TSTP Solution File: SEV260^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV260^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n179.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:57 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV260^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n179.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:38:51 CDT 2014
% % CPUTime : 300.04
% 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 0x1c3cc68>, <kernel.Type object at 0x1c3ccb0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x206d3b0>, <kernel.Type object at 0x1c3c518>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (forall (T:((a->Prop)->Prop)) (S:((b->Prop)->Prop)) (Xf:(a->b)), (((and ((and ((and ((and ((and ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))) (forall (K:((b->Prop)->Prop)) (R:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))) (forall (X:(b->Prop)), ((S X)->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(T Y))))))->(forall (X:(b->Prop)), ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((X Xx)->False)))->(S R)))->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))->(T R))))))))) of role conjecture named cCLOSED_THM1_pme
% Conjecture to prove = (forall (T:((a->Prop)->Prop)) (S:((b->Prop)->Prop)) (Xf:(a->b)), (((and ((and ((and ((and ((and ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))) (forall (K:((b->Prop)->Prop)) (R:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))) (forall (X:(b->Prop)), ((S X)->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(T Y))))))->(forall (X:(b->Prop)), ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((X Xx)->False)))->(S R)))->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))->(T R))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(forall (T:((a->Prop)->Prop)) (S:((b->Prop)->Prop)) (Xf:(a->b)), (((and ((and ((and ((and ((and ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))) (forall (K:((b->Prop)->Prop)) (R:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))) (forall (X:(b->Prop)), ((S X)->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(T Y))))))->(forall (X:(b->Prop)), ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((X Xx)->False)))->(S R)))->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))->(T R)))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Trying to prove (forall (T:((a->Prop)->Prop)) (S:((b->Prop)->Prop)) (Xf:(a->b)), (((and ((and ((and ((and ((and ((and ((and ((and (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> False))->(T R)))) (forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> (False->False)))->(T R))))) (forall (K:((a->Prop)->Prop)) (R:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R))))) (forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))) (forall (K:((b->Prop)->Prop)) (R:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))) (forall (X:(b->Prop)), ((S X)->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(T Y))))))->(forall (X:(b->Prop)), ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((X Xx)->False)))->(S R)))->(forall (Y:(a->Prop)), ((((eq (a->Prop)) Y) (fun (Xb:a)=> (X (Xf Xb))))->(forall (R:(a->Prop)), ((((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))->(T R)))))))))
% Found x2:(((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))
% Instantiate: b0:=(fun (Xx:a)=> ((Y Xx)->False)):(a->Prop)
% Found x2 as proof of (((eq (a->Prop)) R) b0)
% 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)) (R x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (R x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (R 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)) (R x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (R x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found x2:(((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))
% Instantiate: b0:=(fun (Xx:a)=> ((Y Xx)->False)):(a->Prop)
% Found x2 as proof of (((eq (a->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 X0):(((eq (b->Prop)) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found (x00 ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found x2:(((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))
% Instantiate: b0:=(fun (Xx:a)=> ((Y Xx)->False)):(a->Prop)
% Found x2 as proof of (((eq (a->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 X0):(((eq (b->Prop)) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found (x00 ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((eq_sym00 (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((eq_sym0 (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((eq_sym00 (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((eq_sym0 (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found x2:(((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))
% Instantiate: b0:=(fun (Xx:a)=> ((Y Xx)->False)):(a->Prop)
% Found x2 as proof of (((eq (a->Prop)) R) b0)
% Found x20:=(x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5)))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found x10:=(x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5)))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found ((eq_ref0 (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (((eq_ref Prop) (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (((eq_ref Prop) (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x5))) P)) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found x20:=(x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x6:(a->Prop))=> (P (X0 (Xf x5)))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found x10:=(x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x6:(a->Prop))=> (P (X0 (Xf x5)))))) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x5)))->(P (X0 (Xf x5))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found ((eq_ref0 (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (((eq_ref Prop) (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (((eq_ref Prop) (X0 (Xf x5))) P) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x5))) P)) as proof of ((P (X0 (Xf x5)))->(P (Y0 x5)))
% Found eq_ref00:=(eq_ref0 X0):(((eq (b->Prop)) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found (x00 ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found eq_ref00:=(eq_ref0 (f x9)):(((eq Prop) (f x9)) (f x9))
% Found (eq_ref0 (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found (fun (x9:a)=> ((eq_ref Prop) (f x9))) as proof of (((eq Prop) (f x9)) (R x9))
% Found (fun (x9:a)=> ((eq_ref Prop) (f x9))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (f x9)):(((eq Prop) (f x9)) (f x9))
% Found (eq_ref0 (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found ((eq_ref Prop) (f x9)) as proof of (((eq Prop) (f x9)) (R x9))
% Found (fun (x9:a)=> ((eq_ref Prop) (f x9))) as proof of (((eq Prop) (f x9)) (R x9))
