%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU988^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n184.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 13:33:28 EDT 2014

% Result   : Timeout 300.09s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU988^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n184.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:48:01 CDT 2014
% % CPUTime  : 300.09 
% 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 0xe9a830>, <kernel.Type object at 0xe9af38>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1078128>, <kernel.Type object at 0xe9a998>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0xe9add0>, <kernel.DependentProduct object at 0xe9a680>) of role type named f
% Using role type
% Declaring f:(a->(b->Prop))
% FOF formula ((forall (Y:(b->Prop)), ((ex a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))) of role conjecture named cTHM529_pme
% Conjecture to prove = ((forall (Y:(b->Prop)), ((ex a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['((forall (Y:(b->Prop)), ((ex a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter f:(a->(b->Prop)).
% Trying to prove ((forall (Y:(b->Prop)), ((ex a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y))))->((ex ((a->Prop)->(b->Prop))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) (fun (x:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found (eta_expansion00 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind00 (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind0 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (((a->Prop)->(b->Prop))->Prop)) a0) (fun (x:((a->Prop)->(b->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (x:((a->Prop)->(b->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))):(((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) (fun (x:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x0 Xx))) Xx))))))
% Found (eta_expansion_dep00 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x2:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x2:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x2:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found (fun (x3:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found (fun (x3:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found (fun (x3:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))))
% Found (fun (x3:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found (fun (x1:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found (fun (x1:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x0 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found (fun (x1:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))))
% Found (fun (x1:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x0 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality_dep000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((eq_sym10 (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (((eq_sym1 ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality_dep000 (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality_dep00 Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_sym00 (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((eq_sym0 Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x20:=(x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x30:=(x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found x40:=(x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality_dep000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((eq_trans000 Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((eq_trans00 Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((eq_trans0 (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x3:a) (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x3:a) (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (b->Prop)) (f x)) Y)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind00 (fun (x3:a) (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind0 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x3:a) (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x3:(forall (x:a), ((((eq (b->Prop)) (f x)) Y)->P)))=> (((((ex_ind a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y))) P) x3) x2)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x3:a) (x4:(((eq (b->Prop)) (f x3)) Y))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x3 (x4 Xy))):(((eq (b->Prop)) (x3 (x4 Xy))) (x3 (x4 Xy)))
% Found (eq_ref0 (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx))
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))
% Found (ex_ind10 (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x0 (x4 Xy))):(((eq (b->Prop)) (x0 (x4 Xy))) (x0 (x4 Xy)))
% Found (eq_ref0 (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx))
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))
% Found (ex_ind10 (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality_dep00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality_dep0000 x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality0000 x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality000 Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x3 (x4 Xy))) P)) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found x20:=(x2 (fun (x5:(b->Prop))=> (P (x3 (x4 Xy))))):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (x2 (fun (x5:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (x2 (fun (x5:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x3 (x4 Xy)))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind10 (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x5:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x5) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x5:a) (x6:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x5)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x30:=(x3 (fun (x5:(b->Prop))=> (P (x0 (x4 Xy))))):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (x3 (fun (x5:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (x3 (fun (x5:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x0 (x4 Xy)))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x4 Xy))) P)) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((functional_extensionality00 (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x40:=(x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x40:=(x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((functional_extensionality000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((functional_extensionality00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((functional_extensionality_dep000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found (fun (x4:(((eq (b->Prop)) (f x3)) Y))=> x4) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (fun (x:b)=> ((x0 (x1 Xy)) x)))
% Found (eta_expansion_dep00 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eta_expansion_dep0 (fun (x3:b)=> Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion_dep b) (fun (x3:b)=> Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_trans0000 ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((eq_trans000 Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((eq_trans00 Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((eq_trans0 (x0 (x1 Xy))) Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((((eq_trans (b->Prop)) (x0 (x1 Xy))) Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (eq_sym100 ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((eq_sym10 (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (((eq_sym1 ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (x:b), (((eq Prop) (Xy x)) ((x0 (x1 Xy)) x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((functional_extensionality000 (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((functional_extensionality00 Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((((functional_extensionality0 Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (eq_sym000 (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_sym00 (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((eq_sym0 Xy) (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((eq_sym (b->Prop)) Xy) (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((eq_sym10 (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (((eq_sym1 ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality000 (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality00 Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_sym00 (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((eq_sym0 Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x20:=(x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x2 (fun (x5:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (functional_extensionality00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((functional_extensionality0000 x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((functional_extensionality000 Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality00 (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (functional_extensionality_dep00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((functional_extensionality_dep0000 x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((functional_extensionality_dep000 Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found x30:=(x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x3 (fun (x5:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x40:=(x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x4 (fun (x5:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) (fun (x:((a->Prop)->(b->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) (fun (x:((a->Prop)->(b->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion000:=(eta_expansion00 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) (fun (x:((a->Prop)->(b->Prop)))=> (f0 x)))
