TSTP Solution File: SEV234^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV234^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n097.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:55 EDT 2014
% Result : Timeout 300.11s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV234^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n097.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 08:34:41 CDT 2014
% % CPUTime : 300.11
% 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 0xa4d830>, <kernel.DependentProduct object at 0x86a488>) of role type named cOPEN
% Using role type
% Declaring cOPEN:((fofType->Prop)->Prop)
% FOF formula ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (B:(fofType->Prop)), ((forall (Xx:fofType), ((B Xx)->((ex (fofType->Prop)) (fun (D:(fofType->Prop))=> ((and ((and (cOPEN D)) (D Xx))) (forall (Xx0:fofType), ((D Xx0)->(B Xx0))))))))->(cOPEN B)))) of role conjecture named cBLEDSOE_FENG_SV_10_pme
% Conjecture to prove = ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (B:(fofType->Prop)), ((forall (Xx:fofType), ((B Xx)->((ex (fofType->Prop)) (fun (D:(fofType->Prop))=> ((and ((and (cOPEN D)) (D Xx))) (forall (Xx0:fofType), ((D Xx0)->(B Xx0))))))))->(cOPEN B)))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (B:(fofType->Prop)), ((forall (Xx:fofType), ((B Xx)->((ex (fofType->Prop)) (fun (D:(fofType->Prop))=> ((and ((and (cOPEN D)) (D Xx))) (forall (Xx0:fofType), ((D Xx0)->(B Xx0))))))))->(cOPEN B))))']
% Parameter fofType:Type.
% Parameter cOPEN:((fofType->Prop)->Prop).
% Trying to prove ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (B:(fofType->Prop)), ((forall (Xx:fofType), ((B Xx)->((ex (fofType->Prop)) (fun (D:(fofType->Prop))=> ((and ((and (cOPEN D)) (D Xx))) (forall (Xx0:fofType), ((D Xx0)->(B Xx0))))))))->(cOPEN B))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 B):(((eq (fofType->Prop)) B) B)
% Found (eq_ref0 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (B x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (B x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (B x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eta_expansion000:=(eta_expansion00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion0 Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref00:=(eq_ref0 B):(((eq (fofType->Prop)) B) B)
% Found (eq_ref0 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))):(((eq Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref0 (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) B) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref00:=(eq_ref0 B):(((eq (fofType->Prop)) B) B)
% Found (eq_ref0 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eta_expansion000:=(eta_expansion00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion0 Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% 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: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) iff_sym) as proof of (P b)
% Found (((conj0 b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) iff_sym) as proof of (P b)
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) iff_sym) as proof of (P b)
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) iff_sym) as proof of (P b)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found x2:(P B)
% Instantiate: b:=B:(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) b)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) b)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) b)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) b)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) b))) as proof of (P b)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) b))) as proof of (P b)
% Found (((x b) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) b))) as proof of (P b)
% Found (((x b) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) b))) as proof of (P b)
% Found x2:(P B)
% Instantiate: b:=B:(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found x2:(P B)
% Instantiate: f:=B:(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found x2:(P B)
% Instantiate: f:=B:(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq (fofType->Prop)) f) (fun (x:fofType)=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f))) as proof of (P f)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f))) as proof of (P f)
% Found (((x f) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f))) as proof of (P f)
% Found (((x f) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) f))) as proof of (P f)
% Found eq_ref00:=(eq_ref0 f):(((eq (fofType->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) f) as proof of (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) f)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) f))) as proof of (P f)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) f))) as proof of (P f)
% Found (((x f) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) f))) as proof of (P f)
% Found (((x f) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) f) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) f))) as proof of (P f)
% Found x2:(P B)
% Instantiate: f:=B:(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found x2:(P B)
% Instantiate: f:=B:(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x3)))))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eta_expansion000:=(eta_expansion00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion0 Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion_dep00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref000:=(eq_ref00 P):((P B)->(P B))
% Found (eq_ref00 P) as proof of (P0 B)
% Found ((eq_ref0 B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found (((eq_ref (fofType->Prop)) B) P) as proof of (P0 B)
% Found eta_expansion000:=(eta_expansion00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion0 Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion000:=(eta_expansion00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion0 Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion fofType) Prop) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found ((eq_ref (fofType->Prop)) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion_dep00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: b:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 B):(((eq (fofType->Prop)) B) (fun (x:fofType)=> (B x)))
% Found (eta_expansion_dep00 B) as proof of (((eq (fofType->Prop)) B) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) B) as proof of (((eq (fofType->Prop)) B) b)
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P b)->(P (fun (x:fofType)=> (b x))))
% Found (eta_expansion_dep000 P) as proof of (P0 b)
% Found ((eta_expansion_dep00 b) P) as proof of (P0 b)
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) P) as proof of (P0 b)
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found x2:(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))):(fofType->Prop)
% Found x2 as proof of (P0 f)
% Found x3:(P (B x2))
% Instantiate: b:=(B x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found x3:(P (B x2))
% Instantiate: b:=(B x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) (B x3))
% Found (fun (x3:fofType)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (B x)))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found x3:(P (B x2))
% Instantiate: b:=(B x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found x3:(P (B x2))
% Instantiate: b:=(B x2):Prop
% Found x3 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of (P0 b)
% Found ((eq_ref0 b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found (((eq_ref (fofType->Prop)) b) P) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 B):(((eq (fofType->Prop)) B) B)
% Found (eq_ref0 B) as proof of (((eq (fofType->Prop)) B) b0)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b0)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b0)
% Found ((eq_ref (fofType->Prop)) B) as proof of (((eq (fofType->Prop)) B) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion_dep000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) B)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P1):((P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P1 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P1) as proof of (P2 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P):((P (B x2))->(P (B x2)))
% Found (eq_ref00 P) as proof of (P0 (B x2))
% Found ((eq_ref0 (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found (((eq_ref Prop) (B x2)) P) as proof of (P0 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref000:=(eq_ref00 P1):((P1 (B x2))->(P1 (B x2)))
% Found (eq_ref00 P1) as proof of (P2 (B x2))
% Found ((eq_ref0 (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found (((eq_ref Prop) (B x2)) P1) as proof of (P2 (B x2))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (B x2)):(((eq Prop) (B x2)) (B x2))
% Found (eq_ref0 (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found ((eq_ref Prop) (B x2)) as proof of (((eq Prop) (B x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq (fofType->Prop)) b0) (fun (x:fofType)=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found (((eta_expansion fofType) Prop) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion (fofType->Prop)) Prop) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->Prop)->Prop)) a) (fun (x:(fofType->Prop))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found ((eta_expansion_dep0 (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x5:(fofType->Prop))=> Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eta_expansion0000:=(eta_expansion000 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x)))))))
% Found (eta_expansion000 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found ((eq_ref Prop) b) as proof of (P b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (B x2))
% Found eq_ref00:=(eq_ref0 b0):(((eq (fofType->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found ((eq_ref (fofType->Prop)) b0) as proof of (((eq (fofType->Prop)) b0) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=
% EOF
%------------------------------------------------------------------------------