%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV314^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n115.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 13:34:02 EDT 2014

% Result   : Timeout 300.04s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV314^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n115.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:48:56 CDT 2014
% % CPUTime  : 300.04 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1d12e60>, <kernel.Type object at 0x1ef3ab8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1d12ea8>, <kernel.Type object at 0x1ef3c68>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1d123b0>, <kernel.DependentProduct object at 0x1ef3dd0>) of role type named cF
% Using role type
% Declaring cF:((a->(b->Prop))->(a->(b->Prop)))
% FOF formula (<kernel.Constant object at 0x1d12950>, <kernel.DependentProduct object at 0x1ef3dd0>) of role type named cCL
% Using role type
% Declaring cCL:((a->(b->Prop))->Prop)
% FOF formula (((and ((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b), (((R Xa) Xb)->((S Xa) Xb))))->(forall (Xa:a) (Xb:b), ((((cF R) Xa) Xb)->(((cF S) Xa) Xb))))))->((ex (a->(b->Prop))) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb))))))))) of role conjecture named cFP_THM2_pme
% Conjecture to prove = (((and ((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b), (((R Xa) Xb)->((S Xa) Xb))))->(forall (Xa:a) (Xb:b), ((((cF R) Xa) Xb)->(((cF S) Xa) Xb))))))->((ex (a->(b->Prop))) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% We need to prove ['(((and ((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b), (((R Xa) Xb)->((S Xa) Xb))))->(forall (Xa:a) (Xb:b), ((((cF R) Xa) Xb)->(((cF S) Xa) Xb))))))->((ex (a->(b->Prop))) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter cF:((a->(b->Prop))->(a->(b->Prop))).
% Parameter cCL:((a->(b->Prop))->Prop).
% Trying to prove (((and ((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b), (((R Xa) Xb)->((S Xa) Xb))))->(forall (Xa:a) (Xb:b), ((((cF R) Xa) Xb)->(((cF S) Xa) Xb))))))->((ex (a->(b->Prop))) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))))
% Found eq_ref000:=(eq_ref00 (x0 Xa)):(((x0 Xa) Xb)->((x0 Xa) Xb))
% Found (eq_ref00 (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found eq_ref000:=(eq_ref00 (x2 Xa)):(((x2 Xa) Xb)->((x2 Xa) Xb))
% Found (eq_ref00 (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x00000:=(x0000 x00):((Y Xa) Xb)
% Found (x0000 x00) as proof of ((Y Xa) Xb)
% Found ((x000 Y) x00) as proof of ((Y Xa) Xb)
% Found ((x000 Y) x00) as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((x000 Y) x00)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))))
% Found eq_ref000:=(eq_ref00 (x0 Xa)):(((x0 Xa) Xb)->((x0 Xa) Xb))
% Found (eq_ref00 (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found eta_expansion000:=(eta_expansion00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))):(((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) (fun (x:(a->(b->Prop)))=> ((and ((and (cCL x)) (((eq (a->(b->Prop))) (cF x)) x))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x Xa) Xb)->((Y Xa) Xb))))))))
% Found (eta_expansion00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found ((eta_expansion0 Prop) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion (a->(b->Prop))) Prop) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found eq_ref000:=(eq_ref00 (x4 Xa)):(((x4 Xa) Xb)->((x4 Xa) Xb))
% Found (eq_ref00 (x4 Xa)) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x4 Xa)) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x4 Xa)) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x4 Xa)) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x4 Xa))) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x4 Xa))) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x4 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found x0100:=(x010 x00):((Y Xa) Xb)
% Found (x010 x00) as proof of ((Y Xa) Xb)
% Found ((x01 Y) x00) as proof of ((Y Xa) Xb)
% Found ((x01 Y) x00) as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found eq_ref000:=(eq_ref00 (x0 Xa)):(((x0 Xa) Xb)->((x0 Xa) Xb))
% Found (eq_ref00 (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x0 Xa)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x0 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found eq_ref000:=(eq_ref00 (x2 Xa)):(((x2 Xa) Xb)->((x2 Xa) Xb))
% Found (eq_ref00 (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found ((eq_ref0 Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (((eq_ref b) Xb) (x2 Xa)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b)=> (((eq_ref b) Xb) (x2 Xa))) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))):(((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) (fun (x:(a->(b->Prop)))=> ((and ((and (cCL x)) (((eq (a->(b->Prop))) (cF x)) x))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x Xa) Xb)->((Y Xa) Xb))))))))
% Found (eta_expansion_dep00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x3:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found x01:((x4 Xa) Xb)
% Instantiate: x4:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x4 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x4 Xa) Xb)
% Instantiate: x4:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x4 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x4 Xa) Xb)
% Instantiate: x4:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x4 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x4 Xa) Xb)
% Instantiate: x4:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x4 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x000:((x0 Xa) Xb)
% Instantiate: x0:=Y:(a->(b->Prop))
% Found x000 as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> x000) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> x000) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x01:((x2 Xa) Xb)
