%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SET020^7 : TPTP v6.1.0. Released v5.5.0.
% 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 13:29:58 EDT 2014

% Result   : Timeout 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SET020^7 : TPTP v6.1.0. Released v5.5.0.
% % 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 07:56:06 CDT 2014
% % CPUTime  : 300.05 
% 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/LCL015^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x1291cb0>, <kernel.Type object at 0x12915a8>) of role type named mu_type
% Using role type
% Declaring mu:Type
% FOF formula (<kernel.Constant object at 0x1291758>, <kernel.DependentProduct object at 0x1291cb0>) of role type named qmltpeq_type
% Using role type
% Declaring qmltpeq:(mu->(mu->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x12e79e0>, <kernel.DependentProduct object at 0x12e75a8>) of role type named meq_prop_type
% Using role type
% Declaring meq_prop:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) meq_prop) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W)))) of role definition named meq_prop
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) meq_prop) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W))))
% Defined: meq_prop:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W)))
% FOF formula (<kernel.Constant object at 0x12e7b48>, <kernel.DependentProduct object at 0x12e7098>) of role type named mnot_type
% Using role type
% Declaring mnot:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mnot) (fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False))) of role definition named mnot
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mnot) (fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False)))
% Defined: mnot:=(fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False))
% FOF formula (<kernel.Constant object at 0x12e7098>, <kernel.DependentProduct object at 0x12e7878>) of role type named mor_type
% Using role type
% Declaring mor:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W)))) of role definition named mor
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W))))
% Defined: mor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W)))
% FOF formula (<kernel.Constant object at 0x12e7878>, <kernel.DependentProduct object at 0x12e7128>) of role type named mbox_type
% Using role type
% Declaring mbox:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mbox) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V))))) of role definition named mbox
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mbox) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V)))))
% Defined: mbox:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V))))
% FOF formula (<kernel.Constant object at 0x12e7128>, <kernel.DependentProduct object at 0x12e7b48>) of role type named mforall_prop_type
% Using role type
% Declaring mforall_prop:(((fofType->Prop)->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mforall_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W)))) of role definition named mforall_prop
% A new definition: (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mforall_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W))))
% Defined: mforall_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W)))
% FOF formula (<kernel.Constant object at 0x12e79e0>, <kernel.DependentProduct object at 0x1292368>) of role type named mtrue_type
% Using role type
% Declaring mtrue:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) mtrue) (fun (W:fofType)=> True)) of role definition named mtrue
% A new definition: (((eq (fofType->Prop)) mtrue) (fun (W:fofType)=> True))
% Defined: mtrue:=(fun (W:fofType)=> True)
% FOF formula (<kernel.Constant object at 0x12e7128>, <kernel.DependentProduct object at 0x1292a70>) of role type named mfalse_type
% Using role type
% Declaring mfalse:(fofType->Prop)
% FOF formula (((eq (fofType->Prop)) mfalse) (mnot mtrue)) of role definition named mfalse
% A new definition: (((eq (fofType->Prop)) mfalse) (mnot mtrue))
% Defined: mfalse:=(mnot mtrue)
% FOF formula (<kernel.Constant object at 0x12e75f0>, <kernel.DependentProduct object at 0x1292b48>) of role type named mand_type
% Using role type
% Declaring mand:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mand) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi))))) of role definition named mand
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mand) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi)))))
% Defined: mand:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi))))
% FOF formula (<kernel.Constant object at 0x12e75f0>, <kernel.DependentProduct object at 0x12923b0>) of role type named mimplies_type
% Using role type
% Declaring mimplies:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplies) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi))) of role definition named mimplies
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplies) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi)))
% Defined: mimplies:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi))
% FOF formula (<kernel.Constant object at 0x12e79e0>, <kernel.DependentProduct object at 0x1292368>) of role type named mimplied_type
% Using role type
% Declaring mimplied:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplied) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi))) of role definition named mimplied
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mimplied) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi)))
% Defined: mimplied:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi))
% FOF formula (<kernel.Constant object at 0x1292368>, <kernel.DependentProduct object at 0x1292b48>) of role type named mequiv_type
% Using role type
% Declaring mequiv:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mequiv) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi)))) of role definition named mequiv
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mequiv) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi))))
% Defined: mequiv:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi)))
% FOF formula (<kernel.Constant object at 0x1292b48>, <kernel.DependentProduct object at 0x1292e18>) of role type named mxor_type
% Using role type
% Declaring mxor:((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mxor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi)))) of role definition named mxor
% A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) mxor) (fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi))))
% Defined: mxor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi)))
% FOF formula (<kernel.Constant object at 0x1292e18>, <kernel.DependentProduct object at 0x1292998>) of role type named mdia_type
% Using role type
% Declaring mdia:((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mdia) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi))))) of role definition named mdia
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop)))) mdia) (fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi)))))
% Defined: mdia:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi))))
% FOF formula (<kernel.Constant object at 0x1292e18>, <kernel.DependentProduct object at 0xe9b6c8>) of role type named exists_in_world_type
% Using role type
% Declaring exists_in_world:(mu->(fofType->Prop))
% FOF formula (forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V)))) of role axiom named nonempty_ax
% A new axiom: (forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V))))
% FOF formula (<kernel.Constant object at 0x1292998>, <kernel.DependentProduct object at 0xe9b518>) of role type named mforall_ind_type
% Using role type
% Declaring mforall_ind:((mu->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mforall_ind) (fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W))))) of role definition named mforall_ind