% Found (eta_expansion00 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eta_expansion0 Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion ((a->Prop)->(b->Prop))) Prop) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (((a->Prop)->(b->Prop))->Prop)) a0) (fun (x:((a->Prop)->(b->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) a0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (x:((a->Prop)->(b->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eta_expansion_dep0 (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x4:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (((b->Prop)->(a->Prop))->Prop)) a0) (fun (x:((b->Prop)->(a->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion ((b->Prop)->(a->Prop))) Prop) a0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion ((b->Prop)->(a->Prop))) Prop) a0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) a0) b0)
% Found (((eta_expansion ((b->Prop)->(a->Prop))) Prop) a0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (((b->Prop)->(a->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (((b->Prop)->(a->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) b0) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) b0) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) (fun (x:((a->Prop)->(b->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((eq_sym10 (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (((eq_sym1 ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality000 (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality00 Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym0000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (eq_sym0000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> ((eq_sym000 x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((eq_sym00 (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> ((((eq_sym0 Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))):(((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) (fun (x:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x3 Xx))) Xx))))))
% Found (eta_expansion_dep00 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x5:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x5:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) b0)
% Found (((eta_expansion_dep ((b->Prop)->(a->Prop))) (fun (x5:((b->Prop)->(a->Prop)))=> Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x3 Xx))) Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))):(((eq Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))) b0)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((eq_trans000 Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((eq_trans00 Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((eq_trans0 (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x6:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (forall (P:((b->Prop)->Prop)), ((P (x3 (x4 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))):(((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx))))))
% Found (eq_ref0 (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found ((eq_ref (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) as proof of (((eq (((b->Prop)->(a->Prop))->Prop)) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x0 Xx))) Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b0)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x4 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x3 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x3 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x3 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (((eq_sym1 ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x4 (x3 Xx)) x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found ((functional_extensionality000 (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((functional_extensionality00 Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (eq_sym000 (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_sym00 (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((eq_sym0 Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x0 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 (f0 x4)):(((eq Prop) (f0 x4)) (f0 x4))
% Found (eq_ref0 (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found ((eq_ref Prop) (f0 x4)) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (((eq Prop) (f0 x4)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x4 Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x4 Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))))
% Found (fun (x4:((b->Prop)->(a->Prop)))=> ((eq_ref Prop) (f0 x4))) as proof of (forall (x:((b->Prop)->(a->Prop))), (((eq Prop) (f0 x)) ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (x (x0 Xx))) Xx))))))
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (((eq_sym1 ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x4 (x0 Xx)) x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found ((functional_extensionality000 (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((functional_extensionality00 Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_sym00 (x4 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((eq_sym0 Xx) (x4 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x4 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f0 x6)):(((eq Prop) (f0 x6)) (f0 x6))
% Found (eq_ref0 (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found (fun (x6:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x6))) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found (fun (x6:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x6))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 (f0 x6)):(((eq Prop) (f0 x6)) (f0 x6))
% Found (eq_ref0 (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found ((eq_ref Prop) (f0 x6)) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found (fun (x6:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x6))) as proof of (((eq Prop) (f0 x6)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x6 (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x6 Xx))) Xx)))))))
% Found (fun (x6:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x6))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (x Xx))) Xx))))))))
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality0000 x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality000 Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality_dep00000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality_dep0000 x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) x2) P)) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (((eq_sym1 ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality_dep000 (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality_dep00 Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_sym00 (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((eq_sym0 Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((functional_extensionality00 (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((((functional_extensionality0 Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((functional_extensionality_dep00 (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality_dep00 (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality00 (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality0 Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) (f x3))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) (f x3))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) (f x3))
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) (f x3))
% Found (eq_trans0000 ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found ((eq_trans000 Y) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found (((eq_trans00 (f x3)) Y) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found ((((eq_trans0 (x0 (x1 Xy))) (f x3)) Y) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found (((((eq_trans (b->Prop)) (x0 (x1 Xy))) (f x3)) Y) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found (((((eq_trans (b->Prop)) (x0 (x1 Xy))) (f x3)) Y) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) as proof of ((((eq (b->Prop)) (f x3)) Y)->(((eq (b->Prop)) (x0 (x1 Xy))) Xy))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality_dep000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((functional_extensionality000 Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x6:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x1 Xx)):(((eq (a->Prop)) (x1 Xx)) (fun (x:a)=> ((x1 Xx) x)))
% Found (eta_expansion_dep00 (x1 Xx)) as proof of (((eq (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy))))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (x1 Xx)) as proof of (((eq (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx)) as proof of (((eq (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx)) as proof of (((eq (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy))))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx)) as proof of (((eq (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy))))
