%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU990^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n113.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:28 EDT 2014

% Result   : Timeout 300.07s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU990^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n113.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:48:11 CDT 2014
% % CPUTime  : 300.07 
% 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 0x244c8c0>, <kernel.Type object at 0x244cdd0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x231ae60>, <kernel.Type object at 0x244ce18>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) of role conjecture named cTHM610_pme
% Conjecture to prove = (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Trying to prove (((ex ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X))))))->((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))):(((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (eq_ref0 ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((b->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((b->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))):(((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (eq_ref0 ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))):(((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (x:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((x Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eta_expansion00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) (fun (x:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((b->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((b->(a->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found ((eq_ref ((b->(a->Prop))->Prop)) a0) as proof of (((eq ((b->(a->Prop))->Prop)) a0) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))):(((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eq_ref0 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (x:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((x Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eta_expansion00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (b->(a->Prop))) Prop) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x1:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))):(((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (eq_ref0 ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found ((eq_ref Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) as proof of (((eq Prop) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eq_ref0 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq ((a->(b->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found ((eq_ref ((a->(b->Prop))->Prop)) a0) as proof of (((eq ((a->(b->Prop))->Prop)) a0) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (x:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((x Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((or ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (x:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((x Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (x:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((x Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((x Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eta_expansion00 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))):(((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))
% Found (eq_ref0 (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found ((eq_ref ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))):(((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) (fun (x:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((x Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((x Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((x Xx) Xy1)) ((x Xx) Xy2))->(((eq a) Xy1) Xy2))))))
% Found (eta_expansion_dep00 (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found (((eta_expansion_dep (b->(a->Prop))) (fun (x3:(b->(a->Prop)))=> Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) as proof of (((eq ((b->(a->Prop))->Prop)) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))) b0)
% Found eq_ref00:=(eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (eq_ref0 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) (fun (x:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) (fun (x:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (eta_expansion_dep00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) (fun (x:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) (fun (x:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))):(((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) (fun (x:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found (eta_expansion00 (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found (((eta_expansion a) Prop) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) as proof of (((eq (a->Prop)) (fun (Xt:a)=> ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq a) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x1:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x1) x00)) (((eq b) Xy1) Xy2)) (fun (x1:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x1)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (a->(b->Prop))) (fun (Xr:(a->(b->Prop)))=> ((and ((and (forall (Xx:a), ((ex b) (fun (Xy:b)=> ((Xr Xx) Xy))))) (forall (Xy:b), ((ex a) (fun (Xx:a)=> ((Xr Xx) Xy)))))) (forall (Xx:a) (Xy1:b) (Xy2:b), (((and ((Xr Xx) Xy1)) ((Xr Xx) Xy2))->(((eq b) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2))))))))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((x1 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq b) Xy1) Xy2))
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq b) Xy1) Xy2)))
% Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((ex_ind0 (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq a) Xy1) Xy2))
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xy1) Xy2)))
% Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((ex_ind0 (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x2 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq b) Xy1) Xy2)->(((eq b) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq b) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq b) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found ((and_rect0 (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq b) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:b)=> (((((eq_trans b) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_trans00000:=(eq_trans0000 Xy2):((((eq a) Xy1) Xy2)->(((eq a) Xy1) Xy2))
