TSTP Solution File: QUA010^1 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : QUA010^1 : TPTP v6.1.0. Released v4.1.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n190.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 13:29:03 EDT 2014
% Result : Timeout 300.06s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : QUA010^1 : TPTP v6.1.0. Released v4.1.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.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 07:25:26 CDT 2014
% % CPUTime : 300.06
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/QUA001^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0xf38320>, <kernel.DependentProduct object at 0xf38e18>) of role type named emptyset_type
% Using role type
% Declaring emptyset:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False)) of role definition named emptyset_def
% A new definition: (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False))
% Defined: emptyset:=(fun (X:fofType)=> False)
% FOF formula (<kernel.Constant object at 0xf38320>, <kernel.DependentProduct object at 0xf38fc8>) of role type named union_type
% Using role type
% Declaring union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))) of role definition named union_def
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))))
% Defined: union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))
% FOF formula (<kernel.Constant object at 0xf38fc8>, <kernel.DependentProduct object at 0xf38518>) of role type named singleton_type
% Using role type
% Declaring singleton:(fofType->(fofType->Prop))
% FOF formula (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))) of role definition named singleton_def
% A new definition: (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)))
% Defined: singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))
% FOF formula (<kernel.Constant object at 0xf38e18>, <kernel.Single object at 0xf382d8>) of role type named zero_type
% Using role type
% Declaring zero:fofType
% FOF formula (<kernel.Constant object at 0xf38680>, <kernel.DependentProduct object at 0xd60b90>) of role type named sup_type
% Using role type
% Declaring sup:((fofType->Prop)->fofType)
% FOF formula (((eq fofType) (sup emptyset)) zero) of role axiom named sup_es
% A new axiom: (((eq fofType) (sup emptyset)) zero)
% FOF formula (forall (X:fofType), (((eq fofType) (sup (singleton X))) X)) of role axiom named sup_singleset
% A new axiom: (forall (X:fofType), (((eq fofType) (sup (singleton X))) X))
% FOF formula (<kernel.Constant object at 0xf38e18>, <kernel.DependentProduct object at 0xd60cb0>) of role type named supset_type
% Using role type
% Declaring supset:(((fofType->Prop)->Prop)->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->Prop)->(fofType->Prop))) supset) (fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (((eq fofType) (sup Y)) X)))))) of role definition named supset
% A new definition: (((eq (((fofType->Prop)->Prop)->(fofType->Prop))) supset) (fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (((eq fofType) (sup Y)) X))))))
% Defined: supset:=(fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (((eq fofType) (sup Y)) X)))))
% FOF formula (<kernel.Constant object at 0xf38e18>, <kernel.DependentProduct object at 0xd60dd0>) of role type named unionset_type
% Using role type
% Declaring unionset:(((fofType->Prop)->Prop)->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->Prop)->(fofType->Prop))) unionset) (fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (Y X)))))) of role definition named unionset
% A new definition: (((eq (((fofType->Prop)->Prop)->(fofType->Prop))) unionset) (fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (Y X))))))
% Defined: unionset:=(fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (Y X)))))
% FOF formula (forall (X:((fofType->Prop)->Prop)), (((eq fofType) (sup (supset X))) (sup (unionset X)))) of role axiom named sup_set
% A new axiom: (forall (X:((fofType->Prop)->Prop)), (((eq fofType) (sup (supset X))) (sup (unionset X))))
