TSTP Solution File: SEV396^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV396^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n187.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 13:34:08 EDT 2014
% Result : Timeout 300.00s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV396^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.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 09:05:41 CDT 2014
% % CPUTime : 300.00
% 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 0x1a7e200>, <kernel.DependentProduct object at 0x1a9c098>) of role type named cR
% Using role type
% Declaring cR:(fofType->Prop)
% FOF formula (<kernel.Constant object at 0x16c1680>, <kernel.DependentProduct object at 0x1a9c488>) of role type named cS
% Using role type
% Declaring cS:(fofType->Prop)
% FOF formula ((iff (forall (Xx:fofType), ((cR Xx)->(cS Xx)))) (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) cR)) of role conjecture named cTHM31_pme
% Conjecture to prove = ((iff (forall (Xx:fofType), ((cR Xx)->(cS Xx)))) (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) cR)):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((iff (forall (Xx:fofType), ((cR Xx)->(cS Xx)))) (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) cR))']
% Parameter fofType:Type.
% Parameter cR:(fofType->Prop).
% Parameter cS:(fofType->Prop).
% Trying to prove ((iff (forall (Xx:fofType), ((cR Xx)->(cS Xx)))) (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) cR))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: b:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR 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)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR x)))
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: b:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR 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)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR x)))
% Found x0:(P cR)
% Instantiate: b:=cR:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found x0:(P cR)
% Instantiate: f:=cR:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P cR)
% Instantiate: f:=cR:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found x0:(P cR)
% Instantiate: b:=cR:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (cR x)) (cS x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (cR x)) (cS x))))
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x0:(P cR)
% Instantiate: f:=cR:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P cR)
% Instantiate: f:=cR:(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found x00:(P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P2 cR)
% Found x00:(P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P2 cR)
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (cR x)) (cS x))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and (cR x1)) (cS x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((and (cR x)) (cS x))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (fun (x00:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))))=> x00) as proof of (P0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x00:(P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P2 cR)
% Found x00:(P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P1 cR)
% Found (fun (x00:(P1 cR))=> x00) as proof of (P2 cR)
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P0 ((and (cR x0)) (cS x0)))
% Found x10:(P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P0 ((and (cR x0)) (cS x0)))
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) x0)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) cR)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) cR)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) cR)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) cR)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (cR x0))
% Instantiate: b:=(cR x0):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: b:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(cR x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (cR x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (cR x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(cR x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (cR x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (cR x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found x00:(P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P cR)
% Found (fun (x00:(P cR))=> x00) as proof of (P0 cR)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR 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)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=((and (cR x0)) (cS x0)):Prop
% Found x1 as proof of (P0 b)
% Found x10:(P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P0 ((and (cR x0)) (cS x0)))
% Found x10:(P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P ((and (cR x0)) (cS x0)))
% Found (fun (x10:(P ((and (cR x0)) (cS x0))))=> x10) as proof of (P0 ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion_dep00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=(fun (x2:fofType)=> ((and (cR x2)) (cS x2))):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found x1:(P ((and (cR x0)) (cS x0)))
% Instantiate: b:=(fun (x2:fofType)=> ((and (cR x2)) (cS x2))):(fofType->Prop)
% Found x1 as proof of (P0 b)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: a:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 a)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: b:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eta_expansion000:=(eta_expansion00 cR):(((eq (fofType->Prop)) cR) (fun (x:fofType)=> (cR x)))
% Found (eta_expansion00 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eta_expansion0 Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found (((eta_expansion fofType) Prop) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and (cR x0)) (cS x0)))
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found x00:(P b)
% Found (fun (x00:(P b))=> x00) as proof of (P b)
% Found (fun (x00:(P b))=> x00) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (b x0)):(((eq Prop) (b x0)) (b x0))
% Found (eq_ref0 (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found ((eq_ref Prop) (b x0)) as proof of (((eq Prop) (b x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (b x0))
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: f:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 f)
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found x10:(P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P1 (cR x0))
% Found (fun (x10:(P1 (cR x0)))=> x10) as proof of (P2 (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(cR x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (cR x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (cR x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found x1:(P0 b)
% Instantiate: b:=(cR x0):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (cR x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (cR x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b0)
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) cR)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found eq_ref00:=(eq_ref0 (cR x0)):(((eq Prop) (cR x0)) (cR x0))
% Found (eq_ref0 (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found ((eq_ref Prop) (cR x0)) as proof of (((eq Prop) (cR x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and (cR x0)) (cS x0)))
% 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)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR 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)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (cR x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (cR x)))
% Found x0:(P cR)
% Instantiate: b:=cR:(fofType->Prop)
% Found x0 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) (fun (x:fofType)=> ((and (cR x)) (cS x))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx)))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (cS x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (cR x0))
% Found eq_ref00:=(eq_ref0 ((and (cR x0)) (cS x0))):(((eq Prop) ((and (cR x0)) (cS x0))) ((and (cR x0)) (cS x0)))
% Found (eq_ref0 ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found ((eq_ref Prop) ((and (cR x0)) (cS x0))) as proof of (((eq Prop) ((and (cR x0)) (cS x0))) b)
% Found eq_ref00:=(eq_ref0 cR):(((eq (fofType->Prop)) cR) cR)
% Found (eq_ref0 cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found ((eq_ref (fofType->Prop)) cR) as proof of (((eq (fofType->Prop)) cR) b)
% Found x0:(P (fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))))
% Instantiate: a:=(fun (Xx:fofType)=> ((and (cR Xx)) (cS Xx))):(fofType->Prop)
% Found x0 as proof of (P0 a)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P (b x0))
% Found (fun (x10:(P (b x0)))=> x10) as proof of (P0 (b x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P0 (cR x0))
% Found x10:(P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (P (cR x0))
% Found (fun (x10:(P (cR x0)))=> x10) as proof of (
% EOF
%------------------------------------------------------------------------------