.41 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 120.13/120.41 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 120.13/120.41 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 120.13/120.41 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 120.13/120.41 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.13/120.41 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.13/120.41 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.13/120.41 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.13/120.41 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 120.13/120.41 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.13/120.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.13/120.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.13/120.41 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.13/120.41 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.13/120.41 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.13/120.41 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.49 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.49 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.49 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.49 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 120.19/120.49 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.19/120.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.19/120.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.19/120.49 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 120.19/120.49 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 120.19/120.49 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.19/120.50 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.19/120.50 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.19/120.50 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 120.19/120.50 Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.50 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.50 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.50 Found (((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 120.19/120.50 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 125.01/125.30 Found ((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))))
% 125.01/125.30 Found x3:=??:a
% 125.01/125.30 Found x3 as proof of a
% 125.01/125.30 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 125.01/125.30 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (fun (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((S x4)->((ex a) (fun (Xt:a)=> (((eq a) Xz) x3))))
% 125.01/125.30 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of (forall (x:a), ((S x)->((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))))
% 125.01/125.30 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found ((ex_ind1 ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found x3:=??:a
% 125.01/125.30 Found x3 as proof of a
% 125.01/125.30 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 125.01/125.30 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 125.01/125.30 Found ((ex_intro11 x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 125.01/125.30 Found (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found (fun (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of ((S x4)->((ex a) (fun (Xt:a)=> (((eq a) Xz) x3))))
% 128.77/129.04 Found (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz))) as proof of (forall (x:a), ((S x)->((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))))
% 128.77/129.04 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found ((ex_ind1 ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))) (fun (x4:a) (x5:(S x4))=> (((ex_intro1 (fun (Xt:a)=> (((eq a) Xz) x3))) x3) ((eq_ref a) Xz)))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xz) x3)))
% 128.77/129.04 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 128.77/129.04 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found (fun (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of ((S x4)->(((eq a) Xz) x3))
% 128.77/129.04 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x3)))
% 128.77/129.04 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found ((ex_ind1 (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x3)
% 128.77/129.04 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 128.77/129.04 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x3)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 128.77/129.04 Found x10:(P0 (f x0))
% 128.77/129.04 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 128.77/129.04 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 128.77/129.04 Found x10:(P0 (f x0))
% 128.77/129.04 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 128.77/129.04 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 128.77/129.04 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 128.77/129.04 Found (eq_ref0 Xz) as proof of (((eq a) Xz) x1)
% 128.77/129.04 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 128.77/129.04 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) x1)
% 128.77/129.04 Found (fun (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 128.77/129.04 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of ((S x4)->(((eq a) Xz) x1))
% 132.62/132.89 Found (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)) as proof of (forall (x:a), ((S x)->(((eq a) Xz) x1)))
% 132.62/132.89 Found (ex_ind10 (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 132.62/132.89 Found ((ex_ind1 (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 132.62/132.89 Found (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 132.62/132.89 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (((eq a) Xz) x1)
% 132.62/132.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 132.62/132.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> (((fun (P:Prop) (x4:(forall (x:a), ((S x)->P)))=> (((((ex_ind a) (fun (Xu:a)=> (S Xu))) P) x4) x00)) (((eq a) Xz) x1)) (fun (x4:a) (x5:(S x4))=> ((eq_ref a) Xz)))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 132.62/132.89 Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% 132.62/132.89 Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% 132.62/132.89 Found eq_sym0 as proof of b
% 132.62/132.89 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 132.62/132.89 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 132.62/132.89 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 132.62/132.89 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 132.62/132.89 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 132.62/132.89 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 132.62/132.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 132.62/132.89 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 132.62/132.89 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 132.62/132.89 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 132.62/132.89 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 132.62/132.89 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 132.62/132.89 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 132.62/132.89 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 132.62/132.89 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 132.62/132.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 132.62/132.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 132.62/132.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 132.62/132.91 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 132.62/132.91 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 132.62/132.91 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 132.62/132.91 Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 132.62/132.91 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 132.62/132.91 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 133.06/133.30 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 133.06/133.30 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 133.06/133.30 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 133.06/133.30 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 133.06/133.30 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 133.06/133.30 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 133.06/133.30 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 133.06/133.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 133.06/133.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 133.06/133.30 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 133.06/133.30 Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 133.06/133.30 Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 133.06/133.30 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 133.06/133.30 Found (((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 136.03/136.30 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 136.03/136.30 Found ((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 136.03/136.30 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 136.03/136.30 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 136.03/136.30 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 136.03/136.30 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 136.03/136.30 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 136.03/136.30 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 136.03/136.30 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 136.03/136.30 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 136.03/136.30 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 136.03/136.30 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 136.03/136.30 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 136.03/136.30 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 139.49/139.75 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 139.49/139.75 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 139.49/139.75 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 139.49/139.75 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 139.49/139.75 Found x4:(P Xz)
% 139.49/139.75 Instantiate: x3:=Xz:a
% 139.49/139.75 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 139.49/139.75 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 139.49/139.75 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 139.49/139.75 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 139.49/139.75 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 139.49/139.75 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 139.49/139.75 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 139.49/139.75 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 139.49/139.75 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 139.49/139.75 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 139.49/139.75 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 139.49/139.75 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 139.49/139.75 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 139.49/139.75 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 139.49/139.75 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 139.49/139.75 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 139.49/139.75 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 139.49/139.75 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 139.49/139.75 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 139.49/139.75 Found x4:(P Xz)
% 139.49/139.75 Instantiate: x1:=Xz:a
% 139.49/139.75 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 139.49/139.75 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 142.92/143.18 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 142.92/143.18 Found x2:(P Xz)
% 142.92/143.18 Instantiate: x1:=Xz:a
% 142.92/143.18 Found (fun (x2:(P Xz))=> x2) as proof of (P x1)
% 142.92/143.18 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (fun (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 142.92/143.18 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (((eq a) Xz) x1)
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x2:(P Xz))=> x2) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 142.92/143.18 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 142.92/143.18 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 142.92/143.18 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 142.92/143.18 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 142.92/143.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 142.92/143.18 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 142.92/143.18 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 142.92/143.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 142.92/143.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 142.92/143.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 142.92/143.18 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 142.92/143.18 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 142.92/143.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 142.92/143.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 142.92/143.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 142.92/143.18 Found x60:(P Xz)
% 142.92/143.18 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 142.92/143.18 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 142.92/143.18 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 142.92/143.18 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 142.92/143.18 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 142.92/143.18 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 142.92/143.18 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 142.92/143.18 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 142.92/143.18 Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 142.92/143.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 152.46/152.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 152.46/152.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 152.46/152.73 Found x30:(P0 (f x2))
% 152.46/152.73 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 152.46/152.73 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 152.46/152.73 Found x30:(P0 (f x2))
% 152.46/152.73 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 152.46/152.73 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 152.46/152.73 Found x1:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x0 X))))
% 152.46/152.73 Instantiate: b:=(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x0 X)))):Prop
% 152.46/152.73 Found x1 as proof of b
% 152.46/152.73 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 152.46/152.73 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 152.46/152.73 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 152.46/152.73 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 152.46/152.73 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 152.46/152.73 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found x60:(P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 152.46/152.73 Found x60:(P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 152.46/152.73 Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% 152.46/152.73 Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% 152.46/152.73 Found eta_expansion as proof of b
