%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET611^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n090.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 13:30:50 EDT 2014

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

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET611^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.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 10:19:36 CDT 2014
% % CPUTime  : 300.09 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x22dc908>, <kernel.Type object at 0x22dc8c0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> False))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) X))) of role conjecture named cBOOL_PROP_84_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> False))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) X))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> False))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) X)))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)), ((iff (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> False))) (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) X)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x)))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion_dep00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) ((Y x)->False))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))->(P (fun (x:a)=> ((and ((and (X x)) (not (Y x)))) (Y x)))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion00 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx)))) P) as proof of (P0 (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))):(((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) ((and ((and (X x00)) ((Y x00)->False))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))):(((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) ((and ((and (X x00)) ((Y x00)->False))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))):(((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) ((and ((and (X x1)) ((Y x1)->False))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))):(((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) ((and ((and (X x0)) ((Y x0)->False))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))):(((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) ((and ((and (X x0)) ((Y x0)->False))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))):(((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) ((and ((and (X x1)) ((Y x1)->False))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) ((Y x0)->False))):(((eq Prop) ((and (X x0)) ((Y x0)->False))) ((and (X x0)) ((Y x0)->False)))
% Found (eq_ref0 ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found ((eq_ref Prop) ((and (X x0)) ((Y x0)->False))) as proof of (((eq Prop) ((and (X x0)) ((Y x0)->False))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P X))):((P X)->(P X))
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found (x (fun (x0:(a->Prop))=> (P X))) as proof of (P0 X)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> False)))) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))):(((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))):(((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))):(((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))):(((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))):(((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))):(((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (Xx:a)=> False))
% Found (eq_ref0 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (Y x0))):(((eq Prop) ((and (X x0)) (Y x0))) ((and (X x0)) (Y x0)))
% Found (eq_ref0 ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found ((eq_ref Prop) ((and (X x0)) (Y x0))) as proof of (((eq Prop) ((and (X x0)) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and (X x0)) (not (Y x0)))):(((eq Prop) ((and (X x0)) (not (Y x0)))) ((and (X x0)) (not (Y x0))))
% Found (eq_ref0 ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found ((eq_ref Prop) ((and (X x0)) (not (Y x0)))) as proof of (((eq Prop) ((and (X x0)) (not (Y x0)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x0))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x00)) (not (Y x00)))) (Y x00)))->(P ((and ((and (X x00)) (not (Y x00)))) (Y x00))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found ((eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x00)) (not (Y x00)))) (Y x00)))->(P ((and ((and (X x00)) (not (Y x00)))) (Y x00))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found ((eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) P) as proof of (P0 ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x0)) (not (Y x0)))) (Y x0)))->(P ((and ((and (X x0)) (not (Y x0)))) (Y x0))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x1)) (not (Y x1)))) (Y x1)))->(P ((and ((and (X x1)) (not (Y x1)))) (Y x1))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found ((eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x1)) (not (Y x1)))) (Y x1)))->(P ((and ((and (X x1)) (not (Y x1)))) (Y x1))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found ((eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) P) as proof of (P0 ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found eq_ref000:=(eq_ref00 P):((P ((and ((and (X x0)) (not (Y x0)))) (Y x0)))->(P ((and ((and (X x0)) (not (Y x0)))) (Y x0))))
% Found (eq_ref00 P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found ((eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) P) as proof of (P0 ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found x1:=(x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))):((P ((and (X x0)) (Y x0)))->(P ((and (X x0)) (Y x0))))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found (x (fun (x1:(a->Prop))=> (P ((and (X x0)) (Y x0))))) as proof of (P0 ((and (X x0)) (Y x0)))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (x:a)=> ((and (X x)) ((Y x)->False))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) (fun (x:a)=> ((and (X x)) (not (Y x)))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (x:a)=> ((and (X x)) (Y x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) False)
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) False))
% Found x1:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (X x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) (X x)))
