TSTP Solution File: CSR138^1 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : CSR138^1 : TPTP v6.1.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n095.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 13:21:04 EDT 2014
% Result : Timeout 300.02s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : CSR138^1 : TPTP v6.1.0. Released v4.1.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n095.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 10:07:11 CDT 2014
% % CPUTime : 300.02
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x16a9bd8>, <kernel.Type object at 0x16a9cb0>) of role type named numbers
% Using role type
% Declaring num:Type
% FOF formula (<kernel.Constant object at 0x12ce9e0>, <kernel.Constant object at 0x16a9d40>) of role type named lAnna_THFTYPE_i
% Using role type
% Declaring lAnna_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x12ce9e0>, <kernel.Single object at 0x16a9dd0>) of role type named lBen_THFTYPE_i
% Using role type
% Declaring lBen_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x16a9bd8>, <kernel.Single object at 0x16a9a28>) of role type named lBill_THFTYPE_i
% Using role type
% Declaring lBill_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x16a9ab8>, <kernel.Single object at 0x16a9cb0>) of role type named lBob_THFTYPE_i
% Using role type
% Declaring lBob_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x16a9c20>, <kernel.Single object at 0x16a9908>) of role type named lMary_THFTYPE_i
% Using role type
% Declaring lMary_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x16a9bd8>, <kernel.Single object at 0x16a9a70>) of role type named lSue_THFTYPE_i
% Using role type
% Declaring lSue_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x16a9ab8>, <kernel.DependentProduct object at 0x16a9c20>) of role type named likes_THFTYPE_IiioI
% Using role type
% Declaring likes_THFTYPE_IiioI:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x16a9950>, <kernel.DependentProduct object at 0x16a9bd8>) of role type named parent_THFTYPE_IiioI
% Using role type
% Declaring parent_THFTYPE_IiioI:(fofType->(fofType->Prop))
% FOF formula ((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i) of role axiom named ax
% A new axiom: ((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i)
% FOF formula ((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i) of role axiom named ax_001
% A new axiom: ((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i)
% FOF formula ((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI X) Y)))))) of role axiom named ax_002
% A new axiom: ((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI X) Y))))))
% FOF formula ((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI X) Y)))))) of role axiom named ax_003
% A new axiom: ((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI X) Y))))))
% FOF formula ((parent_THFTYPE_IiioI lMary_THFTYPE_i) lBen_THFTYPE_i) of role axiom named ax_004
% A new axiom: ((parent_THFTYPE_IiioI lMary_THFTYPE_i) lBen_THFTYPE_i)
% FOF formula ((parent_THFTYPE_IiioI lSue_THFTYPE_i) lBen_THFTYPE_i) of role axiom named ax_005
% A new axiom: ((parent_THFTYPE_IiioI lSue_THFTYPE_i) lBen_THFTYPE_i)
% FOF formula ((likes_THFTYPE_IiioI lBob_THFTYPE_i) lBill_THFTYPE_i) of role axiom named ax_006
% A new axiom: ((likes_THFTYPE_IiioI lBob_THFTYPE_i) lBill_THFTYPE_i)
% FOF formula ((parent_THFTYPE_IiioI lSue_THFTYPE_i) lAnna_THFTYPE_i) of role axiom named ax_007
% A new axiom: ((parent_THFTYPE_IiioI lSue_THFTYPE_i) lAnna_THFTYPE_i)
% FOF formula ((parent_THFTYPE_IiioI lMary_THFTYPE_i) lAnna_THFTYPE_i) of role axiom named ax_008
% A new axiom: ((parent_THFTYPE_IiioI lMary_THFTYPE_i) lAnna_THFTYPE_i)
% FOF formula ((ex (fofType->(fofType->Prop))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) of role conjecture named con
% Conjecture to prove = ((ex (fofType->(fofType->Prop))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):Prop
% Parameter num_DUMMY:num.
% We need to prove ['((ex (fofType->(fofType->Prop))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))']
% Parameter num:Type.
% Parameter fofType:Type.
% Parameter lAnna_THFTYPE_i:fofType.
% Parameter lBen_THFTYPE_i:fofType.
% Parameter lBill_THFTYPE_i:fofType.
% Parameter lBob_THFTYPE_i:fofType.
% Parameter lMary_THFTYPE_i:fofType.
% Parameter lSue_THFTYPE_i:fofType.
% Parameter likes_THFTYPE_IiioI:(fofType->(fofType->Prop)).
% Parameter parent_THFTYPE_IiioI:(fofType->(fofType->Prop)).
% Axiom ax:((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i).
% Axiom ax_001:((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i).
% Axiom ax_002:((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI X) Y)))))).
% Axiom ax_003:((ex fofType) (fun (X:fofType)=> ((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI X) Y)))))).
% Axiom ax_004:((parent_THFTYPE_IiioI lMary_THFTYPE_i) lBen_THFTYPE_i).
% Axiom ax_005:((parent_THFTYPE_IiioI lSue_THFTYPE_i) lBen_THFTYPE_i).
% Axiom ax_006:((likes_THFTYPE_IiioI lBob_THFTYPE_i) lBill_THFTYPE_i).
% Axiom ax_007:((parent_THFTYPE_IiioI lSue_THFTYPE_i) lAnna_THFTYPE_i).
% Axiom ax_008:((parent_THFTYPE_IiioI lMary_THFTYPE_i) lAnna_THFTYPE_i).
% Trying to prove ((ex (fofType->(fofType->Prop))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% 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)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% 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)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% 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)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x4 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x4) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x4:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eta_expansion_dep00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x7:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x7:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x7:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x7:(fofType->(fofType->Prop)))=> Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x6)):(((eq Prop) (f x6)) (f x6))
% Found (eq_ref0 (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x6)) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (((eq Prop) (f x6)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x6 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x6) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x6:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x6))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x5 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x5) (fun (Z:fofType) (W:fofType)=> True)))))))))
% Found (fun (x5:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x3:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x2 Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x2:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x2))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq ((fofType->(fofType->Prop))->Prop)) a) (fun (x:(fofType->(fofType->Prop)))=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) Q) (fun (Z:fofType) (W:fofType)=> True))))))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x00 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x00) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x00:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x0:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x0:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x4:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x5:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x5:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x5:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x3 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found x20:((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI x1) Y))))
% Found (fun (x20:((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI x1) Y)))))=> x20) as proof of ((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI x1) Y))))
% Found (fun (x20:((ex fofType) (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI x1) Y)))))=> x20) as proof of (P (fun (Y:fofType)=> (not ((parent_THFTYPE_IiioI x1) Y))))
% Found x20:((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI x1) Y))))
% Found (fun (x20:((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI x1) Y)))))=> x20) as proof of ((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI x1) Y))))
% Found (fun (x20:((ex fofType) (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI x1) Y)))))=> x20) as proof of (P (fun (Y:fofType)=> (not ((likes_THFTYPE_IiioI x1) Y))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x3 Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x3) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True)))))))
% Found (fun (x3:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((x Y) lBill_THFTYPE_i)) ((x2 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x2) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True))))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) (fun (Z:fofType) (W:fofType)=> True))))) (not (((eq (fofType->(fofType->Prop))) x1) (fun (Z:fofType) (W:fofType)=> True)))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((and ((R Y) lBill_THFTYPE_i)) ((x1 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType
% EOF
%------------------------------------------------------------------------------