TSTP Solution File: SEU910^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU910^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n104.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:22 EDT 2014
% Result : Timeout 300.03s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU910^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n104.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:42:21 CDT 2014
% % CPUTime : 300.03
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1ba4128>, <kernel.Type object at 0x1ba4170>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1bc4b90>, <kernel.Type object at 0x1ba4320>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1ba4908>, <kernel.DependentProduct object at 0x1ba4248>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1ba4170>, <kernel.DependentProduct object at 0x1ba4128>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula ((and (forall (Xx:(a->Prop)), ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) of role conjecture named cDOMTHM17_pme
% Conjecture to prove = ((and (forall (Xx:(a->Prop)), ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['((and (forall (Xx:(a->Prop)), ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter cA:((a->Prop)->Prop).
% Parameter cB:((b->Prop)->Prop).
% Trying to prove ((and (forall (Xx:(a->Prop)), ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))):(((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found (eq_ref0 (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))))) b0)
% Found x1:(cA Xx)
% Found (fun (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1) as proof of (cA Xx)
% Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect00 (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1)) as proof of (cA Xx)
% Found ((and_rect0 (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1)) as proof of (cA Xx)
% Found (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1)) as proof of (cA Xx)
% Found (fun (x0:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x0:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x1:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x1) x0)) (cA Xx)) (fun (x1:(cA Xx)) (x2:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x1))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found x:(cA Xx)
% Instantiate: x0:=Xx:(a->Prop)
% Found x as proof of (cA x0)
% Found x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))
% Instantiate: x0:=Xx:(a->Prop)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00) as proof of ((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))
% Found (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00) as proof of (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))
% Found (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00) as proof of (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))
% Found ((conj10 x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)) as proof of ((and (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))
% Found (((conj1 (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)) as proof of ((and (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))
% Found ((((conj (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)) as proof of ((and (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))
% Found ((((conj (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)) as proof of ((and (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))
% Found (ex_intro000 ((((conj (cA x0)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (x0 Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))) as proof of ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))
% Found ((ex_intro00 Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))) as proof of ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))
% Found (((ex_intro0 (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))) as proof of ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))
% Found ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))) as proof of ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))
% Found ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))) as proof of ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))
% Found (or_introl00 ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)))) as proof of ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))
% Found ((or_introl0 ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))) ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)))) as proof of ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))
% Found (((or_introl ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))) ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00)))) as proof of ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))
% Found (fun (x:(cA Xx))=> (((or_introl ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))) ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))))) as proof of ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))
% Found (fun (Xx:(a->Prop)) (x:(cA Xx))=> (((or_introl ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))) ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))))) as proof of ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))))
% Found (fun (Xx:(a->Prop)) (x:(cA Xx))=> (((or_introl ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V)))))))))))) ((((ex_intro (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))))) Xx) ((((conj (cA Xx)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))))) x) (fun (G:((a->Prop)->((b->Prop)->Prop))) (x00:((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))=> x00))))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), (((ex a) (fun (Xx0:a)=> ((and (Xx Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False)))))))->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))))
% Found x3:(cA Xx)
% Found (fun (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3) as proof of (cA Xx)
% Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect10 (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3)) as proof of (cA Xx)
% Found (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3)) as proof of (cA Xx)
% Found (fun (x2:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x2:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P:Type) (x3:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P) x3) x2)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x3))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found x0:(cA Xx)
% Found (fun (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of (cA Xx)
% Found (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect00 (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found ((and_rect0 (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found ((conj10 (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)))) iff_refl) as proof of (P b0)
% Found (((conj1 b0) (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)))) iff_refl) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)))) iff_refl) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)))) iff_refl) as proof of (P b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x4:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found x2:(cA Xx)
% Found (fun (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of (cA Xx)
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect10 (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found ((conj10 (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)))) or_ind) as proof of (P b0)
% Found (((conj1 b0) (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)))) or_ind) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)))) or_ind) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)))) or_ind) as proof of (P b0)
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found x2:(cA Xy)
% Found x2 as proof of (cA Xy)
% Found x0:(cA Xx)
% Found (fun (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of (cA Xx)