% Found eq_ref00:=(eq_ref0 X):(((eq (a->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found ((eq_ref (a->Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))):(((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) ((and ((and (X x00)) ((Y x00)->False))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))):(((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) ((and ((and (X x00)) ((Y x00)->False))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) ((Y x00)->False))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))):(((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) ((and ((and (X x1)) ((Y x1)->False))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))):(((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) ((and ((and (X x0)) ((Y x0)->False))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))):(((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) ((and ((and (X x0)) ((Y x0)->False))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) ((Y x0)->False))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))):(((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) ((and ((and (X x1)) ((Y x1)->False))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) ((Y x1)->False))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) ((Y x0)->False)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (P b)
% Found ((eta_expansion0 Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P b)
% Found (((eta_expansion a) Prop) b) as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found eq_ref000:=(eq_ref00 P):((P False)->(P False))
% Found (eq_ref00 P) as proof of (P0 False)
% Found ((eq_ref0 False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found (((eq_ref Prop) False) P) as proof of (P0 False)
% Found eq_ref000:=(eq_ref00 P):((P (X x0))->(P (X x0)))
% Found (eq_ref00 P) as proof of (P0 (X x0))
% Found ((eq_ref0 (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found (((eq_ref Prop) (X x0)) P) as proof of (P0 (X x0))
% Found x1:=(x (fun (x1:(a->Prop))=> (P False))):((P False)->(P False))
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found (x (fun (x1:(a->Prop))=> (P False))) as proof of (P0 False)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x1:(P (fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and ((and (X Xx)) (not (Y Xx)))) (Y Xx))):(a->Prop)
% Found x1 as proof of (P0 b)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))))
% Instantiate: b:=(fun (Xx:a)=> ((and (X Xx)) (not (Y Xx)))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x1:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x1 as proof of (P0 f)
% Found x1:(P (fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and ((and (X Xx)) ((Y Xx)->False))) (Y Xx))):(a->Prop)
% Found x1 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (X Xx)) (Y Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion_dep00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:a)=> False))->(P (fun (Xx:a)=> False)))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eq_ref0 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eq_ref (a->Prop)) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (a->Prop)) X) (fun (x:a)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (a->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion a) Prop) X) as proof of (((eq (a->Prop)) X) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> False)):(((eq (a->Prop)) (fun (Xx:a)=> False)) (fun (x:a)=> False))
% Found (eta_expansion00 (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> False)) as proof of (((eq (a->Prop)) (fun (Xx:a)=> False)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (X Xx)) ((Y Xx)->False)))) b0)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))):(((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))):(((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) ((and ((and (X x00)) (not (Y x00)))) (Y x00)))
% Found (eq_ref0 ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found ((eq_ref Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) as proof of (((eq Prop) ((and ((and (X x00)) (not (Y x00)))) (Y x00))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:a)=> False))->(P (fun (x:a)=> False)))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((eta_expansion00 (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found (((eta_expansion0 Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found ((((eta_expansion a) Prop) (fun (Xx:a)=> False)) P) as proof of (P0 (fun (Xx:a)=> False))
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))):(((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))):(((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) ((and ((and (X x0)) (not (Y x0)))) (Y x0)))
% Found (eq_ref0 ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found ((eq_ref Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) as proof of (((eq Prop) ((and ((and (X x0)) (not (Y x0)))) (Y x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))):(((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))):(((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) ((and ((and (X x1)) (not (Y x1)))) (Y x1)))
% Found (eq_ref0 ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found ((eq_ref Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) as proof of (((eq Prop) ((and ((and (X x1)) (not (Y x1)))) (Y x1))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) False)
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found x0:=(x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))):((P (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))->(P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found (x (fun (x0:(a->Prop))=> (P (fun (Xx:a)=> ((and (X Xx)) (Y Xx)))))) as proof of (P0 (fun (Xx:a)=> ((and (X Xx)) (Y Xx))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (not (Y x0))))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (X x0)) (Y x0)))
% Found eq_ref00:=(eq_ref0 (X x0)):(((eq Prop) (X x0)) (X x0))
% Found (eq_ref0 (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found ((eq_ref Prop) (X x0)) as proof of (((eq Prop) (X x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_r
% EOF
%------------------------------------------------------------------------------