%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : CSR139^1 : TPTP v6.1.0. Released v4.1.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n107.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 13:21:05 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  : CSR139^1 : TPTP v6.1.0. Released v4.1.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n107.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 10:07:21 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 0x1e71488>, <kernel.Type object at 0x1e71368>) of role type named numbers
% Using role type
% Declaring num:Type
% FOF formula (<kernel.Constant object at 0x205b488>, <kernel.Constant object at 0x1e71758>) of role type named lAnna_THFTYPE_i
% Using role type
% Declaring lAnna_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x1e716c8>, <kernel.Single object at 0x1e71ef0>) of role type named lBen_THFTYPE_i
% Using role type
% Declaring lBen_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x1e71488>, <kernel.Single object at 0x1e71368>) of role type named lBill_THFTYPE_i
% Using role type
% Declaring lBill_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x1e717a0>, <kernel.Single object at 0x1e71488>) of role type named lBob_THFTYPE_i
% Using role type
% Declaring lBob_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x1e716c8>, <kernel.Single object at 0x1e71488>) of role type named lMary_THFTYPE_i
% Using role type
% Declaring lMary_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x22c42d8>, <kernel.Single object at 0x1e717a0>) of role type named lSue_THFTYPE_i
% Using role type
% Declaring lSue_THFTYPE_i:fofType
% FOF formula (<kernel.Constant object at 0x22c42d8>, <kernel.DependentProduct object at 0x207a368>) of role type named likes_THFTYPE_IiioI
% Using role type
% Declaring likes_THFTYPE_IiioI:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1e71488>, <kernel.DependentProduct object at 0x207a368>) 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 (not ((likes_THFTYPE_IiioI lSue_THFTYPE_i) lMary_THFTYPE_i)) of role axiom named ax_001
% A new axiom: (not ((likes_THFTYPE_IiioI lSue_THFTYPE_i) lMary_THFTYPE_i))
% FOF formula ((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i) of role axiom named ax_002
% A new axiom: ((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i)
% FOF formula ((parent_THFTYPE_IiioI lMary_THFTYPE_i) lBen_THFTYPE_i) of role axiom named ax_003
% 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_004
% A new axiom: ((parent_THFTYPE_IiioI lSue_THFTYPE_i) lBen_THFTYPE_i)
% FOF formula (not ((parent_THFTYPE_IiioI lBob_THFTYPE_i) lBen_THFTYPE_i)) of role axiom named ax_005
% A new axiom: (not ((parent_THFTYPE_IiioI lBob_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 (not ((parent_THFTYPE_IiioI lBob_THFTYPE_i) lAnna_THFTYPE_i)) of role axiom named ax_009
% A new axiom: (not ((parent_THFTYPE_IiioI lBob_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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))):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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))']
% 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:(not ((likes_THFTYPE_IiioI lSue_THFTYPE_i) lMary_THFTYPE_i)).
% Axiom ax_002:((likes_THFTYPE_IiioI lMary_THFTYPE_i) lBill_THFTYPE_i).
% Axiom ax_003:((parent_THFTYPE_IiioI lMary_THFTYPE_i) lBen_THFTYPE_i).
% Axiom ax_004:((parent_THFTYPE_IiioI lSue_THFTYPE_i) lBen_THFTYPE_i).
% Axiom ax_005:(not ((parent_THFTYPE_IiioI lBob_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).