% 152.46/152.73 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 152.46/152.73 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 152.46/152.73 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 152.46/152.73 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 152.46/152.73 Found x60:(P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 152.46/152.73 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 152.46/152.73 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 152.46/152.73 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 152.46/152.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.46/152.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.46/152.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.46/152.73 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 152.46/152.73 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 152.78/153.06 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 152.78/153.06 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.78/153.06 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.78/153.06 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 152.78/153.06 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 152.78/153.06 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 152.78/153.06 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 157.78/158.09 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (x:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((x Xx) Xy)) ((x Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x Xv) Xz))))))))))
% 157.78/158.09 Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 157.78/158.09 Found ((eta_expansion_dep0 (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 157.78/158.09 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 157.78/158.09 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 157.78/158.09 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 164.77/165.08 Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% 164.77/165.08 Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% 164.77/165.08 Found eta_expansion as proof of b
% 164.77/165.08 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((a->(a->Prop))->Prop)) b) (fun (x:(a->(a->Prop)))=> (b x)))
% 164.77/165.08 Found (eta_expansion_dep00 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 164.77/165.08 Found ((eta_expansion_dep0 (fun (x1:(a->(a->Prop)))=> Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 164.77/165.08 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 164.77/165.08 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 164.77/165.08 Found (((eta_expansion_dep (a->(a->Prop))) (fun (x1:(a->(a->Prop)))=> Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 164.77/165.08 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 164.77/165.08 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 164.77/165.08 Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 164.77/165.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 164.77/165.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 164.77/165.08 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 164.77/165.08 Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% 164.77/165.08 Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% 164.77/165.08 Found eta_expansion as proof of b
% 164.77/165.08 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 164.77/165.08 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 164.77/165.08 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 164.77/165.08 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 164.77/165.08 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 164.77/165.08 Found (eq_sym000 ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 164.77/165.08 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 164.77/165.08 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 164.77/165.08 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 164.77/165.08 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 164.77/165.08 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 164.77/165.08 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 164.77/165.08 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 164.77/165.08 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 164.77/165.08 Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% 164.77/165.08 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 169.88/170.18 Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 169.88/170.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 169.88/170.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 169.88/170.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 169.88/170.18 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 169.88/170.18 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 169.88/170.18 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 169.88/170.18 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 169.88/170.18 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 169.88/170.18 Found (eq_sym000 ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 169.88/170.18 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 169.88/170.18 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 169.88/170.18 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 169.88/170.18 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 169.88/170.18 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 169.88/170.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 169.88/170.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 169.88/170.18 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 169.88/170.18 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 169.88/170.18 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 169.88/170.18 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 169.88/170.18 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 169.88/170.18 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 169.88/170.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 169.88/170.18 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 169.88/170.18 Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% 169.88/170.18 Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% 169.88/170.18 Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% 169.88/170.18 Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% 169.88/170.18 Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% 169.88/170.18 Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% 169.88/170.18 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% 169.88/170.18 Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 169.88/170.18 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% 176.67/176.96 Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 176.67/176.96 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 176.67/176.96 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 176.67/176.96 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 176.67/176.96 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.19/177.47 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.19/177.47 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 177.19/177.47 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 177.19/177.47 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 177.19/177.47 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 177.19/177.47 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 177.19/177.47 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 177.19/177.47 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 177.19/177.47 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 177.19/177.47 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 177.19/177.47 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 177.19/177.47 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 177.19/177.47 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 177.19/177.47 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.19/177.47 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.19/177.47 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.78/178.14 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 177.78/178.14 Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% 177.78/178.14 Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% 177.78/178.14 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% 177.78/178.14 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% 177.78/178.14 Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% 177.78/178.14 Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 177.78/178.14 Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xt:a)=> ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 180.46/180.78 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 180.46/180.78 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 180.46/180.78 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 180.46/180.78 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 180.46/180.78 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 180.46/180.78 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 180.46/180.78 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 180.46/180.78 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 180.46/180.78 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 180.46/180.78 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 180.46/180.78 Found eq_ref00:=(eq_ref0 b):(((eq ((a->(a->Prop))->Prop)) b) b)
% 180.46/180.78 Found (eq_ref0 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 180.46/180.78 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 180.46/180.78 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 180.46/180.78 Found ((eq_ref ((a->(a->Prop))->Prop)) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 180.46/180.78 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 180.46/180.78 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 180.46/180.78 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 180.46/180.78 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 180.46/180.78 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 180.46/180.78 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 180.46/180.78 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 180.46/180.78 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 182.58/182.91 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 182.58/182.91 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 182.58/182.91 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 182.58/182.91 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 182.58/182.91 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 182.58/182.91 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 182.58/182.91 Found x4:(P Xz)
% 182.58/182.91 Instantiate: x3:=Xz:a
% 182.58/182.91 Found (fun (x4:(P Xz))=> x4) as proof of (P x3)
% 182.58/182.91 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 182.58/182.91 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x3)
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 182.58/182.91 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 182.58/182.91 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 182.58/182.91 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 182.58/182.91 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 182.58/182.91 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 182.58/182.91 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 182.58/182.91 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 182.58/182.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 182.58/182.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 182.58/182.91 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 182.58/182.91 Found (eq_sym000 ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 182.58/182.91 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 182.58/182.91 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 182.58/182.91 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 182.58/182.91 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 182.58/182.91 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 184.14/184.44 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 184.14/184.44 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 184.14/184.44 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 184.14/184.44 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 184.14/184.44 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 184.14/184.44 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 184.14/184.44 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 184.14/184.44 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 184.14/184.44 Found (eq_ref0 x5) as proof of (((eq a) x5) b)
% 184.14/184.44 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 184.14/184.44 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 184.14/184.44 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 184.14/184.44 Found x1:(P0 (f x0))
% 184.14/184.44 Instantiate: b:=(f x0):Prop
% 184.14/184.44 Found x1 as proof of (P1 b)
% 184.14/184.44 Found x1:(P0 (f x0))
% 184.14/184.44 Instantiate: b:=(f x0):Prop
% 184.14/184.44 Found x1 as proof of (P1 b)
% 184.14/184.44 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 184.14/184.44 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 184.14/184.44 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 184.14/184.44 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 184.14/184.44 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 184.14/184.44 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 187.30/187.64 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 187.30/187.64 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 187.30/187.64 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 187.30/187.64 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 187.30/187.64 Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 187.30/187.64 Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 187.30/187.64 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 187.92/188.25 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 187.92/188.25 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 187.92/188.25 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 187.92/188.25 Found (eq_ref00 P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 187.92/188.25 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x5)
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 187.92/188.25 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 187.92/188.25 Found (eq_ref00 P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 187.92/188.25 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x5)
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 187.92/188.25 Found x6:(P Xz)
% 187.92/188.25 Instantiate: x5:=Xz:a
% 187.92/188.25 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 187.92/188.25 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 187.92/188.25 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 187.92/188.25 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 187.92/188.25 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 188.86/189.22 Found x6:(P Xz)
% 188.86/189.22 Instantiate: x5:=Xz:a
% 188.86/189.22 Found (fun (x6:(P Xz))=> x6) as proof of (P x5)
% 188.86/189.22 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x5))
% 188.86/189.22 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 188.86/189.22 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x5)
