TSTP Solution File: SEV242^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV242^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n092.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 300.01s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV242^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.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 08:36:01 CDT 2014
% % CPUTime : 300.01
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((forall (X:fofType), ((iff (P X)) (Q X)))->(((eq (fofType->Prop)) P) Q)))->(forall (T:((fofType->Prop)->Prop)) (U:(fofType->Prop)) (V:(fofType->Prop)), ((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))))) of role conjecture named cTHM4A_pme
% Conjecture to prove = ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((forall (X:fofType), ((iff (P X)) (Q X)))->(((eq (fofType->Prop)) P) Q)))->(forall (T:((fofType->Prop)->Prop)) (U:(fofType->Prop)) (V:(fofType->Prop)), ((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((forall (X:fofType), ((iff (P X)) (Q X)))->(((eq (fofType->Prop)) P) Q)))->(forall (T:((fofType->Prop)->Prop)) (U:(fofType->Prop)) (V:(fofType->Prop)), ((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))))))']
% Parameter fofType:Type.
% Trying to prove ((forall (P:(fofType->Prop)) (Q:(fofType->Prop)), ((forall (X:fofType), ((iff (P X)) (Q X)))->(((eq (fofType->Prop)) P) Q)))->(forall (T:((fofType->Prop)->Prop)) (U:(fofType->Prop)) (V:(fofType->Prop)), ((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 T):((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eq_ref000:=(eq_ref00 T):((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))))
% Found (eq_ref00 T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 T):((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 T):((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found x0:(T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Instantiate: P:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))):(fofType->Prop)
% Found x0 as proof of (P0 P)
% Found iff_refl0:=(iff_refl (P X)):((iff (P X)) (P X))
% Found (iff_refl (P X)) as proof of ((iff (P X)) ((or (U X)) (V X)))
% Found (iff_refl (P X)) as proof of ((iff (P X)) ((or (U X)) (V X)))
% Found (fun (X:fofType)=> (iff_refl (P X))) as proof of ((iff (P X)) ((or (U X)) (V X)))
% Found (fun (X:fofType)=> (iff_refl (P X))) as proof of (forall (X:fofType), ((iff (P X)) ((or (U X)) (V X))))
% Found x0:(T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Instantiate: b:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))):(fofType->Prop)
% Found x0 as proof of (P b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) (fun (x:fofType)=> ((or (U x)) (V x))))
% Found (eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found ((eta_expansion0 Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found x0:(T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))):(fofType->Prop)
% Found x0 as proof of (P f)
% 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)) ((or (U x1)) (V x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((or (U x)) (V x))))
% Found x0:(T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Instantiate: f:=(fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))):(fofType->Prop)
% Found x0 as proof of (P f)
% 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)) ((or (U x1)) (V x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((or (U x1)) (V x1)))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) ((or (U x)) (V x))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 T):((T (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(T (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) T) as proof of (P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion0 Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion0 Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) (fun (x:fofType)=> ((or (U x)) (V x))))
% Found (eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eq_ref00:=(eq_ref0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))):(((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (eq_ref0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) as proof of (((eq (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion0 Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion0000:=(eta_expansion000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion0 Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion fofType) Prop) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eq_ref (fofType->Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (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) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))->(P0 (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x)))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found (((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found ((((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx)))))) P0) as proof of (P1 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((or (U x0)) (V x0)))->(P0 ((or (U x0)) (V x0))))
% Found (eq_ref00 P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found ((eq_ref0 ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((or (U x0)) (V x0)))->(P0 ((or (U x0)) (V x0))))
% Found (eq_ref00 P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found ((eq_ref0 ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((or (U x0)) (V x0)))->(P0 ((or (U x0)) (V x0))))
% Found (eq_ref00 P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found ((eq_ref0 ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((or (U x0)) (V x0)))->(P0 ((or (U x0)) (V x0))))
% Found (eq_ref00 P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found ((eq_ref0 ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% Found (((eq_ref Prop) ((or (U x0)) (V x0))) P0) as proof of (P1 ((or (U x0)) (V x0)))
% 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))->(P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))))
% Found (eq_ref00 P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) P0) as proof of (P1 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((or (U x0)) (V x0)))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0))))) b)
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found eq_ref00:=(eq_ref0 ((or (U x0)) (V x0))):(((eq Prop) ((or (U x0)) (V x0))) ((or (U x0)) (V x0)))
% Found (eq_ref0 ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) b)
% Found ((eq_ref Prop) ((or (U x0)) (V x0))) as proof of (((eq Prop) ((or (U x0)) (V x0))) 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 ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and ((or (((eq (fofType->Prop)) S) U)) (((eq (fofType->Prop)) S) V))) (S x0)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found (((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))->(P0 (fun (x:fofType)=> ((or (U x)) (V x)))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz))))
% Found ((eta_expansion_dep00 (fun (Xz:fofType)=> ((or (U Xz)) (V Xz)))) P0) as proof of (P1
% EOF
%------------------------------------------------------------------------------