TSTP Solution File: SEV171^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV171^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n180.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 13:33:50 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV171^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n180.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:19:51 CDT 2014
% % CPUTime : 300.02
% 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 0x11c2248>, <kernel.Type object at 0x1150950>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((ex (a->((a->(a->a))->a))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) of role conjecture named cTHM33_pme
% Conjecture to prove = ((ex (a->((a->(a->a))->a))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((ex (a->((a->(a->a))->a))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))']
% Parameter a:Type.
% Trying to prove ((ex (a->((a->(a->a))->a))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found eq_ref00:=(eq_ref0 (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))):(((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (eq_ref0 (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b)
% Found x0:=??:a
% Found x0 as proof of a
% Found eta_expansion000:=(eta_expansion00 (x X)):(((eq ((a->(a->a))->a)) (x X)) (fun (x0:(a->(a->a)))=> ((x X) x0)))
% Found (eta_expansion00 (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((eta_expansion0 a) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion (a->(a->a))) a) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion (a->(a->a))) a) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion (a->(a->a))) a) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((ex_intro11 x0) (((eta_expansion (a->(a->a))) a) (x X))) as proof of ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) x0) (((eta_expansion (a->(a->a))) a) (x X))) as proof of ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) x0) (((eta_expansion (a->(a->a))) a) (x X))) as proof of ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))
% Found (ex_intro100 (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) x0) (((eta_expansion (a->(a->a))) a) (x X)))) as proof of ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq ((a->((a->(a->a))->a))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((a->((a->(a->a))->a))->Prop)) a0) (fun (x:(a->((a->(a->a))->a)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:(a->((a->(a->a))->a)))=> Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->((a->(a->a))->a))) (fun (x1:(a->((a->(a->a))->a)))=> Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->((a->(a->a))->a))) (fun (x1:(a->((a->(a->a))->a)))=> Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found (((eta_expansion_dep (a->((a->(a->a))->a))) (fun (x1:(a->((a->(a->a))->a)))=> Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->((a->(a->a))->a))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) a0) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))):(((eq Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found (eq_ref0 (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) as proof of (((eq Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) b)
% Found ((eq_ref Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) as proof of (((eq Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) b)
% Found ((eq_ref Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) as proof of (((eq Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) b)
% Found ((eq_ref Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) as proof of (((eq Prop) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))):(((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) (fun (x0:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))))
% Found (eta_expansion_dep00 (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) as proof of (((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) as proof of (((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) as proof of (((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) as proof of (((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) as proof of (((eq (a->Prop)) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found x1:(P Xx)
% Instantiate: b:=Xx:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:a), (((eq Prop) (f x0)) ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x1:(P Xx)
% Instantiate: b:=Xx:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found x1:(P Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found x1:(P Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P0 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Instantiate: b:=((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_sym01 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found x1:(P Xx)
% Instantiate: a0:=Xx:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x00:=(x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((eq_sym00 (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (((((eq_trans0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((eq_sym00 (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> ((((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)))=> (((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) ((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found x1:(P Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))):(((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (eq_ref0 (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b0)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b0)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b0)
% Found ((eq_ref ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) (fun (F:(a->((a->(a->a))->a)))=> ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (F Xx)) (F Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (F X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((a->((a->(a->a))->a))->Prop)) b) (fun (x:(a->((a->(a->a))->a)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) b0)
% Found (((eta_expansion (a->((a->(a->a))->a))) Prop) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) b0)
% Found (((eta_expansion (a->((a->(a->a))->a))) Prop) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) b0)
% Found (((eta_expansion (a->((a->(a->a))->a))) Prop) b) as proof of (((eq ((a->((a->(a->a))->a))->Prop)) b) b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found x1:(P b)
% Found x1 as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b))):((P b)->(P b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b))) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 Xy)->(P1 Xy))
% Found (eq_ref00 P1) as proof of (P2 Xy)
% Found ((eq_ref0 Xy) P1) as proof of (P2 Xy)
% Found (((eq_ref a) Xy) P1) as proof of (P2 Xy)
% Found (((eq_ref a) Xy) P1) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_trans000 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_trans00 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((((eq_trans0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (x00:(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((((eq_trans Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x1:(P Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Instantiate: b:=((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))):Prop
% Found x0 as proof of (P1 b)
% Found x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Instantiate: b:=((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_sym01 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((eq_sym01 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0) as proof of (P0 (f x))