% FOF formula (<kernel.Constant object at 0xd60b00>, <kernel.DependentProduct object at 0xd60fc8>) of role type named addition_type
% Using role type
% Declaring addition:(fofType->(fofType->fofType))
% FOF formula (((eq (fofType->(fofType->fofType))) addition) (fun (X:fofType) (Y:fofType)=> (sup ((union (singleton X)) (singleton Y))))) of role definition named addition_def
% A new definition: (((eq (fofType->(fofType->fofType))) addition) (fun (X:fofType) (Y:fofType)=> (sup ((union (singleton X)) (singleton Y)))))
% Defined: addition:=(fun (X:fofType) (Y:fofType)=> (sup ((union (singleton X)) (singleton Y))))
% FOF formula (<kernel.Constant object at 0xd60fc8>, <kernel.DependentProduct object at 0xd60ea8>) of role type named order_type
% Using role type
% Declaring leq:(fofType->(fofType->Prop))
% FOF formula (forall (X1:fofType) (X2:fofType), ((iff ((leq X1) X2)) (((eq fofType) ((addition X1) X2)) X2))) of role axiom named order_def
% A new axiom: (forall (X1:fofType) (X2:fofType), ((iff ((leq X1) X2)) (((eq fofType) ((addition X1) X2)) X2)))
% FOF formula (<kernel.Constant object at 0xd60dd0>, <kernel.DependentProduct object at 0xd60d40>) of role type named multiplication_type
% Using role type
% Declaring multiplication:(fofType->(fofType->fofType))
% FOF formula (<kernel.Constant object at 0xd608c0>, <kernel.DependentProduct object at 0xd603b0>) of role type named crossmult_type
% Using role type
% Declaring crossmult:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) crossmult) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (A:fofType)=> ((ex fofType) (fun (X1:fofType)=> ((ex fofType) (fun (Y1:fofType)=> ((and ((and (X X1)) (Y Y1))) (((eq fofType) A) ((multiplication X1) Y1))))))))) of role definition named crossmult_def
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) crossmult) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (A:fofType)=> ((ex fofType) (fun (X1:fofType)=> ((ex fofType) (fun (Y1:fofType)=> ((and ((and (X X1)) (Y Y1))) (((eq fofType) A) ((multiplication X1) Y1)))))))))
% Defined: crossmult:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (A:fofType)=> ((ex fofType) (fun (X1:fofType)=> ((ex fofType) (fun (Y1:fofType)=> ((and ((and (X X1)) (Y Y1))) (((eq fofType) A) ((multiplication X1) Y1))))))))
% FOF formula (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((eq fofType) ((multiplication (sup X)) (sup Y))) (sup ((crossmult X) Y)))) of role axiom named multiplication_def
% A new axiom: (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((eq fofType) ((multiplication (sup X)) (sup Y))) (sup ((crossmult X) Y))))
% FOF formula (<kernel.Constant object at 0x11919e0>, <kernel.Single object at 0xd60560>) of role type named one_type
% Using role type
% Declaring one:fofType
% FOF formula (forall (X:fofType), (((eq fofType) ((multiplication X) one)) X)) of role axiom named multiplication_neutral_right
% A new axiom: (forall (X:fofType), (((eq fofType) ((multiplication X) one)) X))
% FOF formula (forall (X:fofType), (((eq fofType) ((multiplication one) X)) X)) of role axiom named multiplication_neutral_left
% A new axiom: (forall (X:fofType), (((eq fofType) ((multiplication one) X)) X))
% FOF formula (forall (X:(fofType->Prop)), (((eq fofType) (sup ((union X) (singleton zero)))) (sup X))) of role conjecture named zero_least
% Conjecture to prove = (forall (X:(fofType->Prop)), (((eq fofType) (sup ((union X) (singleton zero)))) (sup X))):Prop
% We need to prove ['(forall (X:(fofType->Prop)), (((eq fofType) (sup ((union X) (singleton zero)))) (sup X)))']
% Parameter fofType:Type.
% Definition emptyset:=(fun (X:fofType)=> False):(fofType->Prop).
% Definition union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)):(fofType->(fofType->Prop)).
% Parameter zero:fofType.
% Parameter sup:((fofType->Prop)->fofType).
% Axiom sup_es:(((eq fofType) (sup emptyset)) zero).
% Axiom sup_singleset:(forall (X:fofType), (((eq fofType) (sup (singleton X))) X)).
% Definition supset:=(fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (((eq fofType) (sup Y)) X))))):(((fofType->Prop)->Prop)->(fofType->Prop)).
