TSTP Solution File: SEV269^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV269^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n100.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 13:33:58 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV269^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.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 08:40:36 CDT 2014
% % CPUTime : 300.02
% 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 0x1d2a830>, <kernel.Type object at 0x1d2a680>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (forall (S:((b->Prop)->Prop)), (((and ((and ((and (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 (W:(b->Prop)) (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))->(S R))))) of role conjecture named cCLOSURE_THM1_pme
% Conjecture to prove = (forall (S:((b->Prop)->Prop)), (((and ((and ((and (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 (W:(b->Prop)) (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))->(S R))))):Prop
% Parameter b_DUMMY:b.
% We need to prove ['(forall (S:((b->Prop)->Prop)), (((and ((and ((and (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 (W:(b->Prop)) (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))->(S R)))))']
% Parameter b:Type.
% Trying to prove (forall (S:((b->Prop)->Prop)), (((and ((and ((and (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 (W:(b->Prop)) (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))->(S R)))))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(S Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(S Xx))
% Found (((eq_ref (b->Prop)) Xx) K) as proof of ((K Xx)->(S Xx))
% Found (((eq_ref (b->Prop)) Xx) K) as proof of ((K Xx)->(S Xx))
% Found (fun (Xx:(b->Prop))=> (((eq_ref (b->Prop)) Xx) K)) as proof of ((K Xx)->(S Xx))
% Found (fun (Xx:(b->Prop))=> (((eq_ref (b->Prop)) Xx) K)) as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found eq_ref000:=(eq_ref00 K):((K Xx)->(K Xx))
% Found (eq_ref00 K) as proof of ((K Xx)->(S Xx))
% Found ((eq_ref0 Xx) K) as proof of ((K Xx)->(S Xx))
% Found (((eq_ref (b->Prop)) Xx) K) as proof of ((K Xx)->(S Xx))
% Found (((eq_ref (b->Prop)) Xx) K) as proof of ((K Xx)->(S Xx))
% Found (fun (Xx:(b->Prop))=> (((eq_ref (b->Prop)) Xx) K)) as proof of ((K Xx)->(S Xx))
% Found (fun (Xx:(b->Prop))=> (((eq_ref (b->Prop)) Xx) K)) as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))
% Instantiate: K:=(fun (x8:(b->Prop))=> (((eq (b->Prop)) x8) (fun (Xx:b)=> False))):((b->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))
% Instantiate: K:=(fun (x8:(b->Prop))=> (((eq (b->Prop)) x8) (fun (Xx:b)=> False))):((b->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))
% Instantiate: K:=(fun (x8:(b->Prop))=> (((eq (b->Prop)) x8) (fun (Xx:b)=> False))):((b->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))
% Instantiate: K:=(fun (x8:(b->Prop))=> (((eq (b->Prop)) x8) (fun (Xx:b)=> False))):((b->Prop)->Prop)
% Found x5 as proof of (forall (Xx:(b->Prop)), ((K Xx)->(S Xx)))
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) (R 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)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% 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)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) (R 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)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (R x1))
% Found (fun (x1:b)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% 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:b)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:b), (((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:b)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->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:b)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:b), (((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:b)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (R x3))
% Found (fun (x3:b)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> False)):(((eq (b->Prop)) (fun (Xx:b)=> False)) (fun (x:b)=> False))
% Found (eta_expansion00 (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> (False->False))):(((eq (b->Prop)) (fun (Xx:b)=> (False->False))) (fun (x:b)=> (False->False)))
% Found (eta_expansion00 (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))):((P (fun (Xx:b)=> (False->False)))->(P (fun (Xx:b)=> (False->False))))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))):((P (fun (Xx:b)=> False))->(P (fun (Xx:b)=> False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))):((P (fun (Xx:b)=> False))->(P (fun (Xx:b)=> False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))):((P (fun (Xx:b)=> (False->False)))->(P (fun (Xx:b)=> (False->False))))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> False)):(((eq (b->Prop)) (fun (Xx:b)=> False)) (fun (x:b)=> False))
% Found (eta_expansion00 (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) R)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> (False->False))):(((eq (b->Prop)) (fun (Xx:b)=> (False->False))) (fun (x:b)=> (False->False)))
% Found (eta_expansion00 (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% 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:b)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:b)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:b), (((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:b)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:b)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))):((P (fun (Xx:b)=> (False->False)))->(P (fun (Xx:b)=> (False->False))))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))):((P (fun (Xx:b)=> False))->(P (fun (Xx:b)=> False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))):((P (fun (Xx:b)=> False))->(P (fun (Xx:b)=> False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> False)))) as proof of (P0 (fun (Xx:b)=> False))
% Found x00:=(x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))):((P (fun (Xx:b)=> (False->False)))->(P (fun (Xx:b)=> (False->False))))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% Found (x0 (fun (x7:(b->Prop))=> (P (fun (Xx:b)=> (False->False))))) as proof of (P0 (fun (Xx:b)=> (False->False)))
