%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV248^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n099.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 13:33:56 EDT 2014

% Result   : Timeout 300.10s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV248^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n099.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:37:11 CDT 2014
% % CPUTime  : 300.10 
% 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 0x195a6c8>, <kernel.Type object at 0x195a3f8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:a) (Y:a) (U:a) (V:a), ((iff (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))) ((and (((eq a) X) U)) (((eq a) Y) V)))) of role conjecture named cTHM103_pme
% Conjecture to prove = (forall (X:a) (Y:a) (U:a) (V:a), ((iff (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))) ((and (((eq a) X) U)) (((eq a) Y) V)))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:a) (Y:a) (U:a) (V:a), ((iff (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))) ((and (((eq a) X) U)) (((eq a) Y) V))))']
% Parameter a:Type.
% Trying to prove (forall (X:a) (Y:a) (U:a) (V:a), ((iff (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))) ((and (((eq a) X) U)) (((eq a) Y) V))))
% Found iff_refl0:=(iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))):((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (x100 (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V)))))))
% Found (x00 (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))) as proof of ((((eq a) X) U)->((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))))
% Found (and_rect00 (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found ((and_rect0 ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (fun (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (fun (x:((and (((eq a) X) U)) (((eq a) Y) V))) (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x:((and (((eq a) X) U)) (((eq a) Y) V))) (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of (((and (((eq a) X) U)) (((eq a) Y) V))->(forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found iff_refl0:=(iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))):((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))
% Found (x100 (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V))))))
% Found (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) V)))))))
% Found (x00 (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))) as proof of ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))) as proof of ((((eq a) X) U)->((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))))
% Found (and_rect00 (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found ((and_rect0 ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (fun (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))
% Found (fun (x:((and (((eq a) X) U)) (((eq a) Y) V))) (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of (forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))))
% Found (fun (x:((and (((eq a) X) U)) (((eq a) Y) V))) (Xx:(a->Prop))=> (((fun (P:Type) (x0:((((eq a) X) U)->((((eq a) Y) V)->P)))=> (((((and_rect (((eq a) X) U)) (((eq a) Y) V)) P) x0) x)) ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))) (fun (x0:(((eq a) X) U))=> ((x0 (fun (x2:a)=> ((((eq a) Y) V)->((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) x2) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) x2)) (((eq a) W) V))))))))) (fun (x10:(((eq a) Y) V))=> ((x10 (fun (x2:a)=> ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) x2)))))))) (iff_refl ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))))))))) as proof of (((and (((eq a) X) U)) (((eq a) Y) V))->(forall (Xx:(a->Prop)), ((iff ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))) ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))))))
% Found eq_ref00:=(eq_ref0 (((eq a) Y) V)):(((eq Prop) (((eq a) Y) V)) (((eq a) Y) V))
% Found (eq_ref0 (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 (((eq a) Y) V)):(((eq Prop) (((eq a) Y) V)) (((eq a) Y) V))
% Found (eq_ref0 (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% Found ((eq_ref Prop) (((eq a) Y) V)) as proof of (((eq Prop) (((eq a) Y) V)) b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found x0:(P Y)
% Instantiate: b:=Y:a
% Found x0 as proof of (P0 b)
% Found x0:(P X)
% Instantiate: b:=X:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found x0:(P X)
% Instantiate: b:=X:a
% Found x0 as proof of (P0 b)
% Found x0:(P Y)
% Instantiate: b:=Y:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x0:(P V)
% Instantiate: b:=V:a
% Found x0 as proof of (P0 b)
% Found x0:(P U)
% Instantiate: b:=U:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found ((eq_ref a) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found x0:(P V)
% Instantiate: b:=V:a
% Found x0 as proof of (P0 b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x0:(P U)
% Instantiate: b:=U:a
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found ((eq_ref a) X) as proof of (((eq a) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) U)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found eq_ref000:=(eq_ref00 P):((P U)->(P U))
% Found (eq_ref00 P) as proof of (P0 U)
% Found ((eq_ref0 U) P) as proof of (P0 U)
% Found (((eq_ref a) U) P) as proof of (P0 U)
% Found (((eq_ref a) U) P) as proof of (P0 U)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found eq_ref000:=(eq_ref00 P):((P V)->(P V))
% Found (eq_ref00 P) as proof of (P0 V)
% Found ((eq_ref0 V) P) as proof of (P0 V)
% Found (((eq_ref a) V) P) as proof of (P0 V)
% Found (((eq_ref a) V) P) as proof of (P0 V)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x3:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x21):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x21) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x21) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x11):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x11) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x11) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 X):(((eq a) X) X)
% Found (eq_ref0 X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found ((eq_ref a) X) as proof of (((eq a) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found ((eq_ref a) b) as proof of (((eq a) b) U)
% Found eq_ref00:=(eq_ref0 Y):(((eq a) Y) Y)