% Found (eq_sym000 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (x1 Xx))
% Found ((eq_sym00 (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (x1 Xx))
% Found (((eq_sym0 (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (x1 Xx))
% Found ((((eq_sym (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (x1 Xx))
% Found ((((eq_sym (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx))) as proof of (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (x1 Xx))
% Found (ex_intro200 ((((eq_sym (a->Prop)) (x1 Xx)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (x1 Xx)))) as proof of ((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))):(((eq Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy)))
% Found (eq_ref0 (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) as proof of (((eq Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) b0)
% Found ((eq_ref Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) as proof of (((eq Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) b0)
% Found ((eq_ref Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) as proof of (((eq Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) b0)
% Found ((eq_ref Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) as proof of (((eq Prop) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (x0 (x1 Xy))) Xy))) b0)
% Found eq_ref00:=(eq_ref0 (x7 (x6 Xx))):(((eq (a->Prop)) (x7 (x6 Xx))) (x7 (x6 Xx)))
% Found (eq_ref0 (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x7 (x6 Xx)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: b0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (x7 (x3 Xx))):(((eq (a->Prop)) (x7 (x3 Xx))) (x7 (x3 Xx)))
% Found (eq_ref0 (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x7 (x3 Xx)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xx):(((eq (a->Prop)) Xx) (fun (x:a)=> (Xx x)))
% Found (eta_expansion_dep00 Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found x50:=(x5 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))):((P (x7 (x6 Xx)))->(P (x7 (x6 Xx))))
% Found (x5 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (x5 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x5 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx)))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x5 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))):((P (x7 (x6 Xx)))->(P (x7 (x6 Xx))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (x2 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx)))))) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x7 (x6 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x7 (x6 Xx)))->(P (x7 (x6 Xx))))
% Found (eq_ref00 P) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found ((eq_ref0 (x7 (x6 Xx))) P) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x6 Xx))) P) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x6 Xx))) P) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x6 Xx))) P)) as proof of ((P (x7 (x6 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x6 Xx))) P)) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found eta_expansion000:=(eta_expansion00 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (fun (x:b)=> ((x0 (x1 Xy)) x)))
% Found (eta_expansion00 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eta_expansion0 Prop) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion b) Prop) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion b) Prop) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found (((eta_expansion b) Prop) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_trans0000 ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((eq_trans000 Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((eq_trans00 Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((((eq_trans0 (x0 (x1 Xy))) Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((((eq_trans (b->Prop)) (x0 (x1 Xy))) Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((((eq_trans (b->Prop)) (x0 (x1 Xy))) Xy) Xy) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x7 (x0 Xx))):(((eq (a->Prop)) (x7 (x0 Xx))) (x7 (x0 Xx)))
% Found (eq_ref0 (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x7 (x0 Xx)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (eq_sym100 ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((eq_sym10 (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (((eq_sym1 ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (x:b), (((eq Prop) (Xy x)) ((x0 (x1 Xy)) x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((functional_extensionality000 (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((functional_extensionality00 Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((((functional_extensionality0 Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (eq_sym000 (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((eq_sym00 (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found (((eq_sym0 Xy) (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((eq_sym (b->Prop)) Xy) (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found ((((eq_sym (b->Prop)) Xy) (x0 (x1 Xy))) (((((functional_extensionality b) Prop) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: f0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 f0)
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: f0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 f0)
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))):((P (x7 (x3 Xx)))->(P (x7 (x3 Xx))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (x2 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx)))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x7 (x3 Xx)))->(P (x7 (x3 Xx))))
% Found (eq_ref00 P) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found ((eq_ref0 (x7 (x3 Xx))) P) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x3 Xx))) P) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x3 Xx))) P) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x3 Xx))) P)) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x3 Xx))) P)) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found x60:=(x6 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))):((P (x7 (x3 Xx)))->(P (x7 (x3 Xx))))
% Found (x6 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (x6 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx)))))) as proof of ((P (x7 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x7 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind00 (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind0 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x10)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x10)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (fun (x:a)=> ((x1 (x0 Xx)) x)))
% Found (eta_expansion_dep00 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_trans00000 ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_trans00000 ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) b0)) (x3:(((eq (a->Prop)) b0) Xx))=> (((eq_trans0000 x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) b0)) (x3:(((eq (a->Prop)) b0) Xx))=> ((((eq_trans000 Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (x3:(((eq (a->Prop)) Xx) Xx))=> (((((eq_trans00 Xx) Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (x3:(((eq (a->Prop)) Xx) Xx))=> ((((((eq_trans0 (x1 (x0 Xx))) Xx) Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (x3:(((eq (a->Prop)) Xx) Xx))=> (((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (x3:(((eq (a->Prop)) Xx) Xx))=> (((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((fun (x2:(((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (x3:(((eq (a->Prop)) Xx) Xx))=> (((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) x2) x3) P)) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (x7 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x3 Xx))) P) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x3 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))):((P (x4 (x3 Xx)))->(P (x4 (x3 Xx))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (x2 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x60:=(x6 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))):((P (x7 (x0 Xx)))->(P (x7 (x0 Xx))))
% Found (x6 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (x6 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx)))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x7 (x0 Xx)))->(P (x7 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x7 (x0 Xx))) P) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x0 Xx))) P) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x7 (x0 Xx))) P) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x0 Xx))) P)) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x7 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found x30:=(x3 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))):((P (x7 (x0 Xx)))->(P (x7 (x0 Xx))))