% Axiom ax_009:(not ((parent_THFTYPE_IiioI lBob_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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))):(((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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))))
% 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 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% 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 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 (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found ((eta_expansion0 Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) 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)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))):(((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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))))
% Found (eta_expansion00 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b0)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b0)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b0)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b0)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (not (((eq (fofType->(fofType->Prop))) x0) x))):(((eq Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) (not (((eq (fofType->(fofType->Prop))) x0) x)))
% Found (eq_ref0 (not (((eq (fofType->(fofType->Prop))) x0) x))) as proof of (((eq Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) b)
% Found ((eq_ref Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) as proof of (((eq Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) b)
% Found ((eq_ref Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) as proof of (((eq Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) b)
% Found ((eq_ref Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) as proof of (((eq Prop) (not (((eq (fofType->(fofType->Prop))) x0) x))) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) b0)
% Found or_introl:(forall (A:Prop) (B:Prop), (A->((or A) B)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->((or A) B))):Prop
% Found or_introl as proof of b
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found ax:((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i)
% Instantiate: b:=lBill_THFTYPE_i:fofType
% Found ax as proof of (P b)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found eq_ref00:=(eq_ref0 lBen_THFTYPE_i):(((eq fofType) lBen_THFTYPE_i) lBen_THFTYPE_i)
% Found (eq_ref0 lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found eq_ref00:=(eq_ref0 lAnna_THFTYPE_i):(((eq fofType) lAnna_THFTYPE_i) lAnna_THFTYPE_i)
% Found (eq_ref0 lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (not (((eq (fofType->(fofType->Prop))) x0) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not (((eq (fofType->(fofType->Prop))) x0) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not (((eq (fofType->(fofType->Prop))) x0) x)))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (not (((eq (fofType->(fofType->Prop))) x0) x)))
% Found ax:((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i)
% Instantiate: b:=lBill_THFTYPE_i:fofType
% Found ax as proof of (P b)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b)
% Found eq_ref00:=(eq_ref0 lAnna_THFTYPE_i):(((eq fofType) lAnna_THFTYPE_i) lAnna_THFTYPE_i)
% Found (eq_ref0 lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found ((eq_ref fofType) lAnna_THFTYPE_i) as proof of (((eq fofType) lAnna_THFTYPE_i) b)
% Found eq_ref00:=(eq_ref0 lBen_THFTYPE_i):(((eq fofType) lBen_THFTYPE_i) lBen_THFTYPE_i)
% Found (eq_ref0 lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) b)
% Found ((eq_ref fofType) lBen_THFTYPE_i) as proof of (((eq fofType) lBen_THFTYPE_i) 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)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x0:(P2 (f x))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x0:(P2 (f x)))=> x0) as proof of (P2 (f0 x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of ((P2 (f x))->(P2 (f0 x)))
% Found (fun (x:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (((eq Prop) (f x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x0:(P2 (f x)))=> x0) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (x0:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b)
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x0:(fofType->(fofType->Prop))), (((eq Prop) (f0 x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found ax_009:(not ((parent_THFTYPE_IiioI lBob_THFTYPE_i) lAnna_THFTYPE_i))
% Instantiate: a:=(not ((parent_THFTYPE_IiioI lBob_THFTYPE_i) lAnna_THFTYPE_i)):Prop
% Found ax_009 as proof of a
% Found ax_009:(not ((parent_THFTYPE_IiioI lBob_THFTYPE_i) lAnna_THFTYPE_i))
% Instantiate: b:=(((parent_THFTYPE_IiioI lBob_THFTYPE_i) lAnna_THFTYPE_i)->False):Prop
% Found ax_009 as proof of a
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) b0)
% 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 eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) (f x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:fofType), (((eq Prop) (f0 x)) (f x)))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found x11:(P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P2 (f x0))
% Found x11:(P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P2 (f x0))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b0)
% Found ax:((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i)
% Instantiate: a:=lBill_THFTYPE_i:fofType
% Found ax as proof of (P a)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) (fun (x:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% Found (eta_expansion_dep00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))) b)
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% 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) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% 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) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% 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) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (fun (x1:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% 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) (fun (Y:fofType)=> ((and ((and ((x Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x) x0)))))))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) 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_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 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 (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0)))))))
% Found ((eta_expansion_dep0 (fun (x2:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x2:(fofType->(fofType->Prop)))=> Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0)))))))
% Found x100:=(x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))):((P0 lMary_THFTYPE_i)->(P0 lMary_THFTYPE_i))
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f1 x)):(((eq Prop) (f1 x)) (f1 x))
% Found (eq_ref0 (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found ((eq_ref Prop) (f1 x)) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (((eq Prop) (f1 x)) (f0 x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f1 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f1 x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found eta_expansion000:=(eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found ((eta_expansion0 Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found x20:(P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P2 (f x1))
% Found x20:(P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P2 (f x1))
% 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)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found x11:(P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P2 (f x0))
% Found x11:(P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P1 (f x0))
% Found (fun (x11:(P1 (f x0)))=> x11) as proof of (P2 (f x0))
% Found ax:((likes_THFTYPE_IiioI lSue_THFTYPE_i) lBill_THFTYPE_i)
% Instantiate: a:=lBill_THFTYPE_i:fofType
% Found ax as proof of (P a)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b1)
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% 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)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lAnna_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) lBen_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) b)
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x1:(P2 (f x0))