% 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:b)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:b)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:b), (((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:b)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) (R x5))
% Found (fun (x5:b)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) False)
% Found eq_ref00:=(eq_ref0 (R x7)):(((eq Prop) (R x7)) (R x7))
% Found (eq_ref0 (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found ((eq_ref Prop) (R x7)) as proof of (((eq Prop) (R x7)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (False->False))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found eq_ref000:=(eq_ref00 P):((P (R x7))->(P (R x7)))
% Found (eq_ref00 P) as proof of (P0 (R x7))
% Found ((eq_ref0 (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found (((eq_ref Prop) (R x7)) P) as proof of (P0 (R x7))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found ((x6 b0) (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found ((x6 b0) (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% 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:b)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:b)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% 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:b)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:b)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P b0))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P b0))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of (P b0)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of ((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)))->(P b0))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of (((and (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)))->(P b0)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion b) Prop) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P b0))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))) as proof of (P b0)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))) as proof of (P b0)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))) as proof of ((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)))->(P b0))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))) as proof of (((and (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)))->(P b0)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))) as proof of (P b0)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))))) as proof of (P b0)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))))) 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)))->(P b0))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0))))))) as proof of (((and ((and (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)))->(P b0)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))))) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))))) as proof of (P b0)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))))) as proof of (P b0)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 b0) (((eta_expansion b) Prop) b0)))))))) as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found x7:(P R)
% Instantiate: b0:=R:(b->Prop)
% Found x7 as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> (False->False))):(((eq (b->Prop)) (fun (Xx:b)=> (False->False))) (fun (x:b)=> (False->False)))
% Found (eta_expansion00 (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found x7:(P R)
% Instantiate: b0:=R:(b->Prop)
% Found x7 as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> False)):(((eq (b->Prop)) (fun (Xx:b)=> False)) (fun (x:b)=> False))
% Found (eta_expansion00 (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found ((eta_expansion_dep0 (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found (((eta_expansion_dep b) (fun (x6:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> ((ex (b->Prop)) (fun (S0:(b->Prop))=> ((and (K S0)) (S0 Xx))))))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 f):(((eq (b->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found ((x5 f) ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P f))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))) as proof of (P f)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))))) as proof of (P f)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))))) 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)))->(P f))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))))))) as proof of (((and ((and (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)))->(P f)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))))))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 f):(((eq (b->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found ((x6 f) ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P f))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))))) as proof of (P f)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))))) 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)))->(P f))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f))))))) as proof of (((and ((and (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)))->(P f)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) ((eq_ref (b->Prop)) f)))))))) as proof of (P f)
% Found x7:(P R)
% Instantiate: f:=R:(b->Prop)
% Found x7 as proof of (P0 f)
% Found x7:(P R)
% Instantiate: f:=R:(b->Prop)
% Found x7 as proof of (P0 f)
% Found x7:(P R)
% Instantiate: f:=R:(b->Prop)
% Found x7 as proof of (P0 f)
% Found x7:(P R)
% Instantiate: f:=R:(b->Prop)
% Found x7 as proof of (P0 f)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% 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:b)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:b)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% 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:b)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) (R x7))