% Found (x3 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (x3 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx)))))) as proof of ((P (x7 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x7 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind00 (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind0 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind00 (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind0 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P0:Prop) (x2:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x2) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x2:a) (x3:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x2)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found ((functional_extensionality00000 (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> (((functional_extensionality000 Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (fun (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of (P Xx)
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found ((functional_extensionality_dep00000 (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x6:a)=> Prop)) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2) as proof of (P Xx)
% Found (fun (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of (P Xx)
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x6:a)=> Prop)) (x1 (x0 Xx))) Xx) x3) P)) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))) x2)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x30:=(x3 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (x3 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (x3 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x4 (x0 Xx))) P) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x4 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))):((P (x4 (x0 Xx)))->(P (x4 (x0 Xx))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (x7 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) b0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((eq_trans000 Xx) ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((eq_trans00 Xx) Xx) ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((((eq_trans0 (x4 (x3 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((((eq_trans (a->Prop)) (x4 (x3 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((((eq_trans (a->Prop)) (x4 (x3 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x3 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x0 (x1 Xy)))->(P0 (x0 (x1 Xy))))
% Found (eq_ref00 P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (fun (P0:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0)) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found x40:=(x4 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (x4 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (x4 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x4 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x4 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (eq_ref00 P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))):((P (x1 (x0 Xx)))->(P (x1 (x0 Xx))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (x7 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x0 (x1 Xy)))->(P0 (x0 (x1 Xy))))
% Found (eq_ref00 P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (fun (P0:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0)) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x0 (x1 Xy)))->(P0 (x0 (x1 Xy))))
% Found (eq_ref00 P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found (fun (P0:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P0)) as proof of ((P0 (x0 (x1 Xy)))->(P0 Xy))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) b0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((eq_trans000 Xx) ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((eq_trans00 Xx) Xx) ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((((eq_trans0 (x4 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((((eq_trans (a->Prop)) (x4 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((((eq_trans (a->Prop)) (x4 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x4 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (functional_extensionality00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((functional_extensionality0000 x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((functional_extensionality000 Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality00 (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (functional_extensionality_dep00000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((functional_extensionality_dep0000 x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((functional_extensionality_dep000 Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) x2) P)) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (forall (P:((b->Prop)->Prop)), ((P (x0 (x1 Xy)))->(P Xy)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) b0)
% Found ((eq_trans0000 ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((eq_trans000 Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) b0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((eq_trans00 Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((eq_trans0 (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((((eq_trans (a->Prop)) (x1 (x0 Xx))) Xx) Xx) ((eq_ref (a->Prop)) (x1 (x0 Xx)))) ((eq_ref (a->Prop)) Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep1000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep100 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep10 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep1 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x4:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality1000 (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality100 Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality10 (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality1 Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x2:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of ((((eq (b->Prop)) (f x6)) Y0)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx))
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))
% Found (ex_ind10 (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((fun (P:Prop) (x6:(forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->P)))=> (((((ex_ind a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y0))) P) x6) x5)) (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x6 (x7 Xy))):(((eq (b->Prop)) (x6 (x7 Xy))) (x6 (x7 Xy)))
% Found (eq_ref0 (x6 (x7 Xy))) as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x6 (x7 Xy))) as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x6 (x7 Xy))) as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> (((functional_extensionality000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of ((P (x4 (x3 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of ((((eq (b->Prop)) (f x6)) Y0)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx))
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))
% Found (ex_ind10 (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x6:(forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->P)))=> (((((ex_ind a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y0))) P) x6) x5)) (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (eq_sym100 ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((eq_sym10 (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (((eq_sym1 ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq Prop) (Xy x2)) ((x0 (x1 Xy)) x2))
% Found (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (forall (x:b), (((eq Prop) (Xy x)) ((x0 (x1 Xy)) x)))
% Found (functional_extensionality_dep0000 (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((functional_extensionality_dep000 (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((functional_extensionality_dep00 Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found ((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))) as proof of (((eq (b->Prop)) Xy) (x0 (x1 Xy)))
% Found (eq_sym0000 (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (eq_sym0000 (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(((eq (b->Prop)) Xy) (x0 (x1 Xy))))=> ((eq_sym000 x2) P)) (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(((eq (b->Prop)) Xy) (x0 (x1 Xy))))=> (((eq_sym00 (x0 (x1 Xy))) x2) P)) (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(((eq (b->Prop)) Xy) (x0 (x1 Xy))))=> ((((eq_sym0 Xy) (x0 (x1 Xy))) x2) P)) (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((fun (x2:(((eq (b->Prop)) Xy) (x0 (x1 Xy))))=> (((((eq_sym (b->Prop)) Xy) (x0 (x1 Xy))) x2) P)) (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> ((fun (x2:(((eq (b->Prop)) Xy) (x0 (x1 Xy))))=> (((((eq_sym (b->Prop)) Xy) (x0 (x1 Xy))) x2) P)) (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) Xy) (x0 (x1 Xy))) (fun (x2:b)=> ((((eq_sym Prop) ((x0 (x1 Xy)) x2)) (Xy x2)) ((eq_ref Prop) ((x0 (x1 Xy)) x2))))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x7:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x7 (x6 Xx))):(((eq (a->Prop)) (x7 (x6 Xx))) (x7 (x6 Xx)))
% Found (eq_ref0 (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x6 Xx))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x6 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x6 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x7 (x6 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x6 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x6 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x6 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x3 (x7 Xy))):(((eq (b->Prop)) (x3 (x7 Xy))) (x3 (x7 Xy)))