% Definition unionset:=(fun (F:((fofType->Prop)->Prop)) (X:fofType)=> ((ex (fofType->Prop)) (fun (Y:(fofType->Prop))=> ((and (F Y)) (Y X))))):(((fofType->Prop)->Prop)->(fofType->Prop)).
% Axiom sup_set:(forall (X:((fofType->Prop)->Prop)), (((eq fofType) (sup (supset X))) (sup (unionset X)))).
% Definition addition:=(fun (X:fofType) (Y:fofType)=> (sup ((union (singleton X)) (singleton Y)))):(fofType->(fofType->fofType)).
% Parameter leq:(fofType->(fofType->Prop)).
% Axiom order_def:(forall (X1:fofType) (X2:fofType), ((iff ((leq X1) X2)) (((eq fofType) ((addition X1) X2)) X2))).
% Parameter multiplication:(fofType->(fofType->fofType)).
% Definition crossmult:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (A:fofType)=> ((ex fofType) (fun (X1:fofType)=> ((ex fofType) (fun (Y1:fofType)=> ((and ((and (X X1)) (Y Y1))) (((eq fofType) A) ((multiplication X1) Y1)))))))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Axiom multiplication_def:(forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((eq fofType) ((multiplication (sup X)) (sup Y))) (sup ((crossmult X) Y)))).
% Parameter one:fofType.
% Axiom multiplication_neutral_right:(forall (X:fofType), (((eq fofType) ((multiplication X) one)) X)).
% Axiom multiplication_neutral_left:(forall (X:fofType), (((eq fofType) ((multiplication one) X)) X)).
% Trying to prove (forall (X:(fofType->Prop)), (((eq fofType) (sup ((union X) (singleton zero)))) (sup X)))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found eq_ref00:=(eq_ref0 (sup ((union X) (singleton zero)))):(((eq fofType) (sup ((union X) (singleton zero)))) (sup ((union X) (singleton zero))))
% Found (eq_ref0 (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))):((P (sup ((union X) (singleton zero))))->(P (sup ((union X) (singleton zero)))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found (sup_es (fun (x:fofType)=> (P (sup ((union X) (singleton zero)))))) as proof of (P0 (sup ((union X) (singleton zero))))
% Found eq_ref00:=(eq_ref0 (sup ((union X) (singleton zero)))):(((eq fofType) (sup ((union X) (singleton zero)))) (sup ((union X) (singleton zero))))
% Found (eq_ref0 (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton zero)))) as proof of (((eq fofType) (sup ((union X) (singleton zero)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton zero))):(((eq (fofType->Prop)) ((union X) (singleton zero))) (fun (x:fofType)=> (((union X) (singleton zero)) x)))
% Found (eta_expansion00 ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found eq_ref00:=(eq_ref0 (sup ((union X) (singleton (sup emptyset))))):(((eq fofType) (sup ((union X) (singleton (sup emptyset))))) (sup ((union X) (singleton (sup emptyset)))))
% Found (eq_ref0 (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found eq_ref00:=(eq_ref0 (sup ((union X) (singleton (sup emptyset))))):(((eq fofType) (sup ((union X) (singleton (sup emptyset))))) (sup ((union X) (singleton (sup emptyset)))))
% Found (eq_ref0 (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found ((eq_ref fofType) (sup ((union X) (singleton (sup emptyset))))) as proof of (((eq fofType) (sup ((union X) (singleton (sup emptyset))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup X))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found x2:(P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P (sup ((union X) (singleton (sup emptyset)))))
% Found (fun (x2:(P (sup ((union X) (singleton (sup emptyset))))))=> x2) as proof of (P0 (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton zero))):(((eq (fofType->Prop)) ((union X) (singleton zero))) (fun (x:fofType)=> (((union X) (singleton zero)) x)))
% Found (eta_expansion00 ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 ((union X) (singleton zero))):(((eq (fofType->Prop)) ((union X) (singleton zero))) ((union X) (singleton zero)))
% Found (eq_ref0 ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton zero))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton zero))):(((eq (fofType->Prop)) ((union X) (singleton zero))) (fun (x:fofType)=> (((union X) (singleton zero)) x)))
% Found (eta_expansion00 ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton zero))) as proof of (((eq (fofType->Prop)) ((union X) (singleton zero))) b)
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) ((union X) (singleton (sup emptyset))))
% Found (eq_ref0 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) (fun (x:fofType)=> (((union X) (singleton (sup emptyset))) x)))