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 188.86/189.22 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 188.86/189.22 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 188.86/189.22 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 188.86/189.22 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 188.86/189.22 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 188.86/189.22 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 188.86/189.22 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 188.86/189.22 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 188.86/189.22 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 188.86/189.22 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 188.86/189.22 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 188.86/189.22 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found ((fun (x4:(((eq a) x3) Xz))=> ((eq_sym000 x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found ((fun (x4:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found ((fun (x4:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found (fun (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of ((P Xz)->(P x3))
% 188.86/189.22 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 190.06/190.38 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 190.06/190.38 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 190.06/190.38 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 190.06/190.38 Found x4:(P Xz)
% 190.06/190.38 Instantiate: x1:=Xz:a
% 190.06/190.38 Found (fun (x4:(P Xz))=> x4) as proof of (P x1)
% 190.06/190.38 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((P Xz)->(P x1))
% 190.06/190.38 Found (fun (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 190.06/190.38 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (((eq a) Xz) x1)
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 190.06/190.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x4:(P Xz))=> x4) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 190.06/190.38 Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))
% 190.06/190.38 Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 190.48/190.77 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 190.48/190.77 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 190.48/190.77 Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 190.48/190.77 Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->(a->Prop))->Prop)) b) (fun (x:(a->(a->Prop)))=> (b x)))
% 190.48/190.77 Found (eta_expansion00 b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 190.48/190.77 Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 190.48/190.77 Found (((eta_expansion (a->(a->Prop))) Prop) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 190.48/190.77 Found (((eta_expansion (a->(a->Prop))) Prop) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 190.48/190.77 Found (((eta_expansion (a->(a->Prop))) Prop) b) as proof of (((eq ((a->(a->Prop))->Prop)) b) b0)
% 190.48/190.77 Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz))))))))))))
% 190.48/190.77 Found (eq_ref0 (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b0)
% 190.48/190.77 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b0)
% 196.35/196.66 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b0)
% 196.35/196.66 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))))) b0)
% 196.35/196.66 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 196.35/196.66 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 196.35/196.66 Found (eq_ref0 x5) as proof of (((eq a) x5) b)
% 196.35/196.66 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 196.35/196.66 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 196.35/196.66 Found ((eq_ref a) x5) as proof of (((eq a) x5) b)
% 196.35/196.66 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 196.35/196.66 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 196.35/196.66 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 196.35/196.66 Found (eq_ref0 x3) as proof of (((eq a) x3) b)
% 196.35/196.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 196.35/196.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 196.35/196.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 196.35/196.66 Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% 196.35/196.66 Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) b0)
% 196.35/196.66 Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% 196.35/196.66 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% 196.35/196.66 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% 196.35/196.66 Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% 196.35/196.66 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 196.35/196.66 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 196.35/196.66 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 196.35/196.66 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 196.35/196.66 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 196.35/196.66 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 196.35/196.66 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 196.35/196.66 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 196.35/196.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 196.35/196.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 196.35/196.66 Found x6:(P Xz)
% 197.33/197.65 Instantiate: x3:=Xz:a
% 197.33/197.65 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 197.33/197.65 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 197.33/197.65 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 197.33/197.65 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 197.33/197.65 Found (eq_ref00 P) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 197.33/197.65 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x3)
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 197.33/197.65 Found x6:(P Xz)
% 197.33/197.65 Instantiate: x3:=Xz:a
% 197.33/197.65 Found (fun (x6:(P Xz))=> x6) as proof of (P x3)
% 197.33/197.65 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x3))
% 197.33/197.65 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 197.33/197.65 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x3)
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 197.33/197.65 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 197.33/197.65 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 197.33/197.65 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 197.33/197.65 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 197.33/197.65 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 197.33/197.65 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 197.33/197.65 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 197.33/197.65 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 197.33/197.65 Found (eq_ref0 Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 197.33/197.65 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 200.68/201.02 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 200.68/201.02 Found ((eq_ref a) Xz) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 200.68/201.02 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 200.68/201.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 200.68/201.02 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((eq_ref a) Xz)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 200.68/201.02 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 200.68/201.02 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 200.68/201.02 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 200.68/201.02 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 200.68/201.02 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 200.68/201.02 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found ((fun (x4:(((eq a) x1) Xz))=> ((eq_sym000 x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found ((fun (x4:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found ((fun (x4:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found (fun (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of ((P Xz)->(P x1))
% 200.68/201.02 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 200.68/201.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 200.68/201.02 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 200.68/201.02 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 200.68/201.02 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 200.68/201.02 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 200.68/201.02 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 200.68/201.02 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 200.68/201.02 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 200.68/201.02 Found (eq_ref0 x3) as proof of (((eq a) x3) b)
% 200.68/201.02 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 200.68/201.02 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 200.68/201.02 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 200.68/201.02 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 200.68/201.02 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.68/201.02 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.68/201.02 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.68/201.02 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.68/201.02 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 200.68/201.02 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.68/201.02 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.68/201.02 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.74/201.03 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.74/201.03 Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.03 Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.03 Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.03 Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.03 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.04 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.04 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.74/201.04 Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.85/201.18 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 200.85/201.18 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.85/201.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.85/201.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.85/201.18 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 200.85/201.18 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 200.85/201.18 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.85/201.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.85/201.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.85/201.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 200.85/201.18 Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.85/201.18 Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.85/201.18 Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.85/201.18 Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.90/201.18 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.90/201.18 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 200.90/201.18 Found (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 201.70/202.03 Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 201.70/202.03 Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) (fun (x:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((x Xx) Xy)) ((x Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x Xv) Xz))))))))))
% 201.70/202.03 Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 202.24/202.54 Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 202.24/202.54 Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 202.24/202.54 Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 202.24/202.54 Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (forall (Xx:a) (Xy:a), (((and ((R Xx) Xy)) ((R Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((R Xv) Xz)))))))))) b0)
% 202.24/202.54 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))):(((x2 Xy) Xx)->((x2 Xy) Xx))
% 202.24/202.54 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 202.24/202.54 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 202.24/202.54 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 202.24/202.54 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 202.24/202.54 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 202.24/202.54 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xx) Xy)->(((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 202.24/202.54 Found (and_rect00 (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 202.24/202.54 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 202.24/202.54 Found (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 202.24/202.56 Found (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 202.24/202.56 Found (x10 (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 202.24/202.56 Found ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 202.24/202.57 Found (fun (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 202.24/202.57 Found (fun (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy))
% 202.24/202.57 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 202.24/202.57 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 202.24/202.57 Found ((conj00 (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))))) x1) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 202.24/202.57 Found (((conj0 b) (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))))) x1) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 202.24/202.57 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))))) x1) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 203.27/203.59 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))))) x1) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 203.27/203.59 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))))) x1) as proof of (P b)
% 203.27/203.59 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 203.27/203.59 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 203.27/203.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 203.27/203.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 203.27/203.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 203.27/203.59 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 203.27/203.59 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 203.27/203.59 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 203.27/203.59 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found ((((fun (b:a)=> ((eq_trans00 b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 203.27/203.59 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 203.27/203.59 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 203.27/203.59 Found x41:(P Xz)
% 203.27/203.59 Found (fun (x41:(P Xz))=> x41) as proof of (P Xz)
% 203.27/203.59 Found (fun (x41:(P Xz))=> x41) as proof of (P0 Xz)
% 203.27/203.59 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 203.27/203.59 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 203.27/203.59 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 203.27/203.59 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 203.27/203.59 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 203.27/203.59 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 203.27/203.59 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 203.27/203.59 Found (((fun (x4:(((eq a) x3) Xz))=> ((eq_sym000 x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 204.15/204.48 Found (((fun (x4:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 204.15/204.48 Found (((fun (x4:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 204.15/204.48 Found (((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x3))