% Found (eq_ref0 (x3 (x7 Xy))) as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x7 Xy))) as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x7 Xy))) as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: b0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of ((P (x4 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x4 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (b->Prop)) (f x6)) Y0)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind10 (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind1 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x6:(forall (x:a), ((((eq (b->Prop)) (f x)) Y0)->P)))=> (((((ex_ind a) (fun (X:a)=> (((eq (b->Prop)) (f X)) Y0))) P) x6) x5)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x6:a) (x7:(((eq (b->Prop)) (f x6)) Y0))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x2)):(((eq Prop) ((x1 (x0 Xx)) x2)) ((x1 (x0 Xx)) x2))
% Found (eq_ref0 ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x2)) as proof of (((eq Prop) ((x1 (x0 Xx)) x2)) (Xx x2))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((eq_sym10 (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (((eq_sym1 ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (((eq Prop) (Xx x2)) ((x1 (x0 Xx)) x2))
% Found (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality0000 (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality000 (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality00 Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym0000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (eq_sym0000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> ((eq_sym000 x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((eq_sym00 (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> ((((eq_sym0 Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2)))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x2:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x2) P)) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x2:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x2)) (Xx x2)) ((eq_ref Prop) ((x1 (x0 Xx)) x2))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found x5:(((eq (b->Prop)) (f x4)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x6:=(fun (x9:(a->Prop))=> (f x4)):((a->Prop)->(b->Prop))
% Found x5 as proof of (((eq (b->Prop)) (x6 (x7 Xy))) Xy)
% Found eta_expansion000:=(eta_expansion00 Xx):(((eq (a->Prop)) Xx) (fun (x:a)=> (Xx x)))
% Found (eta_expansion00 Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) b0)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x7:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) (fun (x:((a->Prop)->(b->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eq_ref00:=(eq_ref0 (x7 (x3 Xx))):(((eq (a->Prop)) (x7 (x3 Xx))) (x7 (x3 Xx)))
% Found (eq_ref0 (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x3 Xx))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x3 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x3 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x7 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x3 (x4 Xy))):(((eq (b->Prop)) (x3 (x4 Xy))) (x3 (x4 Xy)))
% Found (eq_ref0 (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x0 (x7 Xy))):(((eq (b->Prop)) (x0 (x7 Xy))) (x0 (x7 Xy)))
% Found (eq_ref0 (x0 (x7 Xy))) as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x7 Xy))) as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x7 Xy))) as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (functional_extensionality_dep00000 (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((functional_extensionality_dep0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((functional_extensionality_dep000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop))=> ((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x6 (x7 Xy)))->(P (x6 (x7 Xy))))
% Found (eq_ref00 P) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found ((eq_ref0 (x6 (x7 Xy))) P) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x6 (x7 Xy))) P) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x6 (x7 Xy))) P) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x6 (x7 Xy))) P)) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))):((P (x6 (x7 Xy)))->(P (x6 (x7 Xy))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (x2 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy)))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found x50:=(x5 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))):((P (x6 (x7 Xy)))->(P (x6 (x7 Xy))))
% Found (x5 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (x5 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x5 (fun (x8:(b->Prop))=> (P (x6 (x7 Xy)))))) as proof of ((P (x6 (x7 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) f0)
% Found (eq_ref0 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f0):(((eq (((a->Prop)->(b->Prop))->Prop)) f0) (fun (x:((a->Prop)->(b->Prop)))=> (f0 x)))
% Found (eta_expansion_dep00 f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found ((eta_expansion_dep0 (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found (((eta_expansion_dep ((a->Prop)->(b->Prop))) (fun (x1:((a->Prop)->(b->Prop)))=> Prop)) f0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) f0) b0)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x7 Xy))) Xy)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x7:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq (((a->Prop)->(b->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) b0) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) b0) b00)
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: f0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 f0)
% Found x2:(P (x1 (x0 Xx)))
% Instantiate: f0:=(x1 (x0 Xx)):(a->Prop)
% Found x2 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (x4 (x3 Xx))):(((eq (a->Prop)) (x4 (x3 Xx))) (x4 (x3 Xx)))
% Found (eq_ref0 (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x3 Xx))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x3 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x4 (x3 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x3 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (x7 (x0 Xx))):(((eq (a->Prop)) (x7 (x0 Xx))) (x7 (x0 Xx)))
% Found (eq_ref0 (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x7 (x0 Xx))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x0 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x7 (x0 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x7 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x7 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x7 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x4 Xy))):(((eq (b->Prop)) (x0 (x4 Xy))) (x0 (x4 Xy)))
% Found (eq_ref0 (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (b0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f0 x)) (b0 x)))
% Found x60:=(x6 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))):((P (x3 (x7 Xy)))->(P (x3 (x7 Xy))))
% Found (x6 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (x6 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy)))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))):((P (x3 (x7 Xy)))->(P (x3 (x7 Xy))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x3 (x7 Xy)))))) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x3 (x7 Xy)))->(P (x3 (x7 Xy))))
% Found (eq_ref00 P) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found ((eq_ref0 (x3 (x7 Xy))) P) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x7 Xy))) P) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x7 Xy))) P) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x3 (x7 Xy))) P)) as proof of ((P (x3 (x7 Xy)))->(P Xy))
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))):(((eq Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx)))))))
% Found (eq_ref0 (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) as proof of (((eq Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) as proof of (((eq Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) as proof of (((eq Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) as proof of (((eq Prop) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (x1 Xx))))))) b0)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x7 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x4 (x0 Xx))):(((eq (a->Prop)) (x4 (x0 Xx))) (x4 (x0 Xx)))
% Found (eq_ref0 (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x4 (x0 Xx))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x4 (x0 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x4 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x4 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x6 Xx)) x8)):(((eq Prop) ((x7 (x6 Xx)) x8)) ((x7 (x6 Xx)) x8))
% Found (eq_ref0 ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x6 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (((functional_extensionality00 (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x6 Xx)) x8)):(((eq Prop) ((x7 (x6 Xx)) x8)) ((x7 (x6 Xx)) x8))
% Found (eq_ref0 ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x6 Xx)) x8)) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))) as proof of (((eq Prop) ((x7 (x6 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x6 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x6 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x6 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x6 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x5)):(((eq Prop) ((x4 (x3 Xx)) x5)) ((x4 (x3 Xx)) x5))