% Found (eta_expansion00 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) (fun (x:fofType)=> (((union X) (singleton (sup emptyset))) x)))
% Found (eta_expansion00 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (fofType->Prop)) X) (fun (x:fofType)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion fofType) Prop) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found eq_ref00:=(eq_ref0 X):(((eq (fofType->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion000:=(eta_expansion00 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) (fun (x:fofType)=> (((union X) (singleton (sup emptyset))) x)))
% Found (eta_expansion00 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eta_expansion0 Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion fofType) Prop) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (fofType->Prop)) X) (fun (x:fofType)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% 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) ((union X) (singleton zero)))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found eq_ref00:=(eq_ref0 X):(((eq (fofType->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) (sup ((union X) (singleton (sup emptyset)))))
% Found eq_ref00:=(eq_ref0 (sup X)):(((eq fofType) (sup X)) (sup X))
% Found (eq_ref0 (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found ((eq_ref fofType) (sup X)) as proof of (((eq fofType) (sup X)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found eta_expansion000:=(eta_expansion00 X):(((eq (fofType->Prop)) X) (fun (x:fofType)=> (X x)))
% Found (eta_expansion00 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eta_expansion0 Prop) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion fofType) Prop) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion fofType) Prop) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion fofType) Prop) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found eq_ref00:=(eq_ref0 X):(((eq (fofType->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eq_ref00:=(eq_ref0 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) ((union X) (singleton (sup emptyset))))
% Found (eq_ref0 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eq_ref (fofType->Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found sup_es1:=(sup_es (fun (x:fofType)=> (P (sup X)))):((P (sup X))->(P (sup X)))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found (sup_es (fun (x:fofType)=> (P (sup X)))) as proof of (P0 (sup X))
% Found eq_ref00:=(eq_ref0 X):(((eq (fofType->Prop)) X) X)
% Found (eq_ref0 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eq_ref (fofType->Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 X):(((eq (fofType->Prop)) X) (fun (x:fofType)=> (X x)))
% Found (eta_expansion_dep00 X) as proof of (((eq (fofType->Prop)) X) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) X) as proof of (((eq (fofType->Prop)) X) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) ((union X) (singleton zero)))
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found sup_es0:=(sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))):((P (((union X) (singleton zero)) x))->(P (((union X) (singleton zero)) x)))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found (sup_es (fun (x0:fofType)=> (P (((union X) (singleton zero)) x)))) as proof of (P0 (((union X) (singleton zero)) x))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) X)
% Found eta_expansion_dep000:=(eta_expansion_dep00 ((union X) (singleton (sup emptyset)))):(((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) (fun (x:fofType)=> (((union X) (singleton (sup emptyset))) x)))
% Found (eta_expansion_dep00 ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) ((union X) (singleton (sup emptyset)))) as proof of (((eq (fofType->Prop)) ((union X) (singleton (sup emptyset)))) b)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found sup_es0:=(sup_es (fun (x:fofType)=> (P X))):((P X)->(P X))
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found (sup_es (fun (x:fofType)=> (P X))) as proof of (P0 X)
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref Prop) (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X x))
% Found eq_ref00:=(eq_ref0 (((union X) (singleton zero)) x)):(((eq Prop) (((union X) (singleton zero)) x)) (((union X) (singleton zero)) x))
% Found (eq_ref0 (((union X) (singleton zero)) x)) as proof of (((eq Prop) (((union X) (singleton zero)) x)) b)
% Found ((eq_ref P
% EOF
%------------------------------------------------------------------------------