% 204.15/204.48 Found (fun (P:(a->Prop))=> (((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41))) as proof of ((P Xz)->(P x3))
% 204.15/204.48 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41))) as proof of (((eq a) Xz) x3)
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x3)) (fun (x41:(P Xz))=> x41))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 204.15/204.48 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 204.15/204.48 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 204.15/204.48 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 204.15/204.48 Found x6:(P Xz)
% 204.15/204.48 Instantiate: x1:=Xz:a
% 204.15/204.48 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 204.15/204.48 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 204.15/204.48 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 204.15/204.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 204.15/204.48 Found x6:(P Xz)
% 204.15/204.48 Instantiate: x1:=Xz:a
% 204.15/204.48 Found (fun (x6:(P Xz))=> x6) as proof of (P x1)
% 204.15/204.48 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((P Xz)->(P x1))
% 204.15/204.48 Found (fun (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 207.26/207.59 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (((eq a) Xz) x1)
% 207.26/207.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 207.26/207.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop)) (x6:(P Xz))=> x6) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 207.26/207.59 Found eq_ref000:=(eq_ref00 P):((P Xz)->(P Xz))
% 207.26/207.59 Found (eq_ref00 P) as proof of ((P Xz)->(P x1))
% 207.26/207.59 Found ((eq_ref0 Xz) P) as proof of ((P Xz)->(P x1))
% 207.26/207.59 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 207.26/207.59 Found (((eq_ref a) Xz) P) as proof of ((P Xz)->(P x1))
% 207.26/207.59 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((P Xz)->(P x1))
% 207.26/207.59 Found (fun (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 207.26/207.59 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (((eq a) Xz) x1)
% 207.26/207.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 207.26/207.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((eq_ref a) Xz) P)) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 207.26/207.59 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 207.26/207.59 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 207.26/207.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 207.26/207.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 207.26/207.59 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 207.26/207.59 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 207.26/207.59 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 207.26/207.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 207.26/207.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 207.26/207.59 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 207.26/207.59 Found x41:(P Xz)
% 207.26/207.59 Found (fun (x41:(P Xz))=> x41) as proof of (P Xz)
% 207.26/207.59 Found (fun (x41:(P Xz))=> x41) as proof of (P0 Xz)
% 207.26/207.59 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 207.26/207.59 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 207.26/207.59 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 207.26/207.59 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 207.26/207.59 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 207.26/207.59 Found eq_sym10000:=(eq_sym1000 (x0 Xy)):(((x0 Xy) Xx)->((x0 Xy) Xy))
% 207.26/207.59 Found (eq_sym1000 (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 207.26/207.59 Found ((eq_sym100 x1) (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 207.26/207.59 Found ((eq_sym100 x1) (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 207.26/207.59 Found (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 207.26/207.59 Found (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->(((eq a) Xy) Xx)))
% 207.26/207.59 Found (and_rect00 (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 207.26/207.59 Found ((and_rect0 (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 207.26/207.59 Found (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 208.90/209.22 Found (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 208.90/209.22 Found (eq_sym100 (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 208.90/209.22 Found ((eq_sym10 Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((eq_sym10 Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 208.90/209.22 Found (((eq_sym1 Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((((eq_sym1 Xy) Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 208.90/209.22 Found ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 208.90/209.22 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy)))))) as proof of (((eq a) Xx) Xy)
% 208.90/209.22 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy)))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 208.90/209.22 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))):(((x0 Xy) Xx)->((x0 Xy) Xx))
% 208.90/209.22 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 208.90/209.22 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 208.90/209.22 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 208.90/209.22 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 208.90/209.22 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 208.90/209.22 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 208.90/209.22 Found (and_rect00 (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 208.90/209.22 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 208.90/209.22 Found (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 208.90/209.22 Found (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 208.94/209.24 Found (x20 (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 208.94/209.24 Found ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 208.94/209.24 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 208.94/209.24 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 208.94/209.24 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 208.94/209.24 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 208.94/209.24 Found ((conj00 (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))))) eta_expansion) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 208.94/209.24 Found (((conj0 b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))))) eta_expansion) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 208.94/209.24 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))))) eta_expansion) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 210.93/211.25 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))))) eta_expansion) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 210.93/211.25 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))))) eta_expansion) as proof of (P b)
% 210.93/211.25 Found a_DUMMY:a
% 210.93/211.25 Found a_DUMMY as proof of a
% 210.93/211.25 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 210.93/211.25 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 210.93/211.25 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 210.93/211.25 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 210.93/211.25 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (f x)))
% 210.93/211.25 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 210.93/211.25 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 210.93/211.25 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 210.93/211.25 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 210.93/211.25 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 210.93/211.25 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 210.93/211.25 Found (eq_ref0 b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 210.93/211.25 Found (eq_ref0 b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 210.93/211.25 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found ((((fun (b:a)=> ((eq_trans00 b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 210.93/211.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 210.93/211.25 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 211.46/211.76 Found eq_sym10000:=(eq_sym1000 (x0 Xy)):(((x0 Xy) Xx)->((x0 Xy) Xy))
% 211.46/211.76 Found (eq_sym1000 (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 211.46/211.76 Found ((eq_sym100 x1) (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 211.46/211.76 Found ((eq_sym100 x1) (x0 Xy)) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 211.46/211.76 Found (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))) as proof of (((x0 Xy) Xx)->(((eq a) Xy) Xx))
% 211.46/211.76 Found (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->(((eq a) Xy) Xx)))
% 211.46/211.76 Found (and_rect00 (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 211.46/211.76 Found ((and_rect0 (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 211.46/211.76 Found (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 211.46/211.76 Found (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy)))) as proof of (((eq a) Xy) Xx)
% 211.46/211.76 Found (eq_sym100 (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((eq_sym100 x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 211.46/211.76 Found ((eq_sym10 Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((eq_sym10 Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 211.46/211.76 Found (((eq_sym1 Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> ((((eq_sym1 Xy) Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 211.46/211.76 Found ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy))))) as proof of (((eq a) Xx) Xy)
% 211.46/211.76 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy)))))) as proof of (((eq a) Xx) Xy)
% 211.46/211.76 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((((eq_sym a) Xy) Xx) (((fun (P0:Type) (x1:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x1) x00)) (((eq a) Xy) Xx)) (fun (x1:((x0 Xx) Xy))=> (((((eq_sym a) Xy) Xx) x1) (x0 Xy)))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 211.46/211.76 Found x41:(P Xz)
% 211.46/211.76 Found (fun (x41:(P Xz))=> x41) as proof of (P Xz)
% 211.46/211.76 Found (fun (x41:(P Xz))=> x41) as proof of (P0 Xz)
% 211.46/211.76 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 211.46/211.76 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 211.46/211.76 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 211.46/211.76 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 211.46/211.76 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 211.46/211.76 Found ((eq_sym0000 ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.46/211.76 Found ((eq_sym0000 ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.46/211.76 Found (((fun (x4:(((eq a) x1) Xz))=> ((eq_sym000 x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.46/211.76 Found (((fun (x4:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.46/211.76 Found (((fun (x4:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.46/211.76 Found (((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41)) as proof of ((P Xz)->(P x1))
% 211.47/211.78 Found (fun (P:(a->Prop))=> (((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41))) as proof of ((P Xz)->(P x1))
% 211.47/211.78 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41))) as proof of (((eq a) Xz) x1)
% 211.47/211.78 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 211.47/211.78 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) (fun (x6:a)=> ((P Xz)->(P x6))))) ((eq_ref a) x1)) (fun (x41:(P Xz))=> x41))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 211.47/211.78 Found ex_ind00:=(ex_ind0 (((eq a) Xx) Xy)):((forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xx) Xy)))->(((eq a) Xx) Xy))
% 211.47/211.78 Instantiate: x0:=(fun (x4:a) (x30:a)=> (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) x30) x4)))):(a->(a->Prop))
% 211.47/211.78 Found (fun (x3:((x0 Xx) Xy))=> ex_ind00) as proof of (((x0 Xy) Xx)->(((eq a) Xx) Xy))
% 211.47/211.78 Found (fun (x3:((x0 Xx) Xy))=> ex_ind00) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->(((eq a) Xx) Xy)))
% 211.47/211.78 Found (and_rect00 (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found ((and_rect0 (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq a) Xx) Xy))
% 211.47/211.78 Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xx) Xy)))
% 211.47/211.78 Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ex_ind00)))) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found ((ex_ind0 (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> (ex_ind0 (((eq a) Xx) Xy)))))) as proof of (((eq a) Xx) Xy)
% 211.47/211.78 Found (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))) as proof of (((eq a) Xx) Xy)
% 211.47/211.79 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy))))))) as proof of (((eq a) Xx) Xy)
% 211.47/211.79 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy))))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 211.47/211.79 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy))))))) as proof of (forall (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 211.47/211.79 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 211.51/211.82 Found ((conj00 (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))))) eq_sym0) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 211.51/211.82 Found (((conj0 b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))))) eq_sym0) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 211.51/211.82 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))))) eq_sym0) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 211.51/211.82 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))))) eq_sym0) as proof of ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b)
% 213.46/213.83 Found ((((conj (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) b) (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> (((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) (((eq a) Xx) Xy)) (fun (x3:((x0 Xx) Xy))=> ((fun (P0:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P0)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xf:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xf X)))))) P0) x1) x)) (((eq a) Xx) Xy)))))))) eq_sym0) as proof of (P b)
% 213.46/213.83 Found x3:(P0 (f x2))
% 213.46/213.83 Instantiate: b:=(f x2):Prop
% 213.46/213.83 Found x3 as proof of (P1 b)
% 213.46/213.83 Found x3:(P0 (f x2))
% 213.46/213.83 Instantiate: b:=(f x2):Prop
% 213.46/213.83 Found x3 as proof of (P1 b)
% 213.46/213.83 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 213.46/213.83 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 213.46/213.83 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 213.46/213.83 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 213.46/213.83 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 214.78/215.15 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 214.78/215.15 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 214.78/215.15 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 214.78/215.15 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 214.78/215.15 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 214.78/215.15 Found a_DUMMY:a
