%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV316^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n114.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 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV316^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n114.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 08:49:16 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1417b00>, <kernel.Type object at 0x14187a0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x14177a0>, <kernel.Type object at 0x14187a0>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1417320>, <kernel.Type object at 0x1418cf8>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0x1417200>, <kernel.DependentProduct object at 0x1418d88>) of role type named cF
% Using role type
% Declaring cF:((a->(b->(c->Prop)))->(a->(b->(c->Prop))))
% FOF formula (<kernel.Constant object at 0x1417b00>, <kernel.DependentProduct object at 0x14187e8>) of role type named cCL
% Using role type
% Declaring cCL:((a->(b->(c->Prop)))->Prop)
% FOF formula (((and ((and (forall (S:((a->(b->(c->Prop)))->Prop)), ((forall (Xx:(a->(b->(c->Prop)))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), ((S R)->(((R Xa) Xb) Xc)))))))) (forall (R:(a->(b->(c->Prop)))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->(c->Prop)))) (S:(a->(b->(c->Prop)))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b) (Xc:c), ((((R Xa) Xb) Xc)->(((S Xa) Xb) Xc))))->(forall (Xa:a) (Xb:b) (Xc:c), (((((cF R) Xa) Xb) Xc)->((((cF S) Xa) Xb) Xc))))))->((and ((and (cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))))) of role conjecture named cFP_THM3_INST_pme
% Conjecture to prove = (((and ((and (forall (S:((a->(b->(c->Prop)))->Prop)), ((forall (Xx:(a->(b->(c->Prop)))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), ((S R)->(((R Xa) Xb) Xc)))))))) (forall (R:(a->(b->(c->Prop)))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->(c->Prop)))) (S:(a->(b->(c->Prop)))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b) (Xc:c), ((((R Xa) Xb) Xc)->(((S Xa) Xb) Xc))))->(forall (Xa:a) (Xb:b) (Xc:c), (((((cF R) Xa) Xb) Xc)->((((cF S) Xa) Xb) Xc))))))->((and ((and (cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% Parameter c_DUMMY:c.
% We need to prove ['(((and ((and (forall (S:((a->(b->(c->Prop)))->Prop)), ((forall (Xx:(a->(b->(c->Prop)))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), ((S R)->(((R Xa) Xb) Xc)))))))) (forall (R:(a->(b->(c->Prop)))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->(c->Prop)))) (S:(a->(b->(c->Prop)))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b) (Xc:c), ((((R Xa) Xb) Xc)->(((S Xa) Xb) Xc))))->(forall (Xa:a) (Xb:b) (Xc:c), (((((cF R) Xa) Xb) Xc)->((((cF S) Xa) Xb) Xc))))))->((and ((and (cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter c:Type.
% Parameter cF:((a->(b->(c->Prop)))->(a->(b->(c->Prop)))).
% Parameter cCL:((a->(b->(c->Prop)))->Prop).
% Trying to prove (((and ((and (forall (S:((a->(b->(c->Prop)))->Prop)), ((forall (Xx:(a->(b->(c->Prop)))), ((S Xx)->(cCL Xx)))->(cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), ((S R)->(((R Xa) Xb) Xc)))))))) (forall (R:(a->(b->(c->Prop)))), ((cCL R)->(cCL (cF R)))))) (forall (R:(a->(b->(c->Prop)))) (S:(a->(b->(c->Prop)))), (((and ((and (cCL R)) (cCL S))) (forall (Xa:a) (Xb:b) (Xc:c), ((((R Xa) Xb) Xc)->(((S Xa) Xb) Xc))))->(forall (Xa:a) (Xb:b) (Xc:c), (((((cF R) Xa) Xb) Xc)->((((cF S) Xa) Xb) Xc))))))->((and ((and (cCL (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))))
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))):(((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))))
% Found (eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))):(((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))))
% Found (eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))):(((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))))
% Found (eq_ref0 (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) as proof of (((eq Prop) (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))) b0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found x2:(cCL Y)
% Found x2 as proof of (cCL Y)
% Found x2:(cCL Y)
% Found x2 as proof of (cCL Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eq_ref00:=(eq_ref0 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found (eq_ref0 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_ref00:=(eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found (eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found x30:=(x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))):(((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))) as proof of (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (fun (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))) as proof of (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (fun (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))) as proof of (forall (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))) as proof of (forall (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))) as proof of (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found ((conj10 x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found (((conj1 (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found (x10 ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))))) as proof of (((Y Xa) Xb) Xc)
% Found ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))) as proof of ((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->(((Y Xa) Xb) Xc))
% Found (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))) as proof of ((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->(((Y Xa) Xb) Xc)))
% Found (and_rect00 (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found ((and_rect0 (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))
% Found (fun (Xb:b) (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (Xa:a) (Xb:b) (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (x0:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (Y:(a->(b->(c->Prop)))) (x0:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))
% Found (fun (Y:(a->(b->(c->Prop)))) (x0:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x1:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x2:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x2) x0)) (((Y Xa) Xb) Xc)) (fun (x2:(cCL Y)) (x3:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x1 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x2) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x3 (fun (x4:(a->(b->(c->Prop))))=> (((x4 Xa0) Xb0) Xc0))))))))) as proof of (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion_dep00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b0
% Found eq_ref00:=(eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found (eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found eq_ref00:=(eq_ref0 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))):(((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (eq_ref0 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found ((eq_ref (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion000:=(eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion0 (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion_dep00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->(c->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found eq_ref00:=(eq_ref0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)):(((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (eq_ref0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found ((eq_ref (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found ((eq_ref (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found ((eq_ref (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (b->(c->Prop))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found ((eq_ref (b->(c->Prop))) b0) as proof of (((eq (b->(c->Prop))) b0) (fun (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) Xb) Xc)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)):(((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) (fun (x:b)=> (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) x)))