% Found (fun (x7:b)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:b), (((eq Prop) (f x)) (R x)))
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P b0))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P b0))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))) as proof of (P b0)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of (P b0)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of ((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)))->(P b0))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0))))) as proof of (((and (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)))->(P b0)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 b0) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) b0)))))) as proof of (P b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found (x60 ((eq_ref (b->Prop)) b0)) as proof of (P b0)
% Found ((x6 b0) ((eq_ref (b->Prop)) b0)) as proof of (P b0)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))) as proof of (P b0)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P b0))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))) as proof of ((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)))->(P b0)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))) as proof of (P b0)
% Found ((and_rect2 (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))) as proof of (P b0)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))) as proof of (P b0)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))) as proof of (P b0)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))) as proof of ((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)))->(P b0))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))) as proof of (((and (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)))->(P b0)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))) as proof of (P b0)
% Found ((and_rect1 (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))) as proof of (P b0)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))) as proof of (P b0)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))))) as proof of (P b0)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))))) 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)))->(P b0))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0))))))) as proof of (((and ((and (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)))->(P b0)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))))) as proof of (P b0)
% Found ((and_rect0 (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))))) as proof of (P b0)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))))) as proof of (P b0)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P b0)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P b0)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P b0)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 b0) ((eq_ref (b->Prop)) b0)))))))) as proof of (P b0)
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)))
% Instantiate: b0:=(fun (Xx:b)=> ((forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx)))->False)):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) (False->False))
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:b), (((eq Prop) (f x)) (False->False)))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) False)
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:b), (((eq Prop) (f x)) False))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) (False->False))
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) (False->False))
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:b), (((eq Prop) (f x)) (False->False)))
% Found eq_ref00:=(eq_ref0 (f x8)):(((eq Prop) (f x8)) (f x8))
% Found (eq_ref0 (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found ((eq_ref Prop) (f x8)) as proof of (((eq Prop) (f x8)) False)
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (((eq Prop) (f x8)) False)
% Found (fun (x8:b)=> ((eq_ref Prop) (f x8))) as proof of (forall (x:b), (((eq Prop) (f x)) False))
% Found x7:(P R)
% Instantiate: b0:=R:(b->Prop)
% Found x7 as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> (False->False))):(((eq (b->Prop)) (fun (Xx:b)=> (False->False))) (fun (x:b)=> (False->False)))
% Found (eta_expansion00 (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> (False->False))) as proof of (((eq (b->Prop)) (fun (Xx:b)=> (False->False))) b0)
% Found x7:(P R)
% Instantiate: b0:=R:(b->Prop)
% Found x7 as proof of (P0 b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:b)=> False)):(((eq (b->Prop)) (fun (Xx:b)=> False)) (fun (x:b)=> False))
% Found (eta_expansion00 (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found (((eta_expansion b) Prop) (fun (Xx:b)=> False)) as proof of (((eq (b->Prop)) (fun (Xx:b)=> False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 (False->False)):(((eq Prop) (False->False)) (False->False))
% Found (eq_ref0 (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 (False->False)):(((eq Prop) (False->False)) (False->False))
% Found (eq_ref0 (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 (False->False)):(((eq Prop) (False->False)) (False->False))
% Found (eq_ref0 (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 False):(((eq Prop) False) False)
% Found (eq_ref0 False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found ((eq_ref Prop) False) as proof of (((eq Prop) False) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found eq_ref00:=(eq_ref0 (False->False)):(((eq Prop) (False->False)) (False->False))
% Found (eq_ref0 (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found ((eq_ref Prop) (False->False)) as proof of (((eq Prop) (False->False)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (R x7))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> (False->False)))
% Found x0:(((eq (b->Prop)) R) (fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))))
% Instantiate: b0:=(fun (Xx:b)=> (not (forall (S0:(b->Prop)), (((and (forall (Xx0:b), ((W Xx0)->(S0 Xx0)))) (forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx0:b)=> ((S0 Xx0)->False)))->(S R0))))->(S0 Xx))))):(b->Prop)