% Found (eq_ref0 ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x5)) as proof of (((eq Prop) ((x4 (x3 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (((eq_sym1 ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x4 (x3 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x4 (x3 Xx)) x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found ((functional_extensionality000 (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((functional_extensionality00 Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x3 Xx)))
% Found (eq_sym000 (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((eq_sym00 (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (((eq_sym0 Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x3 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x4 (x3 Xx))) (((((functional_extensionality a) Prop) Xx) (x4 (x3 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x3 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x3 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref000:=(eq_ref00 P):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x3 (x4 Xy))) P)) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy)))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy)))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x7 Xy)))->(P (x0 (x7 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x7 Xy))) P) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x7 Xy))) P) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x7 Xy))) P) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x7 Xy))) P)) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found x30:=(x3 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))):((P (x0 (x7 Xy)))->(P (x0 (x7 Xy))))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy)))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found x60:=(x6 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))):((P (x0 (x7 Xy)))->(P (x0 (x7 Xy))))
% Found (x6 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (x6 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x6 (fun (x8:(b->Prop))=> (P (x0 (x7 Xy)))))) as proof of ((P (x0 (x7 Xy)))->(P Xy))
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x0)):(((eq Prop) (f1 x0)) (f1 x0))
% Found (eq_ref0 (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found ((eq_ref Prop) (f1 x0)) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (((eq Prop) (f1 x0)) (f0 x0))
% Found (fun (x0:((a->Prop)->(b->Prop)))=> ((eq_ref Prop) (f1 x0))) as proof of (forall (x:((a->Prop)->(b->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx))
% Found (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(((eq (a->Prop)) (x1 (x0 Xx))) Xx)))
% Found (ex_ind20 (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((ex_ind2 (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((fun (P:Prop) (x8:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P) x8) x00)) (((eq (a->Prop)) (x1 (x0 Xx))) Xx)) (fun (x8:a) (x9:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x8)) (f Xy)))) Xx))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (f0 x5)):(((eq Prop) (f0 x5)) (f0 x5))
% Found (eq_ref0 (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found ((eq_ref Prop) (f0 x5)) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (((eq Prop) (f0 x5)) (Xx x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f0 x5))) as proof of (forall (x:a), (((eq Prop) (f0 x)) (Xx x)))
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (f0 x5)))) x2) as proof of (P Xx)
% Found ((functional_extensionality00000 (fun (x5:a)=> ((eq_ref Prop) (f0 x5)))) x2) as proof of (P Xx)
% Found (((fun (x5:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> ((functional_extensionality0000 x5) P)) (fun (x5:a)=> ((eq_ref Prop) (f0 x5)))) x2) as proof of (P Xx)
% Found (((fun (x5:(forall (x:a), (((eq Prop) (f0 x)) (Xx x))))=> (((functional_extensionality000 Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) (f0 x5)))) x2) as proof of (P Xx)
% Found (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((functional_extensionality00 (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2) as proof of (P Xx)
% Found (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> (((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2) as proof of (P Xx)
% Found (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2) as proof of (P Xx)
% Found (fun (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)) as proof of (P Xx)
% Found (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)) as proof of ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx)->(P Xx))
% Found (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)) as proof of (forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->(P Xx)))
% Found (ex_ind00 (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2))) as proof of (P Xx)
% Found ((ex_ind0 (P Xx)) (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2))) as proof of (P Xx)
% Found (((fun (P0:Prop) (x3:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x3) x00)) (P Xx)) (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2))) as proof of (P Xx)
% Found (fun (x2:(P (x1 (x0 Xx))))=> (((fun (P0:Prop) (x3:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x3) x00)) (P Xx)) (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)))) as proof of (P Xx)
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (P0:Prop) (x3:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x3) x00)) (P Xx)) (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)))) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (P0:Prop) (x3:(forall (x:a), ((((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x)) (f Xy)))) Xx)->P0)))=> (((((ex_ind a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))) P0) x3) x00)) (P Xx)) (fun (x3:a) (x4:(((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f x3)) (f Xy)))) Xx))=> (((fun (x5:(forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x))))=> ((((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) x5) P)) (fun (x5:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) x2)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x3 Xx)) x8)):(((eq Prop) ((x7 (x3 Xx)) x8)) ((x7 (x3 Xx)) x8))
% Found (eq_ref0 ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (((functional_extensionality00 (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x3 Xx)) x8)):(((eq Prop) ((x7 (x3 Xx)) x8)) ((x7 (x3 Xx)) x8))
% Found (eq_ref0 ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x3 Xx)) x8)) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))) as proof of (((eq Prop) ((x7 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x3 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x5)):(((eq Prop) ((x4 (x0 Xx)) x5)) ((x4 (x0 Xx)) x5))
% Found (eq_ref0 ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x5)) as proof of (((eq Prop) ((x4 (x0 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (((eq_sym1 ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x4 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x4 (x0 Xx)) x)))
% Found (functional_extensionality_dep0000 (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found ((functional_extensionality_dep000 (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((functional_extensionality_dep00 Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found ((((functional_extensionality_dep0 (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x4 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((eq_sym00 (x4 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (((eq_sym0 Xx) (x4 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x4 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x4 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x4 (x0 Xx))) (((((functional_extensionality_dep a) (fun (x7:a)=> Prop)) Xx) (x4 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x4 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x4 (x0 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x4 Xy))) P)) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found x30:=(x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy)))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy)))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x2:(P (x0 (x1 Xy)))
% Instantiate: b0:=(x0 (x1 Xy)):(b->Prop)
% Found x2 as proof of (P0 b0)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x8)):(((eq Prop) ((x4 (x3 Xx)) x8)) ((x4 (x3 Xx)) x8))
% Found (eq_ref0 ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((functional_extensionality00 (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x3 Xx)) x8)):(((eq Prop) ((x4 (x3 Xx)) x8)) ((x4 (x3 Xx)) x8))
% Found (eq_ref0 ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x3 Xx)) x8)) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))) as proof of (((eq Prop) ((x4 (x3 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x4 (x3 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x4 (x3 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x3 Xx)) x8))))) as proof of (((eq (a->Prop)) (x4 (x3 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x0 Xx)) x8)):(((eq Prop) ((x7 (x0 Xx)) x8)) ((x7 (x0 Xx)) x8))