% Instantiate: x2:=Y:(a->(b->Prop))
% Found x01 as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> x01) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> x01) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found x0000000:=(x000000 x00):((Y Xa) Xb)
% Found (x000000 x00) as proof of ((Y Xa) Xb)
% Found ((x00000 Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x0000 Y0) x001) x1)) Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00) as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x001:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x000 Y0) x001) x2)) Y0) x001) x1)) Y) x00)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))))
% Found x0100:=(x010 x00):((Y Xa) Xb)
% Found (x010 x00) as proof of ((Y Xa) Xb)
% Found ((x01 Y) x00) as proof of ((Y Xa) Xb)
% Found ((x01 Y) x00) as proof of ((Y Xa) Xb)
% Found (fun (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of (((x4 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x4 Xa) Xb))=> ((x01 Y) x00)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x4 Xa) Xb)->((Y Xa) Xb)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))):(((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) (fun (x:(a->(b->Prop)))=> ((and ((and (cCL x)) (((eq (a->(b->Prop))) (cF x)) x))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x Xa) Xb)->((Y Xa) Xb))))))))
% Found (eta_expansion_dep00 (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x5:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x5:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found (((eta_expansion_dep (a->(b->Prop))) (fun (x5:(a->(b->Prop)))=> Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) as proof of (((eq ((a->(b->Prop))->Prop)) (fun (X:(a->(b->Prop)))=> ((and ((and (cCL X)) (((eq (a->(b->Prop))) (cF X)) X))) (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((X Xa) Xb)->((Y Xa) Xb)))))))) b0)
% Found x010000:=(x01000 x00):((Y Xa) Xb)
% Found (x01000 x00) as proof of ((Y Xa) Xb)
% Found ((x0100 Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x010 Y0) x000) x4)) Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00) as proof of ((Y Xa) Xb)
% Found (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00) as proof of ((Y Xa) Xb)
% Found (fun (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of (((x2 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of (forall (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of (forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x01:((x2 Xa) Xb))=> (((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> ((((fun (Y0:(a->(b->Prop))) (x000:((and (cCL Y0)) (((eq (a->(b->Prop))) (cF Y0)) Y0)))=> (((x01 Y0) x000) x3)) Y0) x000) x4)) Y) x00)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x2 Xa) Xb)->((Y Xa) Xb)))))
% Found x2:(forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b), (((R Xa) Xb)->((S Xa) Xb))))->(forall (Xa:a) (Xb:b), ((((cF R) Xa) Xb)->(((cF S) Xa) Xb)))))
% Found x2 as proof of (forall (R:(a->(b->Prop))) (S:(a->(b->Prop))), (((and ((and (cCL R)) (cCL S))) (forall (Xa0:a) (Xb0:b), (((R Xa0) Xb0)->((S Xa0) Xb0))))->(forall (Xa0:a) (Xb0:b), ((((cF R) Xa0) Xb0)->(((cF S) Xa0) Xb0)))))
% Found x1:((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))
% Found x1 as proof of ((and (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa0:a) (Xb0:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa0) Xb0)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))
% Found x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))
% Found x3 as proof of (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa0:a) (Xb0:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa0) Xb0)))))))
% Found x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))
% Found x4 as proof of (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))
% Found x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))
% Found x00 as proof of ((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))
% Found (((((x0000 x00) x1) x2) x3) x4) as proof of ((Y Xa) Xb)
% Found ((((((x000 Y) x00) x1) x2) x3) x4) as proof of ((Y Xa) Xb)
% Found ((((((x000 Y) x00) x1) x2) x3) x4) as proof of ((Y Xa) Xb)
% Found (fun (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of ((Y Xa) Xb)
% Found (fun (Xb:b) (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of (((x0 Xa) Xb)->((Y Xa) Xb))
% Found (fun (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of (forall (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of (forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb))))
% Found (fun (Y:(a->(b->Prop))) (x00:((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))) (Xa:a) (Xb:b) (x000:((x0 Xa) Xb))=> ((((((x000 Y) x00) x1) x2) x3) x4)) as proof of (forall (Y:(a->(b->Prop))), (((and (cCL Y)) (((eq (a->(b->Prop))) (cF Y)) Y))->(forall (Xa:a) (Xb:b), (((x0 Xa) Xb)->((Y Xa) Xb)))))
% Found x6:(cCL Xx)
% Found (fun (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of (cCL Xx)
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx))
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx)))
% Found (and_rect20 (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found ((and_rect2 (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (fun (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (cCL Xx)
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx))
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (forall (Xx:(a->(b->Prop))), (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx)))
% Found (x20 (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x4)
% Found ((x2 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x4)
% Found ((x2 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x4)
% Found eta_expansion000:=(eta_expansion00 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (fun (x:a)=> ((cF x0) x)))
