TSTP Solution File: SET598^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET598^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n097.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 13:30:48 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  : SET598^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n097.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 10:14:11 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 0x108d710>, <kernel.Type object at 0x108d830>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), ((iff (((eq (a->Prop)) X) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))))) of role conjecture named cBOOL_PROP_57_pme
% Conjecture to prove = (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), ((iff (((eq (a->Prop)) X) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), ((iff (((eq (a->Prop)) X) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx))))))))']
% Parameter a:Type.
% Trying to prove (forall (X:(a->Prop)) (Y:(a->Prop)) (Z:(a->Prop)), ((iff (((eq (a->Prop)) X) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))) ((and ((and (forall (Xx:a), ((X Xx)->(Y Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx))))))))
% 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_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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 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_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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 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 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))):(((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx))))))
% Found (eq_ref0 (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% 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 (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x5: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 (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z 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)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))):(((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx))))))
% Found (eq_ref0 (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found ((eq_ref Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) as proof of (((eq Prop) (forall (V:(a->Prop)), (((and (forall (Xx:a), ((V Xx)->(Y Xx)))) (forall (Xx:a), ((V Xx)->(Z Xx))))->(forall (Xx:a), ((V Xx)->(X Xx)))))) b)
% Found x0:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x0 as proof of (P0 b)
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 ((and (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x0:(P0 b)
% Instantiate: b:=X:(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 X)
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 X))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x0 as proof of (P0 f)
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% 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 (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) 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 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z 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)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (Y x1)) (Z x1)))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x0:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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 (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found x30:(P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P0 (X x2))
% Found x30:(P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P0 (X x2))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (a->Prop)) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x10:(P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P0 ((and (Y x0)) (Z x0)))
% Found x10:(P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P0 ((and (Y x0)) (Z x0)))
% Found x2:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x30:(P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P0 (X x0))
% Found x30:(P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P0 (X x0))
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x0:(P0 b)
% Instantiate: b:=X:(a->Prop)
% Found (fun (x0:(P0 b))=> x0) as proof of (P0 X)
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of ((P0 b)->(P0 X))
% Found (fun (P0:((a->Prop)->Prop)) (x0:(P0 b))=> x0) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found eq_ref00:=(eq_ref0 (X x2)):(((eq Prop) (X x2)) (X x2))
% Found (eq_ref0 (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found ((eq_ref Prop) (X x2)) as proof of (((eq Prop) (X x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x2)) (Z x2)))
% 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_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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x2:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 ((and (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 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 (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z 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 x0:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(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 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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found x40:(P X)
% Found (fun (x40:(P X))=> x40) as proof of (P X)
% Found (fun (x40:(P X))=> x40) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (X x0))
% Instantiate: b:=(X x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (Y x0)) (Z x0))):(((eq Prop) ((and (Y x0)) (Z x0))) ((and (Y x0)) (Z x0)))
% Found (eq_ref0 ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found ((eq_ref Prop) ((and (Y x0)) (Z x0))) as proof of (((eq Prop) ((and (Y x0)) (Z x0))) b)
% Found x0:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) 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 x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x20:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x20) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (eq_ref0 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b0) as proof of (((eq (a->Prop)) b0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x30:(P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P0 (X x2))
% Found x30:(P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P (X x2))
% Found (fun (x30:(P (X x2)))=> x30) as proof of (P0 (X x2))
% Found x20:(P X)
% Found (fun (x20:(P X))=> x20) as proof of (P X)
% Found (fun (x20:(P X))=> x20) as proof of (P0 X)
% 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 (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x2:(P X)
% Instantiate: b:=X:(a->Prop)
% Found x2 as proof of (P0 b)
% Found x10:(P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P0 ((and (Y x0)) (Z x0)))
% Found x10:(P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P ((and (Y x0)) (Z x0)))
% Found (fun (x10:(P ((and (Y x0)) (Z x0))))=> x10) as proof of (P0 ((and (Y x0)) (Z x0)))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P0 (fun (Xx:a)=> ((and (Y Xx)) (Z 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 (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% 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_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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x2:(P0 b)
% Instantiate: b:=X:(a->Prop)
% Found (fun (x2:(P0 b))=> x2) as proof of (P0 X)
% Found (fun (P0:((a->Prop)->Prop)) (x2:(P0 b))=> x2) as proof of ((P0 b)->(P0 X))
% Found (fun (P0:((a->Prop)->Prop)) (x2:(P0 b))=> x2) as proof of (P b)
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((X Xx)->(Z Xx)))):(((eq Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) (forall (Xx:a), ((X Xx)->(Z Xx))))
% Found (eq_ref0 (forall (Xx:a), ((X Xx)->(Z Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((X Xx)->(Z Xx)))) b)
% Found x30:(P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P0 (X x0))
% Found x30:(P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P (X x0))
% Found (fun (x30:(P (X x0)))=> x30) as proof of (P0 (X x0))
% Found eq_ref00:=(eq_ref0 (X x4)):(((eq Prop) (X x4)) (X x4))
% Found (eq_ref0 (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found eq_ref00:=(eq_ref0 (X x4)):(((eq Prop) (X x4)) (X x4))
% Found (eq_ref0 (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found ((eq_ref Prop) (X x4)) as proof of (((eq Prop) (X x4)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x4)) (Z x4)))
% Found x2:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P X)
% Instantiate: f:=X:(a->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: f:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x0 as proof of (P0 f)
% 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 eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b0)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion a) Prop) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x2:(P0 b)
% Instantiate: b:=X:(a->Prop)
% Found (fun (x2:(P0 b))=> x2) as proof of (P0 X)
% Found (fun (P0:((a->Prop)->Prop)) (x2:(P0 b))=> x2) as proof of ((P0 b)->(P0 X))
% Found (fun (P0:((a->Prop)->Prop)) (x2:(P0 b))=> x2) as proof of (P b)
% Found x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P2 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P2 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P2 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x00:(P1 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x00) as proof of (P2 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% 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 (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) X) as proof of (((eq (a->Prop)) X) b)
% Found (((eta_expansion_dep a) (fun (x3: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 (Y Xx)) (Z Xx))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found eq_ref00:=(eq_ref0 ((and (Y x2)) (Z x2))):(((eq Prop) ((and (Y x2)) (Z x2))) ((and (Y x2)) (Z x2)))
% Found (eq_ref0 ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found ((eq_ref Prop) ((and (Y x2)) (Z x2))) as proof of (((eq Prop) ((and (Y x2)) (Z x2))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x2))
% 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x2:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Instantiate: b:=(fun (Xx:a)=> ((and (Y Xx)) (Z Xx))):(a->Prop)
% Found x2 as proof of (P0 b)
% 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% 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_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))):(((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) (fun (x:a)=> ((and (Y x)) (Z x))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))) b)
% Found x00:(P X)
% Found (fun (x00:(P X))=> x00) as proof of (P X)
% Found (fun (x00:(P X))=> x00) as proof of (P0 X)
% 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found x10:(P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P (X x0))
% Found (fun (x10:(P (X x0)))=> x10) as proof of (P0 (X 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 (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (Y x0)) (Z x0)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and (Y x3)) (Z x3)))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and (Y x)) (Z x))))
% Found x40:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x40:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x40) as proof of (P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx))))
% Found (fun (x40:(P (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))=> x40) as proof of (P0 (fun (Xx:a)=> ((and (Y X
% EOF
%------------------------------------------------------------------------------