% 214.78/215.15 Found a_DUMMY as proof of a
% 214.78/215.15 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 214.78/215.15 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 214.78/215.15 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 214.78/215.15 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 214.78/215.15 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 214.78/215.15 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 214.78/215.15 Found (eq_ref0 b) as proof of (((eq a) b) x1)
% 214.78/215.15 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 214.78/215.15 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 214.78/215.15 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 214.78/215.15 Found x10:(P0 (f x0))
% 214.78/215.15 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 214.78/215.15 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 214.78/215.15 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 214.78/215.15 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 214.78/215.15 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 214.78/215.15 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 214.78/215.15 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 214.78/215.15 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 214.78/215.15 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 214.78/215.15 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 214.86/215.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 214.86/215.18 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 214.86/215.18 Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 214.86/215.18 Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.34 Found x10:(P0 (f x0))
% 215.03/215.34 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 215.03/215.34 Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 215.03/215.34 Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 215.03/215.34 Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 215.03/215.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 215.03/215.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 215.03/215.34 Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 215.03/215.34 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 215.03/215.34 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 215.03/215.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 215.03/215.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 215.03/215.34 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 215.03/215.34 Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.34 Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.34 Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.35 Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.35 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.35 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> ((((((eq_trans0 (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.35 Found ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10)) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 215.03/215.35 Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (x2:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x0)) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))) (fun (x10:(P0 (f x0)))=> x10))) as proof of ((P0 (f x0))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))))
% 222.33/222.70 Found x41:(P Xz)
% 222.33/222.70 Found (fun (x41:(P Xz))=> x41) as proof of (P Xz)
% 222.33/222.70 Found (fun (x41:(P Xz))=> x41) as proof of (P0 Xz)
% 222.33/222.70 Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% 222.33/222.70 Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 222.33/222.70 Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% 222.33/222.70 Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% 222.33/222.70 Found (fun (x0:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 222.33/222.70 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 222.33/222.70 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 222.33/222.70 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 222.33/222.70 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 222.33/222.70 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 222.33/222.70 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found ((fun (x2:(((eq a) x1) Xz))=> ((eq_sym000 x2) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found ((fun (x2:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x2) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found ((fun (x2:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x2) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found ((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found (fun (P:(a->Prop))=> ((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) P)) ((eq_ref a) x1))) as proof of ((P Xz)->(P x1))
% 222.33/222.70 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) P)) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 222.33/222.70 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) P)) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 222.33/222.70 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) P)) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 222.33/222.70 Found a_DUMMY:a
% 222.33/222.70 Found a_DUMMY as proof of a
% 222.33/222.70 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 222.33/222.70 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 222.33/222.70 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 222.33/222.70 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 222.33/222.70 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 222.33/222.70 Found (eq_sym010 ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 222.33/222.70 Found ((eq_sym01 b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 222.33/222.70 Found (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 222.33/222.70 Found (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2))) as proof of (((eq Prop) b) (f x2))
% 222.33/222.70 Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.33/222.71 Found (((eq_trans000 (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) b) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.33/222.71 Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.33/222.71 Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.33/222.71 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.38/222.73 Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 222.38/222.73 Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 222.38/222.73 Found ((eq_sym00 (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 222.38/222.73 Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 225.83/226.19 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) ((((eq_sym Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 225.83/226.19 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) ((((eq_sym Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((eq_ref Prop) (f x2))))) as proof of (forall (P:(Prop->Prop)), ((P (f x2))->(P ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 225.83/226.19 Found a_DUMMY:a
% 225.83/226.19 Found a_DUMMY as proof of a
% 225.83/226.19 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 226.93/227.30 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 226.93/227.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 226.93/227.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 226.93/227.30 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 226.93/227.30 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 226.93/227.30 Found (eq_ref0 b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 226.93/227.30 Found (eq_ref0 b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_ref a) b) as proof of (((eq a) b) x1)
% 226.93/227.30 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found ((((fun (b:a)=> ((eq_trans00 b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x1)
% 226.93/227.30 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 226.93/227.30 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x1)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 226.93/227.30 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 226.93/227.30 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 226.93/227.30 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 226.93/227.30 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 226.93/227.30 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 229.03/229.36 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 229.03/229.36 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 229.03/229.36 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))
% 229.03/229.36 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 229.03/229.36 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.03/229.36 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.03/229.36 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.03/229.36 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.03/229.36 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 229.03/229.36 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.03/229.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.03/229.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.03/229.36 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.03/229.36 Found ((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found ((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((eq_trans0000 x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.03/229.37 Found (fun (P0:(Prop->Prop))=> (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 229.18/229.55 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.18/229.55 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.18/229.55 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.18/229.55 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 229.18/229.55 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 229.18/229.55 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.18/229.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.18/229.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.18/229.55 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 229.18/229.55 Found ((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found ((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found (((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((eq_trans0000 x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found (((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.18/229.55 Found (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.48/229.81 Found (fun (P0:(Prop->Prop))=> (((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) P0)) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 229.48/229.81 Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) (fun (x:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 229.48/229.81 Found (eta_expansion00 (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.48/229.81 Found ((eta_expansion0 Prop) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.54/229.89 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.54/229.89 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.54/229.89 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.54/229.89 Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 229.54/229.89 Found (eq_ref0 (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 229.54/229.89 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 230.82/231.17 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 230.82/231.17 Found ((eq_ref (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))) b)
% 230.82/231.17 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 230.82/231.17 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 230.82/231.17 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 230.82/231.17 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 230.82/231.17 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 230.82/231.17 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 230.82/231.17 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 230.82/231.17 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 230.82/231.17 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 230.82/231.17 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 232.62/232.98 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 232.62/232.98 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 232.62/232.98 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 232.62/232.98 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 232.62/232.98 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 232.62/232.98 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 232.62/232.98 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 232.62/232.98 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 232.62/232.98 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 232.62/232.98 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 232.62/232.98 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 232.62/232.98 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 232.62/232.98 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 232.62/232.98 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 232.62/232.98 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 232.62/232.98 Found (eq_sym000 ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 232.62/232.98 Found ((eq_sym00 Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 232.62/232.98 Found (((eq_sym0 x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 232.62/232.98 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 232.62/232.98 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 232.62/232.98 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 232.62/232.98 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 234.42/234.82 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 234.42/234.82 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 234.42/234.82 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 234.42/234.82 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 234.42/234.82 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 234.42/234.82 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 234.42/234.82 Found (eq_sym000 ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 234.42/234.82 Found ((eq_sym00 Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 234.42/234.82 Found (((eq_sym0 x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 234.42/234.82 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 234.42/234.82 Found ((((eq_sym a) x5) Xz) ((eq_ref a) x5)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x5)))
% 234.42/234.82 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 234.42/234.82 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 234.42/234.82 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x5) Xz) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 234.42/234.82 Found x20:(P Xz)
% 234.42/234.82 Found (fun (x20:(P Xz))=> x20) as proof of (P Xz)
% 234.42/234.82 Found (fun (x20:(P Xz))=> x20) as proof of (P0 Xz)