% Found (eta_expansion_dep00 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found ((eta_expansion_dep0 (fun (x2:b)=> (c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> (c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> (c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found (((eta_expansion_dep b) (fun (x2:b)=> (c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) as proof of (((eq (b->(c->Prop))) ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)) b0)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_ref00:=(eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found (eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P0 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P0 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion000:=(eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion0 (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found x4:(cCL Y)
% Found x4 as proof of (cCL Y)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))):(((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) (fun (x:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x) Xb) Xc)))))
% Found (eta_expansion_dep00 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:a)=> (b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion_dep a) (fun (x3:a)=> (b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found (fun (x00:(P (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))=> x00) as proof of (P0 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion000:=(eta_expansion00 (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)):(((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) (fun (x:c)=> ((((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y) x)))
% Found (eta_expansion00 (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) as proof of (((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) b0)
% Found ((eta_expansion0 Prop) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) as proof of (((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) b0)
% Found (((eta_expansion c) Prop) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) as proof of (((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) b0)
% Found (((eta_expansion c) Prop) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) as proof of (((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) b0)
% Found (((eta_expansion c) Prop) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) as proof of (((eq (c->Prop)) (((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0) y)) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (c->Prop)) b0) (fun (x:c)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (c->Prop)) b0) (fun (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) y) Xc)))))
% Found ((eta_expansion_dep0 (fun (x2:c)=> Prop)) b0) as proof of (((eq (c->Prop)) b0) (fun (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) y) Xc)))))
% Found (((eta_expansion_dep c) (fun (x2:c)=> Prop)) b0) as proof of (((eq (c->Prop)) b0) (fun (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) y) Xc)))))
% Found (((eta_expansion_dep c) (fun (x2:c)=> Prop)) b0) as proof of (((eq (c->Prop)) b0) (fun (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) y) Xc)))))
% Found (((eta_expansion_dep c) (fun (x2:c)=> Prop)) b0) as proof of (((eq (c->Prop)) b0) (fun (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x0) y) Xc)))))
% Found x4:(cCL Y)
% Found x4 as proof of (cCL Y)
% Found eq_ref00:=(eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Found (eq_ref0 (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eq_ref Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq Prop) (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion000:=(eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion0 (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found x0:(P (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))))
% Instantiate: b0:=(cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))):(a->(b->(c->Prop)))
% Found x0 as proof of (P0 b0)
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found (fun (x10:(P ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0)))=> x10) as proof of (P0 ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x0))
% Found eta_expansion000:=(eta_expansion00 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))):(((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) (fun (x:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R x) Xb) Xc)))))
% Found (eta_expansion00 (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found ((eta_expansion0 (b->(c->Prop))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) as proof of (((eq (a->(b->(c->Prop)))) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) b0)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found x50:=(x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))):(((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))) as proof of (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (fun (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))) as proof of (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))
% Found (fun (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))) as proof of (forall (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))) as proof of (forall (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))) as proof of (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))
% Found ((conj10 x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found (((conj1 (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))) as proof of ((and (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0))))
% Found (x30 ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))))) as proof of (((Y Xa) Xb) Xc)
% Found ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))) as proof of ((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->(((Y Xa) Xb) Xc))
% Found (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))) as proof of ((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->(((Y Xa) Xb) Xc)))
% Found (and_rect10 (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found ((and_rect1 (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0)))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (((Y Xa) Xb) Xc)
% Found (fun (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))
% Found (fun (Xb:b) (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (Xa:a) (Xb:b) (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (x2:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))
% Found (fun (Y:(a->(b->(c->Prop)))) (x2:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc))))
% Found (fun (Y:(a->(b->(c->Prop)))) (x2:((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))) (Xa:a) (Xb:b) (Xc:c) (x3:(forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))=> (((fun (P:Type) (x4:((cCL Y)->((((eq (a->(b->(c->Prop)))) (cF Y)) Y)->P)))=> (((((and_rect (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y)) P) x4) x2)) (((Y Xa) Xb) Xc)) (fun (x4:(cCL Y)) (x5:(((eq (a->(b->(c->Prop)))) (cF Y)) Y))=> ((x3 Y) ((((conj (cCL Y)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF Y) Xa0) Xb0) Xc0)->(((Y Xa0) Xb0) Xc0)))) x4) (fun (Xa0:a) (Xb0:b) (Xc0:c)=> (x5 (fun (x6:(a->(b->(c->Prop))))=> (((x6 Xa0) Xb0) Xc0))))))))) as proof of (forall (Y:(a->(b->(c->Prop)))), (((and (cCL Y)) (((eq (a->(b->(c->Prop)))) (cF Y)) Y))->(forall (Xa:a) (Xb:b) (Xc:c), ((forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))->(((Y Xa) Xb) Xc)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (a->(b->(c->Prop)))) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found ((eq_ref (a->(b->(c->Prop)))) b0) as proof of (((eq (a->(b->(c->Prop)))) b0) (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))
% Found eta_expansion000:=(eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))):(((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) (fun (x:a)=> ((cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc))))) x)))
% Found (eta_expansion00 (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found ((eta_expansion0 (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) b0)
% Found (((eta_expansion a) (b->(c->Prop))) (cF (fun (Xa:a) (Xb:b) (Xc:c)=> (forall (R:(a->(b->(c->Prop)))), (((and (cCL R)) (forall (Xa0:a) (Xb0:b) (Xc0:c), (((((cF R) Xa0) Xb0) Xc0)->(((R Xa0) Xb0) Xc0))))->(((R Xa) Xb) Xc)))))) as proof of (((eq (a->(b->(c->Prop)))) (cF (fun (Xa:a)
% EOF
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