% Instantiate: f0:=f:((fofType->(fofType->Prop))->Prop)
% Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (f0 x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (f0 x0)))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (f0 x0))
% Found (fun (x0:(fofType->(fofType->Prop))) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f x)) (f0 x)))
% Found x100:=(x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))):((P0 lMary_THFTYPE_i)->(P0 lMary_THFTYPE_i))
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found x100:=(x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))):((P0 lMary_THFTYPE_i)->(P0 lMary_THFTYPE_i))
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (eq_ref0 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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 ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found eta_expansion000:=(eta_expansion00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eta_expansion0 Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))):(((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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q)))))))))
% Found (eq_ref0 (fun (Q:(fofType->(fofType->Prop)))=> ((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b1)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b1)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b1)
% 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) 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 ((R Y) lBill_THFTYPE_i)) ((Q Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) Q))))))))) b1)
% 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 (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% 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 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 eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (x:(fofType->(fofType->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) b) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% 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 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_expansion000:=(eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found ((eta_expansion0 Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found (((eta_expansion fofType) Prop) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% 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) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found x20:(P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P2 (f x1))
% Found x20:(P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P1 (f x1))
% Found (fun (x20:(P1 (f x1)))=> x20) as proof of (P2 (f x1))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):(((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) (fun (x1:fofType)=> ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (eta_expansion_dep00 (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found ((eta_expansion_dep0 (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found (((eta_expansion_dep fofType) (fun (x2:fofType)=> Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) as proof of (((eq (fofType->Prop)) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% 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)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x1:fofType), (((eq Prop) (f x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x1:fofType), (((eq Prop) (f0 x1)) ((and ((and ((x0 x1) lBill_THFTYPE_i)) ((x x1) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) lMary_THFTYPE_i)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x1 Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x1) x0)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x1))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% 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) b0)
% Found ((eta_expansion0 Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x)))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found x1:(P1 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found eta_expansion000:=(eta_expansion00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq ((fofType->(fofType->Prop))->Prop)) a0) (fun (x:(fofType->(fofType->Prop)))=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) a0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eq_ref ((fofType->(fofType->Prop))->Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) a)
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:(fofType->(fofType->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:(fofType->(fofType->Prop))), (((eq Prop) (f0 x)) (f x)))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found x01:(P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P2 b)
% Found x01:(P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P2 b)
% Found x01:(P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P1 b)
% Found (fun (x01:(P1 b))=> x01) as proof of (P2 b)
% Found x12:(P1 (f x0))
% Found (fun (x12:(P1 (f x0)))=> x12) as proof of (P1 (f x0))
% Found (fun (x12:(P1 (f x0)))=> x12) as proof of (P2 (f x0))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found x12:(P1 (f x0))
% Found (fun (x12:(P1 (f x0)))=> x12) as proof of (P1 (f x0))
% Found (fun (x12:(P1 (f x0)))=> x12) as proof of (P2 (f x0))
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found eq_ref00:=(eq_ref0 b0):(((eq fofType) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found ((eq_ref fofType) b0) as proof of (((eq fofType) b0) b)
% Found eq_ref00:=(eq_ref0 lMary_THFTYPE_i):(((eq fofType) lMary_THFTYPE_i) lMary_THFTYPE_i)
% Found (eq_ref0 lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found ((eq_ref fofType) lMary_THFTYPE_i) as proof of (((eq fofType) lMary_THFTYPE_i) b0)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found x1:(P2 b)
% Instantiate: b:=((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (f x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (f x0)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))):(((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found (eq_ref0 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) as proof of (((eq Prop) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))) b)
% Found x1:(P2 b)
% Instantiate: b:=((ex (fofType->(fofType->Prop))) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x0 Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x0))))))):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (f x0))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (f x0)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))
% Found x100:=(x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))):((P0 lMary_THFTYPE_i)->(P0 lMary_THFTYPE_i))
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found (x10 (fun (x2:(fofType->(fofType->Prop)))=> (P0 lMary_THFTYPE_i))) as proof of (P1 lMary_THFTYPE_i)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))):(((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) (fun (x0:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x)))))))
% Found (eta_expansion00 (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found ((eta_expansion0 Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found (((eta_expansion (fofType->(fofType->Prop))) Prop) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) (fun (R:(fofType->(fofType->Prop)))=> ((ex fofType) (fun (Y:fofType)=> ((and ((and ((R Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) R) x))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq ((fofType->(fofType->Prop))->Prop)) b0) (fun (x:(fofType->(fofType->Prop)))=> (b0 x)))
% Found (eta_expansion_dep00 b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found (((eta_expansion_dep (fofType->(fofType->Prop))) (fun (x1:(fofType->(fofType->Prop)))=> Prop)) b0) as proof of (((eq ((fofType->(fofType->Prop))->Prop)) b0) b)
% Found x1:(P1 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Instantiate: b:=((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Instantiate: b:=((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))):Prop
% Found x1 as proof of (P2 b)
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 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)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill_THFTYPE_i)) ((x Y) lAnna_THFTYPE_i))) (not (((eq (fofType->(fofType->Prop))) x0) x))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((ex fofType) (fun (Y:fofType)=> ((and ((and ((x0 Y) lBill
% EOF
%------------------------------------------------------------------------------