% Found (eq_ref0 ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x7 (x0 Xx)) x8)):(((eq Prop) ((x7 (x0 Xx)) x8)) ((x7 (x0 Xx)) x8))
% Found (eq_ref0 ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x7 (x0 Xx)) x8)) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))) as proof of (((eq Prop) ((x7 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x7 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x7 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x7 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x7 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x5)):(((eq Prop) ((x1 (x0 Xx)) x5)) ((x1 (x0 Xx)) x5))
% Found (eq_ref0 ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x5)) as proof of (((eq Prop) ((x1 (x0 Xx)) x5)) (Xx x5))
% Found (eq_sym100 ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found ((eq_sym10 (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (((eq_sym1 ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (((eq Prop) (Xx x5)) ((x1 (x0 Xx)) x5))
% Found (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))) as proof of (forall (x:a), (((eq Prop) (Xx x)) ((x1 (x0 Xx)) x)))
% Found (functional_extensionality0000 (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((functional_extensionality000 (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((functional_extensionality00 Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found ((((functional_extensionality0 Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))) as proof of (((eq (a->Prop)) Xx) (x1 (x0 Xx)))
% Found (eq_sym000 (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_sym00 (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (((eq_sym0 Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5)))))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) (((((functional_extensionality a) Prop) Xx) (x1 (x0 Xx))) (fun (x5:a)=> ((((eq_sym Prop) ((x1 (x0 Xx)) x5)) (Xx x5)) ((eq_ref Prop) ((x1 (x0 Xx)) x5))))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x40:=(x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xy):(((eq (b->Prop)) Xy) (fun (x:b)=> (Xy x)))
% Found (eta_expansion_dep00 Xy) as proof of (((eq (b->Prop)) Xy) b0)
% Found ((eta_expansion_dep0 (fun (x4:b)=> Prop)) Xy) as proof of (((eq (b->Prop)) Xy) b0)
% Found (((eta_expansion_dep b) (fun (x4:b)=> Prop)) Xy) as proof of (((eq (b->Prop)) Xy) b0)
% Found (((eta_expansion_dep b) (fun (x4:b)=> Prop)) Xy) as proof of (((eq (b->Prop)) Xy) b0)
% Found (((eta_expansion_dep b) (fun (x4:b)=> Prop)) Xy) as proof of (((eq (b->Prop)) Xy) b0)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x8)):(((eq Prop) ((x4 (x0 Xx)) x8)) ((x4 (x0 Xx)) x8))
% Found (eq_ref0 ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x4 (x0 Xx)) x8)):(((eq Prop) ((x4 (x0 Xx)) x8)) ((x4 (x0 Xx)) x8))
% Found (eq_ref0 ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x4 (x0 Xx)) x8)) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))) as proof of (((eq Prop) ((x4 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x4 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x4 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x4 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x4 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x3 (x4 Xy))):(((eq (b->Prop)) (x3 (x4 Xy))) (x3 (x4 Xy)))
% Found (eq_ref0 (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x3 (x4 Xy))) as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref00:=(eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))):(((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx))))))))
% Found (eq_ref0 (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found ((eq_ref (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) as proof of (((eq (((a->Prop)->(b->Prop))->Prop)) (fun (Xg:((a->Prop)->(b->Prop)))=> ((ex ((b->Prop)->(a->Prop))) (fun (Xh:((b->Prop)->(a->Prop)))=> ((and ((and ((and (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))) (forall (Xx:(b->Prop)), (True->((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) (Xh Xx)))))))) (forall (Xy:(b->Prop)), (((eq (b->Prop)) (Xg (Xh Xy))) Xy)))) (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->(((eq (a->Prop)) (Xh (Xg Xx))) Xx)))))))) b00)
% Found x4:(((eq (b->Prop)) (f x3)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x7:(a->Prop))=> (f x3)):((a->Prop)->(b->Prop))
% Found x4 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x2:(P (x0 (x1 Xy)))
% Instantiate: f0:=(x0 (x1 Xy)):(b->Prop)
% Found x2 as proof of (P0 f0)
% Found x2:(P (x0 (x1 Xy)))
% Instantiate: f0:=(x0 (x1 Xy)):(b->Prop)
% Found x2 as proof of (P0 f0)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x8)):(((eq Prop) ((x1 (x0 Xx)) x8)) ((x1 (x0 Xx)) x8))
% Found (eq_ref0 ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality0000 (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality00 (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality0 Prop) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality a) Prop) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 ((x1 (x0 Xx)) x8)):(((eq Prop) ((x1 (x0 Xx)) x8)) ((x1 (x0 Xx)) x8))
% Found (eq_ref0 ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found ((eq_ref Prop) ((x1 (x0 Xx)) x8)) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))) as proof of (((eq Prop) ((x1 (x0 Xx)) x8)) (Xx x8))
% Found (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))) as proof of (forall (x:a), (((eq Prop) ((x1 (x0 Xx)) x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((functional_extensionality_dep00 (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((((functional_extensionality_dep0 (fun (x10:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> (((((functional_extensionality_dep a) (fun (x10:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x8:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x8))))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x4 Xy))):(((eq (b->Prop)) (x0 (x4 Xy))) (x0 (x4 Xy)))
% Found (eq_ref0 (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x4 Xy))) as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy)))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x3 (x4 Xy))) P) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x3 (x4 Xy))) P)) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found x20:=(x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))):((P (x3 (x4 Xy)))->(P (x3 (x4 Xy))))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x2 (fun (x8:(b->Prop))=> (P (x3 (x4 Xy)))))) as proof of ((P (x3 (x4 Xy)))->(P Xy))
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) Xy)
% Found eq_ref00:=(eq_ref0 (x0 (x1 Xy))):(((eq (b->Prop)) (x0 (x1 Xy))) (x0 (x1 Xy)))
% Found (eq_ref0 (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_ref (b->Prop)) (x0 (x1 Xy))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) b0)
% Found ((eq_trans00000 ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_trans00000 ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) b0)) (x3:(((eq (b->Prop)) b0) Xy))=> (((eq_trans0000 x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) b0)) (x3:(((eq (b->Prop)) b0) Xy))=> ((((eq_trans000 Xy) x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) b0)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) Xy)) (x3:(((eq (b->Prop)) Xy) Xy))=> (((((eq_trans00 Xy) Xy) x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) Xy)) (x3:(((eq (b->Prop)) Xy) Xy))=> ((((((eq_trans0 (x0 (x1 Xy))) Xy) Xy) x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) Xy)) (x3:(((eq (b->Prop)) Xy) Xy))=> (((((((eq_trans (b->Prop)) (x0 (x1 Xy))) Xy) Xy) x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((fun (x2:(((eq (b->Prop)) (x0 (x1 Xy))) Xy)) (x3:(((eq (b->Prop)) Xy) Xy))=> (((((((eq_trans (b->Prop)) (x0 (x1 Xy))) Xy) Xy) x2) x3) P)) ((eq_ref (b->Prop)) (x0 (x1 Xy)))) ((eq_ref (b->Prop)) Xy))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found x3:(((eq (b->Prop)) (f x2)) Y)
% Instantiate: Y:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x2)):((a->Prop)->(b->Prop))
% Found x3 as proof of (((eq (b->Prop)) (x0 (x4 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x4 Xy))) P) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x4 Xy))) P)) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found x30:=(x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x3 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy)))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))):((P (x0 (x4 Xy)))->(P (x0 (x4 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x0 (x4 Xy)))))) as proof of ((P (x0 (x4 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (Xy x)))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (Xy x)))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality_dep000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (forall (P:((a->Prop)->Prop)), ((P (x1 (x0 Xx)))->(P Xx)))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found x7:(((eq (b->Prop)) (f x6)) Y0)
% Instantiate: Y0:=Xy:(b->Prop);x0:=(fun (x9:(a->Prop))=> (f x6)):((a->Prop)->(b->Prop))
% Found x7 as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2:(((eq (b->Prop)) (f x1)) Y)
% Instantiate: Y:=Xy:(b->Prop);x3:=(fun (x9:(a->Prop))=> (f x1)):((a->Prop)->(b->Prop))