% 234.42/234.82 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 234.42/234.82 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 234.42/234.82 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 234.42/234.82 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 234.42/234.82 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 234.42/234.82 Found ((eq_sym0000 ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found ((eq_sym0000 ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (((fun (x2:(((eq a) x1) Xz))=> ((eq_sym000 x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (((fun (x2:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (((fun (x2:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20)) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (fun (P:(a->Prop))=> (((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20))) as proof of ((P Xz)->(P x1))
% 234.42/234.82 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20))) as proof of (((eq a) Xz) x1)
% 234.42/234.82 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 234.42/234.82 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x2:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x2) (fun (x4:a)=> ((P Xz)->(P x4))))) ((eq_ref a) x1)) (fun (x20:(P Xz))=> x20))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 237.82/238.18 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) (fun (x:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 237.82/238.18 Found (eta_expansion_dep00 (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 237.82/238.18 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 237.82/238.18 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 237.82/238.18 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 237.82/238.18 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 237.82/238.18 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) (fun (x:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 239.06/239.41 Found (eta_expansion_dep00 (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 239.06/239.41 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 239.06/239.41 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 239.06/239.41 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 239.06/239.41 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 239.06/239.41 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 239.06/239.41 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.06/239.41 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.06/239.41 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.06/239.41 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.06/239.41 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 239.06/239.41 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 239.60/239.94 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.60/239.94 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.60/239.94 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.60/239.94 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 239.60/239.94 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 239.60/239.94 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 239.60/239.94 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found (eq_sym000 ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 239.60/239.94 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 239.60/239.94 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 239.60/239.94 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 239.60/239.94 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found (eq_sym000 ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 239.60/239.94 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 239.60/239.94 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 239.60/239.94 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 239.60/239.94 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 239.60/239.94 Found (eq_sym000 ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((eq_sym00 Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (((eq_sym0 x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found ((((eq_sym a) x3) Xz) ((eq_ref a) x3)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x3)))
% 239.60/239.94 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 242.20/242.57 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 242.20/242.57 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x3) Xz) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 242.20/242.57 Found x30:(P0 (f x2))
% 242.20/242.57 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 242.20/242.57 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 242.20/242.57 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 242.20/242.57 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.20/242.57 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.20/242.57 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.20/242.57 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.20/242.57 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 242.20/242.57 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.20/242.57 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.20/242.57 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.20/242.57 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.20/242.57 Found (((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.57 Found (((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.57 Found ((((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((eq_trans0000 x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.57 Found ((((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.58 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.58 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.20/242.58 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.86 Found (fun (P0:(Prop->Prop))=> ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.86 Found x30:(P0 (f x2))
% 242.50/242.86 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 242.50/242.86 Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 242.50/242.86 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 242.50/242.86 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.50/242.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.50/242.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.50/242.86 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 242.50/242.86 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 242.50/242.86 Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.50/242.86 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.50/242.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.50/242.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 242.50/242.89 Found (((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.89 Found (((eq_trans00000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.89 Found ((((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((eq_trans0000 x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.89 Found ((((fun (x3:(((eq Prop) (f x2)) b)) (x4:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.89 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.90 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((((eq_trans0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.90 Found ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30)) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 242.50/242.90 Found (fun (P0:(Prop->Prop))=> ((((fun (x3:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (x4:(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((((eq_trans Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x3) x4) (fun (x6:Prop)=> ((P0 (f x2))->(P0 x6))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))) (fun (x30:(P0 (f x2)))=> x30))) as proof of ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 248.22/248.57 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) (fun (x:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 248.22/248.57 Found (eta_expansion_dep00 (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.22/248.57 Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.22/248.57 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) (fun (x:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 248.31/248.70 Found (eta_expansion00 (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.31/248.70 Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.74/249.09 Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))) b)
% 248.74/249.09 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 248.74/249.09 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found (eq_sym000 ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 248.74/249.09 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 248.74/249.09 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 248.74/249.09 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 248.74/249.09 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found (eq_sym000 ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 248.74/249.09 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 248.74/249.09 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 248.74/249.09 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 248.74/249.09 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 248.74/249.09 Found (eq_sym000 ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((eq_sym00 Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found (((eq_sym0 x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 248.74/249.09 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 253.00/253.38 Found ((((eq_sym a) x1) Xz) ((eq_ref a) x1)) as proof of (forall (P:(a->Prop)), ((P Xz)->(P x1)))
% 253.00/253.38 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 253.00/253.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 253.00/253.38 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((eq_sym a) x1) Xz) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 253.00/253.38 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 253.00/253.38 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 253.00/253.38 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 253.00/253.38 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 253.00/253.38 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) b)
% 253.00/253.38 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 253.00/253.38 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.00/253.38 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.67 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.67 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.67 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 253.30/253.68 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 253.30/253.68 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 253.30/253.68 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 253.30/253.68 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 253.30/253.68 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 257.50/257.90 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 257.50/257.90 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 257.50/257.90 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 257.50/257.90 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 257.50/257.90 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 257.50/257.90 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz)))))))))))
% 257.50/257.90 Found (fun (x2:a)=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (forall (x20:(a->(a->Prop))), (((eq Prop) (f x20)) ((and (forall (Xx:a) (Xy:a), (((and ((x20 Xx) Xy)) ((x20 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x20 Xv) Xz))))))))))))
% 257.50/257.90 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 257.50/257.90 Found (eq_ref0 x3) as proof of (((eq a) x3) b)
% 257.50/257.90 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 257.50/257.90 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 257.50/257.90 Found ((eq_ref a) x3) as proof of (((eq a) x3) b)
% 257.50/257.90 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 257.50/257.90 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 257.50/257.90 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 257.50/257.90 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 257.50/257.90 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 257.50/257.90 Found x1:(P0 (f x0))
% 257.50/257.90 Instantiate: b:=(f x0):Prop
% 257.50/257.90 Found x1 as proof of (P1 b)
% 257.50/257.90 Found x1:(P0 (f x0))
% 257.50/257.90 Instantiate: b:=(f x0):Prop
% 257.50/257.90 Found x1 as proof of (P1 b)
% 257.50/257.90 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 257.50/257.90 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 257.50/257.90 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 257.50/257.90 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 257.50/257.90 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 257.50/257.90 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 257.50/257.90 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 257.50/257.90 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 259.01/259.37 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 259.01/259.37 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 259.01/259.37 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% 259.01/259.37 Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))
% 259.01/259.37 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 259.01/259.40 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 259.01/259.40 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.01/259.40 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 259.69/260.10 Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) (fun (x:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 259.69/260.10 Found (eta_expansion00 (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 259.69/260.10 Found ((eta_expansion0 Prop) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 259.69/260.10 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 259.69/260.10 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 259.69/260.10 Found (((eta_expansion a) Prop) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 259.69/260.10 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))):(((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) (fun (x:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 261.02/261.39 Found (eta_expansion_dep00 (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 261.02/261.39 Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 261.02/261.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 261.02/261.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 261.02/261.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))) b)
% 261.02/261.39 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 261.02/261.39 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 261.50/261.89 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 261.50/261.89 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 261.50/261.89 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 261.50/261.89 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 261.50/261.89 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.52 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.52 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 262.14/262.53 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.53 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.53 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.53 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 262.14/262.53 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 262.14/262.53 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 262.14/262.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 262.14/262.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 262.14/262.53 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 262.14/262.53 Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz)))))))))