% Found x0 as proof of (((eq (b->Prop)) R) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) b0) as proof of (((eq (b->Prop)) b0) (fun (Xx:b)=> False))
% Found x00:=(x0 (fun (x8:(b->Prop))=> (P False))):((P False)->(P False))
% Found (x0 (fun (x8:(b->Prop))=> (P False))) as proof of (P0 False)
% Found (x0 (fun (x8:(b->Prop))=> (P False))) as proof of (P0 False)
% Found x00:=(x0 (fun (x8:(b->Prop))=> (P False))):((P False)->(P False))
% Found (x0 (fun (x8:(b->Prop))=> (P False))) as proof of (P0 False)
% Found (x0 (fun (x8:(b->Prop))=> (P False))) as proof of (P0 False)
% Found x00:=(x0 (fun (x8:(b->Prop))=> (P (False->False)))):((P (False->False))->(P (False->False)))
% Found (x0 (fun (x8:(b->Prop))=> (P (False->False)))) as proof of (P0 (False->False))
% Found (x0 (fun (x8:(b->Prop))=> (P (False->False)))) as proof of (P0 (False->False))
% Found x00:=(x0 (fun (x8:(b->Prop))=> (P (False->False)))):((P (False->False))->(P (False->False)))
% Found (x0 (fun (x8:(b->Prop))=> (P (False->False)))) as proof of (P0 (False->False))
% Found (x0 (fun (x8:(b->Prop))=> (P (False->False)))) as proof of (P0 (False->False))
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref000:=(eq_ref00 P):((P R)->(P R))
% Found (eq_ref00 P) as proof of (P0 R)
% Found ((eq_ref0 R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found (((eq_ref (b->Prop)) R) P) as proof of (P0 R)
% Found eq_ref00:=(eq_ref0 f):(((eq (b->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found ((x6 f) ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x6 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion_dep0 (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion_dep b) (fun (x8:b)=> Prop)) f)))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x5 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 f):(((eq (b->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found ((eq_ref (b->Prop)) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> False))
% Found (x50 ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found ((x5 f) ((eq_ref (b->Prop)) f)) as proof of (P f)
% Found (fun (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of (P f)
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of ((forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0)))->(P f))
% Found (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> False))->(S R0)))) (x6:(forall (R0:(b->Prop)), ((((eq (b->Prop)) R0) (fun (Xx:b)=> (False->False)))->(S R0))))=> ((x5 f) ((eq_ref (b->Prop)) f)))))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of (P f)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P f))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) as proof of (P f)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) 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)))->(P f))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) as proof of (((and ((and (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)))->(P f)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 f):(((eq (b->Prop)) f) (fun (x:b)=> (f x)))
% Found (eta_expansion00 f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found ((eta_expansion0 Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (((eta_expansion b) Prop) f) as proof of (((eq (b->Prop)) f) (fun (Xx:b)=> (False->False)))
% Found (x60 (((eta_expansion b) Prop) f)) as proof of (P f)
% Found ((x6 f) (((eta_expansion b) Prop) f)) as proof of (P f)
% Found (fun (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of (P f)
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R)))->(P f))
% Found (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))) as proof of ((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)))->(P f)))
% Found (and_rect20 (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found ((and_rect2 (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))) as proof of (P f)
% Found (fun (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of (P f)
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of ((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)))->(P f))
% Found (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))) as proof of (((and (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)))->(P f)))
% Found (and_rect10 (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))) as proof of (P f)
% Found (fun (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) as proof of (P f)
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) 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)))->(P f))
% Found (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f))))))) as proof of (((and ((and (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)))->(P f)))
% Found (and_rect00 (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found ((and_rect0 (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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)))->P0)))=> (((((and_rect ((and (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)))) P0) x3) x1)) (P f)) (fun (x3:((and (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))))) (x4:(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))))=> (((fun (P0:Type) (x5:((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)))->P0)))=> (((((and_rect (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)))) P0) x5) x3)) (P f)) (fun (x5:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> False))->(S R)))) (x6:(forall (R:(b->Prop)), ((((eq (b->Prop)) R) (fun (Xx:b)=> (False->False)))->(S R))))=> ((x6 f) (((eta_expansion b) Prop) f)))))))) as proof of (P f)
% Found (((fun (P0:Type) (x1:(((and ((and (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)))->P0)))=> (((((and_rect ((and ((and (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)))) P0) x1) x)) (P f)) (fun (x1:((and ((and (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))))) (x2:(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))))=> (((fun (P0:Type) (x3:(((and (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->
% EOF
%------------------------------------------------------------------------------