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found x2 as proof of (((eq (b->Prop)) (x3 (x4 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (Xy x)))
% Found ((functional_extensionality_dep00000 (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found ((functional_extensionality_dep00000 (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) (f0 x)) (Xy x))))=> ((functional_extensionality_dep0000 x3) P)) (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) (f0 x)) (Xy x))))=> (((functional_extensionality_dep000 Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality_dep0 (fun (x6:b)=> Prop)) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x6:b)=> Prop)) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (fun (x2:(P (x0 (x1 Xy))))=> (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x6:b)=> Prop)) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2)) as proof of (P Xy)
% Found (fun (P:((b->Prop)->Prop)) (x2:(P (x0 (x1 Xy))))=> (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality_dep b) (fun (x6:b)=> Prop)) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 (f0 x3)):(((eq Prop) (f0 x3)) (f0 x3))
% Found (eq_ref0 (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found ((eq_ref Prop) (f0 x3)) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (((eq Prop) (f0 x3)) (Xy x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f0 x3))) as proof of (forall (x:b), (((eq Prop) (f0 x)) (Xy x)))
% Found ((functional_extensionality00000 (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found ((functional_extensionality00000 (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) (f0 x)) (Xy x))))=> ((functional_extensionality0000 x3) P)) (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) (f0 x)) (Xy x))))=> (((functional_extensionality000 Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) (f0 x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((functional_extensionality00 (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> (((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2) as proof of (P Xy)
% Found (fun (x2:(P (x0 (x1 Xy))))=> (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2)) as proof of (P Xy)
% Found (fun (P:((b->Prop)->Prop)) (x2:(P (x0 (x1 Xy))))=> (((fun (x3:(forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x))))=> ((((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) x3) P)) (fun (x3:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x3)))) x2)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref000:=(eq_ref00 P):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (eq_ref00 P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found ((eq_ref0 (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (((eq_ref (b->Prop)) (x0 (x1 Xy))) P) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (((eq_ref (b->Prop)) (x0 (x1 Xy))) P)) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x70:=(x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x7 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found x40:=(x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))):((P (x0 (x1 Xy)))->(P (x0 (x1 Xy))))
% Found (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found (fun (P:((b->Prop)->Prop))=> (x4 (fun (x8:(b->Prop))=> (P (x0 (x1 Xy)))))) as proof of ((P (x0 (x1 Xy)))->(P Xy))
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep1000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality_dep100 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality_dep10 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality_dep1 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality0 Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality1000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality100 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality10 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality1 Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality b) Prop) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 ((x0 (x1 Xy)) x2)):(((eq Prop) ((x0 (x1 Xy)) x2)) ((x0 (x1 Xy)) x2))
% Found (eq_ref0 ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found ((eq_ref Prop) ((x0 (x1 Xy)) x2)) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (((eq Prop) ((x0 (x1 Xy)) x2)) (Xy x2))
% Found (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2))) as proof of (forall (x:b), (((eq Prop) ((x0 (x1 Xy)) x)) (Xy x)))
% Found (functional_extensionality_dep0000 (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((functional_extensionality_dep000 Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((functional_extensionality_dep00 (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found ((((functional_extensionality_dep0 (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found (((((functional_extensionality_dep b) (fun (x4:b)=> Prop)) (x0 (x1 Xy))) Xy) (fun (x2:b)=> ((eq_ref Prop) ((x0 (x1 Xy)) x2)))) as proof of (((eq (b->Prop)) (x0 (x1 Xy))) Xy)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (x1 (x0 Xx))):(((eq (a->Prop)) (x1 (x0 Xx))) (x1 (x0 Xx)))
% Found (eq_ref0 (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found ((eq_ref (a->Prop)) (x1 (x0 Xx))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> ((eq_ref (a->Prop)) (x1 (x0 Xx)))) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (x10:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found eq_ref00:=(eq_ref0 (b0 x3)):(((eq Prop) (b0 x3)) (b0 x3))
% Found (eq_ref0 (b0 x3)) as proof of (((eq Prop) (b0 x3)) (Xx x3))
% Found ((eq_ref Prop) (b0 x3)) as proof of (((eq Prop) (b0 x3)) (Xx x3))
% Found ((eq_ref Prop) (b0 x3)) as proof of (((eq Prop) (b0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (b0 x3))) as proof of (((eq Prop) (b0 x3)) (Xx x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (b0 x3))) as proof of (forall (x:a), (((eq Prop) (b0 x)) (Xx x)))
% Found (functional_extensionality_dep0000 (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((functional_extensionality_dep000 Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found (((functional_extensionality_dep00 b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found ((((functional_extensionality_dep0 (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))) as proof of (((eq (a->Prop)) b0) Xx)
% Found (eq_sym010 (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3))))) as proof of (((eq (a->Prop)) Xx) b0)
% Found ((eq_sym01 Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3))))) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eq_sym0 b0) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3))))) as proof of (((eq (a->Prop)) Xx) b0)
% Found (((eq_sym0 b0) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3))))) as proof of (((eq (a->Prop)) Xx) b0)
% Found ((eq_sym0000 (((eq_sym0 b0) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))))) x2) as proof of (P Xx)
% Found ((eq_sym0000 (((eq_sym0 b0) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))))) x2) as proof of (P Xx)
% Found (((fun (x3:(((eq (a->Prop)) Xx) b0))=> ((eq_sym000 x3) P)) (((eq_sym0 b0) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) b0) Xx) (fun (x3:a)=> ((eq_ref Prop) (b0 x3)))))) x2) as proof of (P Xx)
% Found (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((eq_sym00 (x1 (x0 Xx))) x3) P)) (((eq_sym0 (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2) as proof of (P Xx)
% Found (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> ((((eq_sym0 Xx) (x1 (x0 Xx))) x3) P)) (((eq_sym0 (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2) as proof of (P Xx)
% Found (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x3) P)) ((((eq_sym (a->Prop)) (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2) as proof of (P Xx)
% Found (fun (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x3) P)) ((((eq_sym (a->Prop)) (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2)) as proof of (P Xx)
% Found (fun (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x3) P)) ((((eq_sym (a->Prop)) (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2)) as proof of ((P (x1 (x0 Xx)))->(P Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))) (P:((a->Prop)->Prop)) (x2:(P (x1 (x0 Xx))))=> (((fun (x3:(((eq (a->Prop)) Xx) (x1 (x0 Xx))))=> (((((eq_sym (a->Prop)) Xx) (x1 (x0 Xx))) x3) P)) ((((eq_sym (a->Prop)) (x1 (x0 Xx))) Xx) (((((functional_extensionality_dep a) (fun (x5:a)=> Prop)) (x1 (x0 Xx))) Xx) (fun (x3:a)=> ((eq_ref Prop) ((x1 (x0 Xx)) x3)))))) x2)) as proof of (((eq (a->Prop)) (x1 (x0 Xx))) Xx)
% Found I:True
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of True
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True)
% Found (fun (Xx:(a->Prop)) (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx))))=> I) as proof of (forall (Xx:(a->Prop)), (((ex a) (fun (Xx0:a)=> (((eq (a->Prop)) (fun (Xy:a)=> (((eq (b->Prop)) (f Xx0)) (f Xy)))) Xx)))->True))
% Found eq_ref000:=(eq_ref00 P0):((P0 (x1 (x0 Xx)))->(P0 (x1 (x0 Xx))))
% Found (eq_ref00 P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found ((eq_ref0 (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (P0:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) (x1 (x0 Xx))) P0)) as proof of ((P0 (x1 (x0 Xx)))->(P0 Xx))
% Found (fun (x00:((ex a) (fun (Xx0:a)=> (((eq (a->Prop)
% EOF
%------------------------------------------------------------------------------