% 265.29/265.73 Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 265.29/265.73 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 265.29/265.73 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 265.29/265.73 Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x0 Xv) Xz))))))))) b)
% 265.29/265.73 Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 265.29/265.73 Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 265.29/265.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 265.29/265.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 265.29/265.73 Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 265.29/265.73 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 265.29/265.73 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 265.29/265.73 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 265.29/265.73 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 265.29/265.73 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 265.29/265.73 Found (eq_sym0000 ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found (eq_sym0000 ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found ((fun (x6:(((eq a) x5) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found ((fun (x6:(((eq a) x5) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found ((fun (x6:(((eq a) x5) Xz))=> ((((eq_sym0 x5) Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of ((P Xz)->(P x5))
% 265.29/265.73 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 265.29/265.73 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 265.29/265.73 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 266.08/266.48 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 266.08/266.48 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 266.08/266.48 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 266.08/266.48 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 266.08/266.48 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 266.08/266.48 Found (eq_sym0000 ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found (eq_sym0000 ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found ((fun (x6:(((eq a) x5) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found ((fun (x6:(((eq a) x5) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found ((fun (x6:(((eq a) x5) Xz))=> ((((eq_sym0 x5) Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5)) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of ((P Xz)->(P x5))
% 266.08/266.48 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of (((eq a) Xz) x5)
% 266.08/266.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 266.08/266.48 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) P)) ((eq_ref a) x5))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 266.08/266.48 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 266.08/266.48 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.08/266.48 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.08/266.48 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.08/266.48 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.08/266.48 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 266.08/266.48 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 266.08/266.48 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.08/266.48 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 266.27/266.70 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 266.27/266.70 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 266.27/266.70 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 266.27/266.70 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 266.27/266.70 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 268.94/269.31 Found x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 268.94/269.31 Instantiate: b:=((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))):Prop
% 268.94/269.31 Found x3 as proof of (P1 b)
% 268.94/269.31 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 268.94/269.31 Found (eq_ref0 b) as proof of (((eq a) b) Xz)
% 268.94/269.31 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 268.94/269.31 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 268.94/269.31 Found ((eq_ref a) b) as proof of (((eq a) b) Xz)
% 268.94/269.31 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 268.94/269.31 Found (eq_ref0 x1) as proof of (((eq a) x1) b)
% 268.94/269.31 Found ((eq_ref a) x1) as proof of (((eq a) x1) b)
% 268.94/269.31 Found ((eq_ref a) x1) as proof of (((eq a) x1) b)
% 268.94/269.31 Found ((eq_ref a) x1) as proof of (((eq a) x1) b)
% 268.94/269.31 Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 268.94/269.31 Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 268.94/269.31 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 268.94/269.31 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 268.94/269.31 Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 268.94/269.31 Found ((eq_sym0100 ((eq_ref Prop) (f x2))) x3) as proof of (P0 (f x2))
% 268.94/269.31 Found ((eq_sym0100 ((eq_ref Prop) (f x2))) x3) as proof of (P0 (f x2))
% 268.94/269.31 Found (((fun (x4:(((eq Prop) (f x2)) b))=> ((eq_sym010 x4) P0)) ((eq_ref Prop) (f x2))) x3) as proof of (P0 (f x2))
% 268.94/269.31 Found (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((eq_sym01 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3) as proof of (P0 (f x2))
% 268.94/269.31 Found (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3) as proof of (P0 (f x2))
% 268.94/269.31 Found (fun (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3)) as proof of (P0 (f x2))
% 268.94/269.31 Found (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3)) as proof of ((P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))->(P0 (f x2)))
% 268.97/269.35 Found (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3)) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2))
% 268.97/269.35 Found (eq_sym000 (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 268.97/269.35 Found ((eq_sym00 (f x2)) (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 268.97/269.35 Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> ((((eq_sym0 (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 274.60/274.98 Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) (f x2)) (fun (P0:(Prop->Prop)) (x3:(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((fun (x4:(((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))=> (((((eq_sym Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))) x4) P0)) ((eq_ref Prop) (f x2))) x3))) as proof of (((eq Prop) (f x2)) ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))
% 274.60/274.98 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 274.60/274.98 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 274.60/274.98 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 274.60/274.98 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 274.60/274.98 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 274.60/274.98 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 274.60/274.98 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 274.60/274.98 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 274.60/274.98 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 274.60/274.98 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 274.60/274.98 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 274.60/274.98 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 274.60/274.98 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.60/274.98 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.60/274.98 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.60/274.98 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found ((fun (x6:(((eq a) x3) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found ((fun (x6:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found ((fun (x6:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of ((P Xz)->(P x3))
% 274.60/274.98 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 274.60/274.98 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 274.60/274.98 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 274.77/275.19 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 274.77/275.19 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 274.77/275.19 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.77/275.19 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.77/275.19 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 274.77/275.19 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found ((fun (x6:(((eq a) x3) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found ((fun (x6:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found ((fun (x6:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of ((P Xz)->(P x3))
% 274.77/275.19 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 274.77/275.19 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 274.77/275.19 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) P)) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 274.77/275.19 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))):(((x2 Xy) Xx)->((x2 Xy) Xx))
% 274.77/275.19 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 274.77/275.19 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 274.77/275.19 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 274.77/275.19 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 274.77/275.19 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 274.77/275.19 Found (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))) as proof of (((x2 Xx) Xy)->(((x2 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 274.77/275.19 Found (and_rect00 (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 274.77/275.19 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 274.77/275.19 Found (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 274.77/275.19 Found (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 274.77/275.19 Found (x10 (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 278.17/278.56 Found ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 278.17/278.56 Found (fun (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 278.17/278.56 Found (fun (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy))
% 278.17/278.56 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 278.17/278.56 Found (fun (Xx:a) (Xy:a) (x00:((and ((x2 Xx) Xy)) ((x2 Xy) Xx)))=> ((x1 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x2 Xx) Xy)->(((x2 Xy) Xx)->P0)))=> (((((and_rect ((x2 Xx) Xy)) ((x2 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x2 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x2 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))
% 278.17/278.56 Found x60:(P Xz)
% 278.17/278.56 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 278.17/278.56 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 278.17/278.56 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 278.17/278.56 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.17/278.56 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.17/278.56 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.17/278.56 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.17/278.56 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 278.30/278.69 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 278.30/278.69 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 278.30/278.69 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 278.30/278.69 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 278.30/278.69 Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% 278.30/278.69 Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 278.30/278.69 Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 279.77/280.18 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 279.77/280.18 Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) ((P0 (f x2))->(P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))))
% 279.77/280.18 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 279.77/280.18 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 279.77/280.18 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 279.77/280.18 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> ((eq_trans00 b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 279.77/280.18 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 279.77/280.18 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 279.77/280.18 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 279.77/280.18 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 279.77/280.18 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 279.77/280.18 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 279.77/280.18 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> ((eq_trans00 b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x5)
% 279.77/280.18 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x5)
% 282.18/282.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 282.18/282.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x5)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 282.18/282.59 Found x60:(P Xz)
% 282.18/282.59 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 282.18/282.59 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 282.18/282.59 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 282.18/282.59 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_sym0000 ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found ((eq_sym0000 ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> ((eq_sym000 x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> (((eq_sym00 Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> ((((eq_sym0 x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (fun (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of (((eq a) Xz) x5)
% 282.18/282.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 282.18/282.59 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 282.18/282.59 Found x60:(P Xz)
% 282.18/282.59 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 282.18/282.59 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 282.18/282.59 Found eq_ref00:=(eq_ref0 x5):(((eq a) x5) x5)
% 282.18/282.59 Found (eq_ref0 x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_ref a) x5) as proof of (((eq a) x5) Xz)
% 282.18/282.59 Found ((eq_sym0000 ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found ((eq_sym0000 ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> ((eq_sym000 x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> (((eq_sym00 Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 282.18/282.59 Found (((fun (x6:(((eq a) x5) Xz))=> ((((eq_sym0 x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 286.22/286.62 Found (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x5))
% 286.22/286.62 Found (fun (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of ((P Xz)->(P x5))
% 286.22/286.62 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of (((eq a) Xz) x5)