% Found (eta_expansion00 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eta_expansion0 (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found x6:(cCL Xx)
% Found (fun (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of (cCL Xx)
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx))
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx)))
% Found (and_rect20 (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found ((and_rect2 (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (fun (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (cCL Xx)
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx))
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (forall (Xx:(a->(b->Prop))), (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx)))
% Found (x30 (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x0)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x0)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x0)
% Found x6:(cCL Xx)
% Found (fun (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of (cCL Xx)
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx))
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx)))
% Found (and_rect20 (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found ((and_rect2 (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (fun (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (cCL Xx)
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx))
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (forall (Xx:(a->(b->Prop))), (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx)))
% Found (x30 (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x2)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x2)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x2)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found eq_ref00:=(eq_ref0 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (cF x0))
% Found (eq_ref0 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x4)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x4)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x4)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x4)
% Found eq_ref00:=(eq_ref0 (cF x4)):(((eq (a->(b->Prop))) (cF x4)) (cF x4))
% Found (eq_ref0 (cF x4)) as proof of (((eq (a->(b->Prop))) (cF x4)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x4)) as proof of (((eq (a->(b->Prop))) (cF x4)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x4)) as proof of (((eq (a->(b->Prop))) (cF x4)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x4)) as proof of (((eq (a->(b->Prop))) (cF x4)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cF x2)):(((eq (a->(b->Prop))) (cF x2)) (fun (x:a)=> ((cF x2) x)))
% Found (eta_expansion_dep00 (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found ((eta_expansion_dep0 (fun (x4:a)=> (b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> (b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> (b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion_dep a) (fun (x4:a)=> (b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->(b->Prop))) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eta_expansion0 (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found eta_expansion000:=(eta_expansion00 (cF x2)):(((eq (a->(b->Prop))) (cF x2)) (fun (x:a)=> ((cF x2) x)))
% Found (eta_expansion00 (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found ((eta_expansion0 (b->Prop)) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found eq_ref00:=(eq_ref0 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (cF x0))
% Found (eq_ref0 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->(b->Prop))) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eta_expansion0 (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found x6:(cCL Xx)
% Found (fun (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of (cCL Xx)
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx))
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx)))
% Found (and_rect20 (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found ((and_rect2 (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (fun (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (cCL Xx)
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx))
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (forall (Xx:(a->(b->Prop))), (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx)))
% Found (x30 (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x0)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x0)
% Found (fun (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of (cCL x0)
% Found (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of ((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->(cCL x0))
% Found (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of ((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->(cCL x0)))
% Found (and_rect10 (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x0)
% Found ((and_rect1 (cCL x0)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x0)
% Found (((fun (P:Type) (x3:((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->P)))=> (((((and_rect (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))) P) x3) x1)) (cCL x0)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x0)
% Found (((fun (P:Type) (x3:((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->P)))=> (((((and_rect (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))) P) x3) x1)) (cCL x0)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found eq_ref00:=(eq_ref0 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (cF x0))
% Found (eq_ref0 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found x6:(cCL Xx)
% Found (fun (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of (cCL Xx)
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx))
% Found (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6) as proof of ((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->(cCL Xx)))