% Found (fun (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (P0 (f x))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of ((P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))->(P0 (f x)))
% Found (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((eq_sym00 (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found (((eq_sym0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> ((((eq_sym0 (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found ((((eq_sym Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) (f x)) (fun (P0:(Prop->Prop)) (x0:(P0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))))=> (((((eq_sym Prop) (f x)) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) x00) P0)) ((eq_ref Prop) (f x))) x0))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))))
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found ((eq_ref a) Xy) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b00)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b00)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b00)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b00)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P b0)
% Found ((eq_ref a) Xx) as proof of (P b0)
% Found ((eq_ref a) Xx) as proof of (P b0)
% Found ((eq_ref a) Xx) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found x1:(P Xx)
% Instantiate: b0:=Xx:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xx)
% Instantiate: a0:=Xx:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->((a->(a->a))->a)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->((a->(a->a))->a)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(a->((a->(a->a))->a))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->((a->(a->a))->a)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(a->((a->(a->a))->a)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(a->((a->(a->a))->a))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xy)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))):(((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found (eq_ref0 ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found ((eq_ref Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) as proof of (((eq Prop) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y)))))))))) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a0
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xx))):((P Xx)->(P Xx))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xx))) as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found ((eq_ref a) b) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b1)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x00:=(x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 b00):(((eq a) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found ((eq_ref a) b00) as proof of (((eq a) b00) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b00)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b00)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b00)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b00)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P Xy))):((P Xy)->(P Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P Xy))) as proof of (P0 Xy)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:a
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found x1:(P Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found x1:(P Xy)
% Instantiate: a0:=Xy:a
% Found x1 as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found x010:=(x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))):((P0 Xx)->(P0 Xx))
% Found (x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found (x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x X)):(((eq ((a->(a->a))->a)) (x X)) (fun (x0:(a->(a->a)))=> ((x X) x0)))
% Found (eta_expansion_dep00 (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((ex_intro11 a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (b x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (b x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (b x0)
% Found (ex_intro100 (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)))) as proof of ((ex a) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found x010:=(x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))):((P0 Xx)->(P0 Xx))
% Found (x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found (x01 (fun (x0:((a->(a->a))->a))=> (P0 Xx))) as proof of (P1 Xx)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 b))):((P1 b)->(P1 b))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 b))) as proof of (P2 b)
% Found eq_ref000:=(eq_ref00 P1):((P1 b)->(P1 b))
% Found (eq_ref00 P1) as proof of (P2 b)
% Found ((eq_ref0 b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found (((eq_ref a) b) P1) as proof of (P2 b)
% Found x1:(P b)
% Found x1 as proof of (P0 Xx)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found a_DUMMY:a
% Found a_DUMMY as proof of a
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x X)):(((eq ((a->(a->a))->a)) (x X)) (fun (x0:(a->(a->a)))=> ((x X) x0)))
% Found (eta_expansion_dep00 (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((ex_intro11 a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (ex_intro100 (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)))) as proof of ((ex a) f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x X)):(((eq ((a->(a->a))->a)) (x X)) (fun (x0:(a->(a->a)))=> ((x X) x0)))
% Found (eta_expansion_dep00 (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((eta_expansion_dep0 (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)) as proof of (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0)))
% Found ((ex_intro11 a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X))) as proof of (f x0)
% Found (ex_intro100 (((ex_intro1 (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) x0))))) a_DUMMY) (((eta_expansion_dep (a->(a->a))) (fun (x2:(a->(a->a)))=> a)) (x X)))) as proof of ((ex a) f)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found x1:(P1 Xy)
% Instantiate: b:=Xy:a
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found x00:=(x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))):((P1 Xy)->(P1 Xy))
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found (x0 (fun (x2:((a->(a->a))->a))=> (P1 Xy))) as proof of (P2 Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found ((eq_ref a) b0) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P b0))):((P b0)->(P b0))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P b0))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found ((eq_ref a) Xx) as proof of (P Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found x1:(P Xy)
% Instantiate: b0:=Xy:a
% Found x1 as proof of (P0 b0)
% Found x00:=(x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))):((P0 Xy)->(P0 Xy))
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found (x0 (fun (x1:((a->(a->a))->a))=> (P0 Xy))) as proof of (P1 Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found x1:(P Xy)
% Found x1 as proof of (P0 Xy)
% Found x1:(P Xy)
% Found x1 as proof of (P0 Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:(P0 (f x))
% Instantiate: a0:=(f x):Prop
% Found x0 as proof of (P1 a0)
% Found x0:(P0 (f x))
% Instantiate: a0:=(f x):Prop
% Found x0 as proof of (P1 a0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (forall (Xx:a) (Xy:a), ((((eq ((a->(a->a))->a)) (x Xx)) (x Xy))->(((eq a) Xx) Xy)))) (forall (X:a), ((ex a) (fun (Y:a)=> ((ex a) (fun (Z:a)=> (((eq ((a->(a->a))->a)) (x X)) (fun (G:(a->(a->a)))=> ((G X) Y))))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found (
% EOF
%------------------------------------------------------------------------------