% 286.22/286.62 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5))
% 286.22/286.62 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x5) Xz))=> (((((eq_sym a) x5) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x5)) (fun (x60:(P Xz))=> x60))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x5)))
% 286.22/286.62 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 286.22/286.62 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 286.22/286.62 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 286.22/286.62 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 286.22/286.62 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 286.22/286.62 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 286.22/286.62 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 286.22/286.62 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 286.22/286.62 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 286.22/286.62 Found (eq_ref0 b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found ((eq_ref a) b) as proof of (((eq a) b) x5)
% 286.22/286.62 Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))):(((x0 Xy) Xx)->((x0 Xy) Xx))
% 286.22/286.62 Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 286.22/286.62 Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 286.22/286.62 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 286.22/286.62 Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 286.76/287.20 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy))))
% 286.76/287.20 Found (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))) as proof of (((x0 Xx) Xy)->(((x0 Xy) Xx)->((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))))
% 286.76/287.20 Found (and_rect00 (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 286.76/287.20 Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 286.76/287.20 Found (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 286.76/287.20 Found (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))
% 286.76/287.20 Found (x20 (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 286.76/287.20 Found ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx)))))) as proof of (((eq a) Xx) Xy)
% 286.76/287.20 Found (fun (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((eq a) Xx) Xy)
% 286.76/287.20 Found (fun (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy))
% 286.76/287.20 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 286.76/287.20 Found (fun (Xx:a) (Xy:a) (x00:((and ((x0 Xx) Xy)) ((x0 Xy) Xx)))=> ((x2 (fun (x3:a)=> (((eq a) Xx) Xy))) (((fun (P0:Type) (x3:(((x0 Xx) Xy)->(((x0 Xy) Xx)->P0)))=> (((((and_rect ((x0 Xx) Xy)) ((x0 Xy) Xx)) P0) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xx) Xy)))) (fun (x3:((x0 Xx) Xy))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xx) Xy))) (fun (x4:(a->Prop))=> ((x0 Xy) Xx))))))) as proof of (forall (Xx:a) (Xy:a), (((and ((x0 Xx) Xy)) ((x0 Xy) Xx))->(((eq a) Xx) Xy)))
% 286.76/287.20 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 286.76/287.20 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 286.76/287.20 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 286.76/287.20 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 287.02/287.42 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found ((fun (x4:(((eq a) x3) Xz))=> ((eq_sym000 x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found ((fun (x4:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found ((fun (x4:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3)) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found (fun (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of ((P Xz)->(P x3))
% 287.02/287.42 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of (((eq a) Xz) x3)
% 287.02/287.42 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 287.02/287.42 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x4) P)) ((eq_ref a) x3))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 287.02/287.42 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 287.02/287.42 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 287.02/287.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 287.02/287.42 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 287.02/287.42 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 287.02/287.42 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> ((eq_sym000 x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 287.02/287.42 Found ((fun (x6:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 290.55/290.95 Found ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 290.55/290.95 Found (fun (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of ((P Xz)->(P x1))
% 290.55/290.95 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 290.55/290.95 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1))
% 290.55/290.95 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x6:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x6) P)) ((eq_ref a) x1))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x1)))
% 290.55/290.95 Found x60:(P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 290.55/290.95 Found x60:(P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 290.55/290.95 Found x60:(P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 290.55/290.95 Found x60:(P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 290.55/290.95 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 290.55/290.95 Found x00:((ex a) (fun (Xu:a)=> (S Xu)))
% 290.55/290.95 Instantiate: b:=(fun (Xu:a)=> (S Xu)):(a->Prop)
% 290.55/290.95 Found x00 as proof of (P b)
% 290.55/290.95 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (((eq a) Xz) x3))):(((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) (fun (x:a)=> (((eq a) Xz) x3)))
% 290.55/290.95 Found (eta_expansion_dep00 (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 290.55/290.95 Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 290.55/290.95 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 290.55/290.95 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 290.55/290.95 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 290.55/290.95 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 290.55/290.95 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 290.55/290.95 Found eq_ref00:=(eq_ref0 (f0 x2)):(((eq Prop) (f0 x2)) (f0 x2))
% 290.55/290.95 Found (eq_ref0 (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found ((eq_ref Prop) (f0 x2)) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (((eq Prop) (f0 x2)) (f x2))
% 290.55/290.95 Found (fun (x2:(a->(a->Prop)))=> ((eq_ref Prop) (f0 x2))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f0 x)) (f x)))
% 290.55/290.95 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 290.55/290.95 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 290.55/290.95 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 290.55/290.95 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 290.55/290.95 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 290.55/290.95 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 290.55/290.95 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 290.55/290.95 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 290.55/290.95 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 290.55/290.95 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 291.47/291.88 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> ((eq_trans00 b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 291.47/291.88 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 291.47/291.88 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 291.47/291.88 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 291.47/291.88 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 291.47/291.88 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> ((eq_trans00 b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 291.47/291.88 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 291.47/291.88 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 291.47/291.88 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 291.47/291.88 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 291.47/291.88 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 291.47/291.88 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 291.47/291.88 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 292.97/293.39 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 292.97/293.39 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 292.97/293.39 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 292.97/293.39 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 292.97/293.39 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 292.97/293.39 Found ((eq_trans0000 ((eq_ref a) Xz)) ((eq_ref a) b)) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found (((eq_trans000 Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found ((((fun (b:a)=> ((eq_trans00 b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found ((((fun (b:a)=> (((eq_trans0 Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz)) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (((eq a) Xz) x3)
% 292.97/293.39 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 292.97/293.39 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X)))))=> ((((fun (b:a)=> ((((eq_trans a) Xz) b) x3)) Xz) ((eq_ref a) Xz)) ((eq_ref a) Xz))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 292.97/293.39 Found x00:((ex a) (fun (Xu:a)=> (S Xu)))
% 292.97/293.39 Instantiate: b:=(fun (Xu:a)=> (S Xu)):(a->Prop)
% 292.97/293.39 Found x00 as proof of (P b)
% 292.97/293.39 Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> (((eq a) Xz) x3))):(((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) (fun (x:a)=> (((eq a) Xz) x3)))
% 292.97/293.39 Found (eta_expansion_dep00 (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 292.97/293.39 Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 292.97/293.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 292.97/293.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 292.97/293.39 Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> (((eq a) Xz) x3))) b)
% 292.97/293.39 Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% 292.97/293.39 Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 292.97/293.39 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 292.97/293.39 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 292.97/293.39 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 292.97/293.39 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 293.26/293.72 Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% 293.26/293.72 Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 293.26/293.72 Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% 293.26/293.72 Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 293.26/293.72 Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% 293.26/293.72 Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 293.26/293.72 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz))))))))))
% 295.23/295.66 Found (fun (x4:a)=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (P0 ((and (forall (Xx:a) (Xy:a), (((and ((x2 Xx) Xy)) ((x2 Xy) Xx))->(((eq a) Xx) Xy)))) (forall (S:(a->Prop)), (((ex a) (fun (Xu:a)=> (S Xu)))->((ex a) (fun (Xv:a)=> ((and (S Xv)) (forall (Xz:a), ((x2 Xv) Xz)))))))))))
% 295.23/295.66 Found x60:(P Xz)
% 295.23/295.66 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 295.23/295.66 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 295.23/295.66 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 295.23/295.66 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> ((eq_sym000 x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (fun (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of (((eq a) Xz) x3)
% 295.23/295.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 295.23/295.66 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 295.23/295.66 Found x60:(P Xz)
% 295.23/295.66 Found (fun (x60:(P Xz))=> x60) as proof of (P Xz)
% 295.23/295.66 Found (fun (x60:(P Xz))=> x60) as proof of (P0 Xz)
% 295.23/295.66 Found eq_ref00:=(eq_ref0 x3):(((eq a) x3) x3)
% 295.23/295.66 Found (eq_ref0 x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_ref a) x3) as proof of (((eq a) x3) Xz)
% 295.23/295.66 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found ((eq_sym0000 ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> ((eq_sym000 x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> (((eq_sym00 Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> ((((eq_sym0 x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 295.23/295.66 Found (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60)) as proof of ((P Xz)->(P x3))
% 297.71/298.13 Found (fun (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of ((P Xz)->(P x3))
% 297.71/298.13 Found (fun (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of (((eq a) Xz) x3)
% 297.71/298.13 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3))
% 297.71/298.13 Found (fun (x300:((a->Prop)->a)) (x400:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> (((fun (x6:(((eq a) x3) Xz))=> (((((eq_sym a) x3) Xz) x6) (fun (x8:a)=> ((P Xz)->(P x8))))) ((eq_ref a) x3)) (fun (x60:(P Xz))=> x60))) as proof of (forall (x300:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))->(((eq a) Xz) x3)))
% 297.71/298.13 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 297.71/298.13 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 297.71/298.13 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 297.71/298.13 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 297.71/298.13 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 Xz):(((eq a) Xz) Xz)
% 297.71/298.13 Found (eq_ref0 Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found ((eq_ref a) Xz) as proof of (((eq a) Xz) b)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% 297.71/298.13 Found (eq_ref0 b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found ((eq_ref a) b) as proof of (((eq a) b) x3)
% 297.71/298.13 Found eq_ref00:=(eq_ref0 x1):(((eq a) x1) x1)
% 297.71/298.13 Found (eq_ref0 x1) as proof of (((eq a) x1) Xz)
% 297.71/298.13 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 297.71/298.13 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 297.71/298.13 Found ((eq_ref a) x1) as proof of (((eq a) x1) Xz)
% 297.71/298.13 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found (eq_sym0000 ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found ((fun (x4:(((eq a) x1) Xz))=> ((eq_sym000 x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found ((fun (x4:(((eq a) x1) Xz))=> (((eq_sym00 Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found ((fun (x4:(((eq a) x1) Xz))=> ((((eq_sym0 x1) Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1)) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found (fun (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of ((P Xz)->(P x1))
% 297.71/298.13 Found (fun (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of (((eq a) Xz) x1)
% 297.71/298.13 Found (fun (x300:((a->Prop)->a)) (x40:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x300 X))))) (P:(a->Prop))=> ((fun (x4:(((eq a) x1) Xz))=> (((((eq_sym a) x1) Xz) x4) P)) ((eq_ref a) x1))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))-
%------------------------------------------------------------------------------