% Found (and_rect20 (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found ((and_rect2 (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)) as proof of (cCL Xx)
% Found (fun (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (cCL Xx)
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx))
% Found (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))) as proof of (forall (Xx:(a->(b->Prop))), (((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx))->(cCL Xx)))
% Found (x30 (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x2)
% Found ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))) as proof of (cCL x2)
% Found (fun (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of (cCL x2)
% Found (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of ((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->(cCL x2))
% Found (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6))))) as proof of ((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->(cCL x2)))
% Found (and_rect10 (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x2)
% Found ((and_rect1 (cCL x2)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x2)
% Found (((fun (P:Type) (x3:((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->P)))=> (((((and_rect (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))) P) x3) x0)) (cCL x2)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x2)
% Found (((fun (P:Type) (x3:((forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))->((forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))->P)))=> (((((and_rect (forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R))))) P) x3) x0)) (cCL x2)) (fun (x3:(forall (S:((a->(b->Prop))->Prop)), ((forall (Xx:(a->(b->Prop))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b)=> (forall (R:(a->(b->Prop))), ((S R)->((R Xa) Xb)))))))) (x4:(forall (R:(a->(b->Prop))), ((cCL R)->(cCL (cF R)))))=> ((x3 (fun (x9:(a->(b->Prop)))=> ((and (cCL x9)) (((eq (a->(b->Prop))) (cF x9)) x9)))) (fun (Xx:(a->(b->Prop))) (x5:((and (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)))=> (((fun (P:Type) (x6:((cCL Xx)->((((eq (a->(b->Prop))) (cF Xx)) Xx)->P)))=> (((((and_rect (cCL Xx)) (((eq (a->(b->Prop))) (cF Xx)) Xx)) P) x6) x5)) (cCL Xx)) (fun (x6:(cCL Xx)) (x7:(((eq (a->(b->Prop))) (cF Xx)) Xx))=> x6)))))) as proof of (cCL x2)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) (cF x2))
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) (cF x2))
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) (cF x2))
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) (cF x2))
% Found eq_ref00:=(eq_ref0 x2):(((eq (a->(b->Prop))) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq (a->(b->Prop))) x2) b0)
% Found ((eq_ref (a->(b->Prop))) x2) as proof of (((eq (a->(b->Prop))) x2) b0)
% Found ((eq_ref (a->(b->Prop))) x2) as proof of (((eq (a->(b->Prop))) x2) b0)
% Found ((eq_ref (a->(b->Prop))) x2) as proof of (((eq (a->(b->Prop))) x2) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found eq_ref00:=(eq_ref0 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (cF x0))
% Found (eq_ref0 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x2)
% Found eq_ref00:=(eq_ref0 (cF x2)):(((eq (a->(b->Prop))) (cF x2)) (cF x2))
% Found (eq_ref0 (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found ((eq_ref (a->(b->Prop))) (cF x2)) as proof of (((eq (a->(b->Prop))) (cF x2)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eq_ref (a->(b->Prop))) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found eta_expansion000:=(eta_expansion00 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (fun (x:a)=> ((cF x0) x)))
% Found (eta_expansion00 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eta_expansion0 (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eta_expansion000:=(eta_expansion00 (cF x0)):(((eq (a->(b->Prop))) (cF x0)) (fun (x:a)=> ((cF x0) x)))
% Found (eta_expansion00 (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found ((eta_expansion0 (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found (((eta_expansion a) (b->Prop)) (cF x0)) as proof of (((eq (a->(b->Prop))) (cF x0)) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (a->(b->Prop))) b0) (fun (x:a)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found ((eta_expansion0 (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found (((eta_expansion a) (b->Prop)) b0) as proof of (((eq (a->(b->Prop))) b0) x0)
% Found eta_expansion000:=(eta_expansion00 ((cF x2) x3)):(((eq (b->Prop)) ((cF x2) x3)) (fun (x:b)=> (((cF x2) x3) x)))
% Found (eta_expansion00 ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found ((eta_expansion0 Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found (((eta_expansion b) Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found (((eta_expansion b) Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found (((eta_expansion b) Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (b->Prop)) b0) (fun (x:b)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (b->Prop)) b0) (x2 x3))
% Found ((eta_expansion_dep0 (fun (x5:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (x2 x3))
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (x2 x3))
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (x2 x3))
% Found (((eta_expansion_dep b) (fun (x5:b)=> Prop)) b0) as proof of (((eq (b->Prop)) b0) (x2 x3))
% Found eta_expansion000:=(eta_expansion00 ((cF x2) x3)):(((eq (b->Prop)) ((cF x2) x3)) (fun (x:b)=> (((cF x2) x3) x)))
% Found (eta_expansion00 ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found ((eta_expansion0 Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF x2) x3)) b0)
% Found (((eta_expansion b) Prop) ((cF x2) x3)) as proof of (((eq (b->Prop)) ((cF
% EOF
%------------------------------------------------------------------------------