% Found (fun (x9:a)=> ((eq_ref Prop) (f x9))) as proof of (forall (x:a), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((eq_sym00 (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found (((eq_sym0 (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((((eq_sym Prop) (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((eq_sym00 (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found (((eq_sym0 (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((((eq_sym Prop) (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found x2:(((eq (a->Prop)) R) (fun (Xx:a)=> ((Y Xx)->False)))
% Instantiate: b0:=(fun (Xx:a)=> ((Y Xx)->False)):(a->Prop)
% Found x2 as proof of (((eq (a->Prop)) R) b0)
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((eq_sym00 (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found (((eq_sym0 (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((((eq_sym Prop) (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((eq_sym00 (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found (((eq_sym0 (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found ((((eq_sym Prop) (X0 (Xf x7))) (Y0 x7)) ((eq_ref Prop) (X0 (Xf x7)))) as proof of (((eq Prop) (Y0 x7)) (X0 (Xf x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x7))):(((eq Prop) (X0 (Xf x7))) (X0 (Xf x7)))
% Found (eq_ref0 (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found ((eq_ref Prop) (X0 (Xf x7))) as proof of (((eq Prop) (X0 (Xf x7))) (Y0 x7))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((eq_sym00 (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((eq_sym0 (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (fun (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of ((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (Y0 x5)) (X0 (Xf x5))))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (Y0 x5)) (X0 (Xf x5)))))
% Found (and_rect10 (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((and_rect1 (((eq Prop) (Y0 x5)) (X0 (Xf x5)))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((fun (P:Type) (x6:(((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))) P) x6) x3)) (((eq Prop) (Y0 x5)) (X0 (Xf x5)))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (eq_sym000 ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((eq_sym00 (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((eq_sym0 (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (fun (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of ((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (Y0 x5)) (X0 (Xf x5))))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (Y0 x5)) (X0 (Xf x5)))))
% Found (and_rect10 (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found ((and_rect1 (((eq Prop) (Y0 x5)) (X0 (Xf x5)))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found (((fun (P:Type) (x6:(((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))) P) x6) x3)) (((eq Prop) (Y0 x5)) (X0 (Xf x5)))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((((eq_sym Prop) (X0 (Xf x5))) (Y0 x5)) ((eq_ref Prop) (X0 (Xf x5)))))) as proof of (((eq Prop) (Y0 x5)) (X0 (Xf x5)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x10:=(x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x1 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref000:=(eq_ref00 P):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (eq_ref00 P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found ((eq_ref0 (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (((eq_ref Prop) (X0 (Xf x7))) P) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (X0 (Xf x7))) P)) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found x20:=(x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))):((P (X0 (Xf x7)))->(P (X0 (Xf x7))))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found (fun (P:(Prop->Prop))=> (x2 (fun (x8:(a->Prop))=> (P (X0 (Xf x7)))))) as proof of ((P (X0 (Xf x7)))->(P (Y0 x7)))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (fun (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of ((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (X0 (Xf x5))) (Y0 x5)))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (X0 (Xf x5))) (Y0 x5))))
% Found (and_rect10 (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((and_rect1 (((eq Prop) (X0 (Xf x5))) (Y0 x5))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (((fun (P:Type) (x6:(((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))) P) x6) x3)) (((eq Prop) (X0 (Xf x5))) (Y0 x5))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found eq_ref00:=(eq_ref0 (X0 (Xf x5))):(((eq Prop) (X0 (Xf x5))) (X0 (Xf x5)))
% Found (eq_ref0 (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((eq_ref Prop) (X0 (Xf x5))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (fun (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of ((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (X0 (Xf x5))) (Y0 x5)))
% Found (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5)))) as proof of (((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->(((eq Prop) (X0 (Xf x5))) (Y0 x5))))
% Found (and_rect10 (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found ((and_rect1 (((eq Prop) (X0 (Xf x5))) (Y0 x5))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found (((fun (P:Type) (x6:(((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))->((forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0)))) P) x6) x3)) (((eq Prop) (X0 (Xf x5))) (Y0 x5))) (fun (x6:((and ((and ((and ((and ((and ((and (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> False))->(T R0)))) (forall (R0:(a->Prop)), ((((eq (a->Prop)) R0) (fun (Xx:a)=> (False->False)))->(T R0))))) (forall (K:((a->Prop)->Prop)) (R0:(a->Prop)), (((and (forall (Xx:(a->Prop)), ((K Xx)->(T Xx)))) (((eq (a->Prop)) R0) (fun (Xx:a)=> ((ex (a->Prop)) (fun (S0:(a->Prop))=> ((and (K S0)) (S0 Xx)))))))->(T R0))))) (forall (Y:(a->Prop)) (Z:(a->Prop)) (S0:(a->Prop)), (((and ((and (T Y)) (T Z))) (((eq (a->Prop)) S0) (fun (Xx:a)=> ((and (Y Xx)) (Z Xx)))))->(T S0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0))))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))) (forall (K:((b->Prop)->Prop)) (R0:(b->Prop)), (((and (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))) (((eq (b->Prop)) R0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx)))))))->(S R0))))) (x7:(forall (Y:(b->Prop)) (Z:(b->Prop)) (S0:(b->Prop)), (((and ((and (S Y)) (S Z))) (((eq (b->Prop)) S0) (fun (Xx:b)=> ((and (Y Xx)) (Z Xx)))))->(S S0))))=> ((eq_ref Prop) (X0 (Xf x5))))) as proof of (((eq Prop) (X0 (Xf x5))) (Y0 x5))
% Found eq_ref00:=(eq_ref0 X0):(((eq (b->Prop)) X0) X0)
% Found (eq_ref0 X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found ((eq_ref (b->Prop)) X0) as proof of (((eq (b->Prop)) X0) (fun (Xx:b)=> ((X Xx)->False)))
% Found (x00 ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found ((x0 X0) ((eq_ref (b->Prop)) X0)) as proof of (S X0)
% Found eq_ref00:=(eq_ref0 (f x11)):(((eq Prop) (f x11)) (f x11))
% Found (eq_ref0 (f x11)) as proof of (((eq Prop) (f x11)) (R x11))
% Found ((eq_ref Prop) (f x11)) as proof of (((eq Prop) (f x11)) (R x11))
% Found ((eq_ref Prop) (f x11)) as proof of (((eq Prop) (f x11)) (R x11))
% Found (fun (x11:a)=> ((eq_re
% EOF
%------------------------------------------------------------------------------