% Found (eq_ref0 Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found ((eq_ref a) Y) as proof of (((eq a) Y) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found ((eq_ref a) b) as proof of (((eq a) b) V)
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x11):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x11) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x11) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx0):(((eq (a->Prop)) Xx0) Xx0)
% Found (eq_ref0 Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x6:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% Found (((eq_ref a) b) P) as proof of (P0 b)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq a) Y) V))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Y) V))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Y) V))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq a) Y) V))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20 as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b0)
% Found ((eq_ref a) U) as proof of (((eq a) U) b0)
% Found ((eq_ref a) U) as proof of (((eq a) U) b0)
% Found ((eq_ref a) U) as proof of (((eq a) U) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b0)
% Found ((eq_ref a) V) as proof of (((eq a) V) b0)
% Found ((eq_ref a) V) as proof of (((eq a) V) b0)
% Found ((eq_ref a) V) as proof of (((eq a) V) b0)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xx0):(((eq (a->Prop)) Xx0) (fun (x:a)=> (Xx0 x)))
% Found (eta_expansion_dep00 Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x7:a)=> Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x7:a)=> Prop)) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eta_expansion000:=(eta_expansion00 Xx0):(((eq (a->Prop)) Xx0) (fun (x:a)=> (Xx0 x)))
% Found (eta_expansion00 Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx0) as proof of (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx0)) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx0 W)):((iff (Xx0 W)) (Xx0 W))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P X)->(P X))
% Found (eq_ref00 P) as proof of (P0 X)
% Found ((eq_ref0 X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% Found (((eq_ref a) X) P) as proof of (P0 X)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref000:=(eq_ref00 P):((P Y)->(P Y))
% Found (eq_ref00 P) as proof of (P0 Y)
% Found ((eq_ref0 Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% Found (((eq_ref a) Y) P) as proof of (P0 Y)
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x3 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x3 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x3 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x3 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion_dep a) (fun (x4:a)=> Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq (a->Prop)) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eq_ref (a->Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_introl00 ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_introl0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx)) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 (((or_introl (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) ((eq_ref (a->Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found ((eq_ref a) b) as proof of (((eq a) b) Y)
% Found eq_ref00:=(eq_ref0 V):(((eq a) V) V)
% Found (eq_ref0 V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found ((eq_ref a) V) as proof of (((eq a) V) b)
% Found eq_ref00:=(eq_ref0 U):(((eq a) U) U)
% Found (eq_ref0 U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found ((eq_ref a) U) as proof of (((eq a) U) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% Found ((eq_ref a) b) as proof of (((eq a) b) X)
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))
% Found (or_intror00 (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (((eta_expansion_dep a) (fun (x5:a)=> Prop)) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% 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) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found ((eta_expansion0 Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (((eta_expansion a) Prop) Xx) as proof of (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))
% Found (or_intror00 (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found ((or_intror0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx)) as proof of ((or (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy))))
% Found (or_comm_i00 (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_comm_i0 (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_comm_i (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((or_intror (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (((eta_expansion a) Prop) Xx))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx0 W)):((iff (Xx0 W)) (Xx0 W))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx0 W)):((iff (Xx0 W)) (Xx0 W))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx0 W)) as proof of ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx0 W))) as proof of (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx0 W)))) as proof of ((or (((eq (a->Prop)) Xx0) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx0 W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found iff_refl0:=(iff_refl (Xx W)):((iff (Xx W)) (Xx W))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (iff_refl (Xx W)) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))
% Found (fun (W:a)=> (iff_refl (Xx W))) as proof of (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))
% Found (or_intror00 (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found ((or_intror0 (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (((or_intror (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y))))) (fun (W:a)=> (iff_refl (Xx W)))) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x20:=(x2 x10):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found (x2 x10) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) X) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) X)) (((eq a) W) Y)))))
% Found x10:=(x1 x20):((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> (((eq a) U) Xy)))) (forall (W:a), ((iff (Xx W)) ((or (((eq a) W) U)) (((eq a) W) V)))))
% Found (x1 x20) as proof of ((or (((eq (a->Prop)) Xx) (fun (Xy:a)=> ((
% EOF
%------------------------------------------------------------------------------