% Found (eq_trans0000 Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((eq_trans000 c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((eq_trans00 Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> ((((eq_trans0 Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy2)->(((eq a) Xy1) Xy2))
% Found (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->(((eq a) Xy1) Xy2)))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found ((and_rect0 (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) (((eq a) Xy1) Xy2)) (fun (x3:((x0 Xx) Xy1))=> ((fun (c:a)=> (((((eq_trans a) Xy1) Xy1) c) x3)) Xy2)))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found x10:(P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P0 Xy1)
% Found x10:(P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P0 Xy1)
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (eq_sym000 ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((eq_sym b) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found (eq_sym000 ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((((eq_sym a) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> ((((eq_sym a) Xy2) Xy1) ((x2 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq a) Xy1) Xy2))
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq a) Xy1) Xy2)))
% Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((ex_ind0 (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((eq a) Xy1) Xy2)
% Found (fun (Xy2:a) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq a) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq a) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq a) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))):(((x0 Xx) Xy2)->((x0 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2))))
% Found (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))) as proof of (((x0 Xx) Xy1)->(((x0 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))))
% Found (and_rect00 (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))
% Found (x20 (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X))))->(((eq b) Xy1) Xy2))
% Found (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))) as proof of (forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->(((eq b) Xy1) Xy2)))
% Found (ex_ind00 (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((ex_ind0 (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2)))=> (((fun (P:Prop) (x1:(forall (x:((a->Prop)->a)), ((forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x X))))->P)))=> (((((ex_ind ((a->Prop)->a)) (fun (Xc:((a->Prop)->a))=> (forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (Xc X)))))) P) x1) x)) (((eq b) Xy1) Xy2)) (fun (x1:((a->Prop)->a)) (x2:(forall (X:(a->Prop)), (((ex a) (fun (Xt:a)=> (X Xt)))->(X (x1 X)))))=> ((x2 (fun (x3:a)=> (((eq b) Xy1) Xy2))) (((fun (P:Type) (x3:(((x0 Xx) Xy1)->(((x0 Xx) Xy2)->P)))=> (((((and_rect ((x0 Xx) Xy1)) ((x0 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy1) Xy2)))) (fun (x3:((x0 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy1) Xy2))) (fun (x4:(a->Prop))=> ((x0 Xx) Xy2))))))))) as proof of (((and ((x0 Xx) Xy1)) ((x0 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found x10:(P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P0 Xy1)
% Found x10:(P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P Xy1)
% Found (fun (x10:(P Xy1))=> x10) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq a) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found ((eq_ref a) Xy1) as proof of (((eq a) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xy2)
% Found eq_ref00:=(eq_ref0 Xy1):(((eq b) Xy1) Xy1)
% Found (eq_ref0 Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found ((eq_ref b) Xy1) as proof of (((eq b) Xy1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq b) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found ((eq_ref b) b0) as proof of (((eq b) b0) Xy2)
% Found x30:(P Xy1)
% Found (fun (x30:(P Xy1))=> x30) as proof of (P Xy1)
% Found (fun (x30:(P Xy1))=> x30) as proof of (P0 Xy1)
% Found x30:(P Xy1)
% Found (fun (x30:(P Xy1))=> x30) as proof of (P Xy1)
% Found (fun (x30:(P Xy1))=> x30) as proof of (P0 Xy1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex (b->(a->Prop))) (fun (Xs:(b->(a->Prop)))=> ((and ((and (forall (Xx:b), ((ex a) (fun (Xy:a)=> ((Xs Xx) Xy))))) (forall (Xy:a), ((ex b) (fun (Xx:b)=> ((Xs Xx) Xy)))))) (forall (Xx:b) (Xy1:a) (Xy2:a), (((and ((Xs Xx) Xy1)) ((Xs Xx) Xy2))->(((eq a) Xy1) Xy2)))))))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq b) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((eq b) Xy1) Xy2)
% Found (fun (Xy2:b) (x00:((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2)))=> ((((eq_sym b) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq b) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq b) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq b) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))))) as proof of (((and ((x2 Xx) Xy1)) ((x2 Xx) Xy2))->(((eq b) Xy1) Xy2))
% Found eq_ref000:=(eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))):(((x2 Xx) Xy2)->((x2 Xx) Xy2))
% Found (eq_ref00 (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found ((eq_ref0 (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1))))
% Found (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))) as proof of (((x2 Xx) Xy1)->(((x2 Xx) Xy2)->((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))))
% Found (and_rect00 (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found ((and_rect0 ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))) as proof of ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))
% Found (x10 (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2)))))) as proof of (((eq a) Xy2) Xy1)
% Found (eq_sym000 ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((eq_sym00 Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found (((eq_sym0 Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> (((eq_ref (a->Prop)) (fun (Xt:a)=> (((eq a) Xy2) Xy1))) (fun (x4:(a->Prop))=> ((x2 Xx) Xy2))))))) as proof of (((eq a) Xy1) Xy2)
% Found ((((eq_sym a) Xy2) Xy1) ((x1 (fun (x3:a)=> (((eq a) Xy2) Xy1))) (((fun (P:Type) (x3:(((x2 Xx) Xy1)->(((x2 Xx) Xy2)->P)))=> (((((and_rect ((x2 Xx) Xy1)) ((x2 Xx) Xy2)) P) x3) x00)) ((ex a) (fun (Xt:a)=> (((eq a) Xy2) Xy1)))) (fun (x3:((x2 Xx) Xy1))=> 
% EOF
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