% Found (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect00 (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found ((and_rect0 (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x0:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x0) x3)) (cA Xx)) (fun (x0:(cA Xx)) (x1:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x0))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x6:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x6:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eq_ref000:=(eq_ref00 x5):((x5 Xx0)->(x5 Xx0))
% Found (eq_ref00 x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of (forall (Xx0:a), ((x5 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eta_expansion00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found (((eta_expansion (a->Prop)) Prop) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))
% Found (eq_ref0 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False))))))))))))))) b0)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found x2:(cA Xx)
% Found (fun (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of (cA Xx)
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect10 (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x5)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found x2:(cA Xy)
% Found (fun (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2) as proof of (cA Xy)
% Found (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2) as proof of ((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->(cA Xy))
% Found (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2) as proof of ((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->(cA Xy)))
% Found (and_rect00 (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2)) as proof of (cA Xy)
% Found ((and_rect0 (cA Xy)) (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2)) as proof of (cA Xy)
% Found (((fun (P:Type) (x2:((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) P) x2) x00)) (cA Xy)) (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2)) as proof of (cA Xy)
% Found (((fun (P:Type) (x2:((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) P) x2) x00)) (cA Xy)) (fun (x2:(cA Xy)) (x3:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x2)) as proof of (cA Xy)
% Found x4:(cA Xy)
% Found x4 as proof of (cA Xy)
% Found x4:(cA Xy)
% Found x4 as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 x5):((x5 Xx0)->(x5 Xx0))
% Found (eq_ref00 x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of (forall (Xx0:a), ((x5 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x5):((x5 Xx0)->(x5 Xx0))
% Found (eq_ref00 x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x5) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of ((x5 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x5)) as proof of (forall (Xx0:a), ((x5 Xx0)->(Xx Xx0)))
% Found x2:(cA Xx)
% Found (fun (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of (cA Xx)
% Found (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx))
% Found (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2) as proof of ((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->(cA Xx)))
% Found (and_rect10 (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x3)) (cA Xx)) (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x3)) (cA Xx)) (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x3)) (cA Xx)) (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))))=> (((fun (P0:Type) (x2:((cA Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False)))))))))) P0) x2) x3)) (cA Xx)) (fun (x2:(cA Xx)) (x4:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))=> x2))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))
% Found x4:(cA Xy)
% Found x4 as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found x4:(cA Xy)
% Found x4 as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))))
% Found (eq_ref00 (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))
% Found ((eq_ref0 (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x1):((x1 Xx0)->(x1 Xx0))
% Found (eq_ref00 x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x1) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of ((x1 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x1)) as proof of (forall (Xx0:a), ((x1 Xx0)->(Xx Xx0)))
% Found x4:(cA Xy)
% Found x4 as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((or ((ex (a->Prop)) (fun (X:(a->Prop))=> ((and (cA X)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex a) (fun (Xx0:a)=> ((and (X Xx0)) (forall (U:(a->Prop)), ((U Xx0)->((G U) (fun (Xy:b)=> False))))))))))))) ((ex (b->Prop)) (fun (Y:(b->Prop))=> ((and (cB Y)) (forall (G:((a->Prop)->((b->Prop)->Prop))), ((S G)->((ex b) (fun (Xy:b)=> ((and (Y Xy)) (forall (V:(b->Prop)), ((V Xy)->((G (fun (Xx0:a)=> False)) V))))))))))))) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xx Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy:b)=> False))))))))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex a) (fun (Xx1:a)=> ((and (Xy Xx1)) (forall (U:(a->Prop)), ((U Xx1)->((Xx0 U) (fun (Xy0:b)=> False)))))))))))))))))))))
% Found x4:(cA Xy)
% Found (fun (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of (cA Xy)
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy))
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy)))
% Found (and_rect10 (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found ((and_rect1 (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found x4:(cA Xy)
% Found (fun (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of (cA Xy)
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy))
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy)))
% Found (and_rect10 (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found ((and_rect1 (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found x4:(cA Xy)
% Found (fun (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4) as proof of (cA Xy)
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4) as proof of ((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->(cA Xy))
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4) as proof of ((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->(cA Xy)))
% Found (and_rect10 (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found ((and_rect1 (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x1 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x1 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found eq_ref000:=(eq_ref00 x3):((x3 Xx0)->(x3 Xx0))
% Found (eq_ref00 x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x3) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of ((x3 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x3)) as proof of (forall (Xx0:a), ((x3 Xx0)->(Xx Xx0)))
% Found x6:(cA Xy)
% Found x6 as proof of (cA Xy)
% Found x4:(cA Xy)
% Found (fun (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of (cA Xy)
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy))
% Found (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4) as proof of ((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->(cA Xy)))
% Found (and_rect10 (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found ((and_rect1 (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found (((fun (P:Type) (x4:((cA Xy)->((forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))->P)))=> (((((and_rect (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) P) x4) x00)) (cA Xy)) (fun (x4:(cA Xy)) (x5:(forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))=> x4)) as proof of (cA Xy)
% Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))))
% Found (eq_ref00 (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found ((eq_ref0 (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found (((eq_ref Prop) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found eq_ref000:=(eq_ref00 (and (cA Xy))):(((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))))
% Found (eq_ref00 (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found ((eq_ref0 (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0)))) (and (cA Xy))) as proof of (P (forall (Xx0:a), ((x3 Xx0)->(Xy Xx0))))
% Found (((
% EOF
%------------------------------------------------------------------------------