% A new definition: (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mforall_ind) (fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W)))))
% Defined: mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W))))
% FOF formula (<kernel.Constant object at 0xe9b710>, <kernel.DependentProduct object at 0xe9b4d0>) of role type named mexists_ind_type
% Using role type
% Declaring mexists_ind:((mu->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mexists_ind) (fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X))))))) of role definition named mexists_ind
% A new definition: (((eq ((mu->(fofType->Prop))->(fofType->Prop))) mexists_ind) (fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X)))))))
% Defined: mexists_ind:=(fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X))))))
% FOF formula (<kernel.Constant object at 0xe9b4d0>, <kernel.DependentProduct object at 0xe9b878>) of role type named mexists_prop_type
% Using role type
% Declaring mexists_prop:(((fofType->Prop)->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mexists_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P))))))) of role definition named mexists_prop
% A new definition: (((eq (((fofType->Prop)->(fofType->Prop))->(fofType->Prop))) mexists_prop) (fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P)))))))
% Defined: mexists_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P))))))
% FOF formula (<kernel.Constant object at 0xe9b518>, <kernel.DependentProduct object at 0xe9bcb0>) of role type named mreflexive_type
% Using role type
% Declaring mreflexive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S)))) of role definition named mreflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mreflexive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S))))
% Defined: mreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S)))
% FOF formula (<kernel.Constant object at 0xe9bcb0>, <kernel.DependentProduct object at 0xe9bc20>) of role type named msymmetric_type
% Using role type
% Declaring msymmetric:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) msymmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S))))) of role definition named msymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) msymmetric) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S)))))
% Defined: msymmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S))))
% FOF formula (<kernel.Constant object at 0xe9bc20>, <kernel.DependentProduct object at 0xe9bcf8>) of role type named mserial_type
% Using role type
% Declaring mserial:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mserial) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))))) of role definition named mserial
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mserial) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T))))))
% Defined: mserial:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))))
% FOF formula (<kernel.Constant object at 0xe9bcf8>, <kernel.DependentProduct object at 0xe9bf38>) of role type named mtransitive_type
% Using role type
% Declaring mtransitive:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mtransitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))))) of role definition named mtransitive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mtransitive) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U)))))
% Defined: mtransitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))))
% FOF formula (<kernel.Constant object at 0xe9bf38>, <kernel.DependentProduct object at 0xe9bd40>) of role type named meuclidean_type
% Using role type
% Declaring meuclidean:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) meuclidean) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))))) of role definition named meuclidean
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) meuclidean) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U)))))
% Defined: meuclidean:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))))
% FOF formula (<kernel.Constant object at 0xe9bd40>, <kernel.DependentProduct object at 0xe9bf80>) of role type named mpartially_functional_type
% Using role type
% Declaring mpartially_functional:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mpartially_functional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U))))) of role definition named mpartially_functional
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mpartially_functional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U)))))
% Defined: mpartially_functional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U))))
% FOF formula (<kernel.Constant object at 0xe9bf80>, <kernel.DependentProduct object at 0xe9b3f8>) of role type named mfunctional_type
% Using role type
% Declaring mfunctional:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mfunctional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U))))))))) of role definition named mfunctional
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mfunctional) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U)))))))))
% Defined: mfunctional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U))))))))
% FOF formula (<kernel.Constant object at 0xe9b3f8>, <kernel.DependentProduct object at 0xe9bcb0>) of role type named mweakly_dense_type
% Using role type
% Declaring mweakly_dense:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_dense) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T))))))))) of role definition named mweakly_dense
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_dense) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T)))))))))
% Defined: mweakly_dense:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T))))))))
% FOF formula (<kernel.Constant object at 0xe9bcb0>, <kernel.DependentProduct object at 0xe9bd40>) of role type named mweakly_connected_type
% Using role type
% Declaring mweakly_connected:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_connected) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T)))))) of role definition named mweakly_connected
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_connected) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T))))))
% Defined: mweakly_connected:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T)))))
% FOF formula (<kernel.Constant object at 0xe9bd40>, <kernel.DependentProduct object at 0xe9bf38>) of role type named mweakly_directed_type
% Using role type
% Declaring mweakly_directed:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_directed) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V)))))))) of role definition named mweakly_directed
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) mweakly_directed) (fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V))))))))
% Defined: mweakly_directed:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V)))))))
% FOF formula (<kernel.Constant object at 0xe9b4d0>, <kernel.DependentProduct object at 0xe9bcf8>) of role type named mvalid_type
% Using role type
% Declaring mvalid:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) mvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W)))) of role definition named mvalid
% A new definition: (((eq ((fofType->Prop)->Prop)) mvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W))))
% Defined: mvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W)))
% FOF formula (<kernel.Constant object at 0xe9bd40>, <kernel.DependentProduct object at 0x1280050>) of role type named msatisfiable_type
% Using role type
% Declaring msatisfiable:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) msatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W))))) of role definition named msatisfiable
% A new definition: (((eq ((fofType->Prop)->Prop)) msatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W)))))
% Defined: msatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W))))
% FOF formula (<kernel.Constant object at 0xe9b440>, <kernel.DependentProduct object at 0x1280050>) of role type named mcountersatisfiable_type
% Using role type
% Declaring mcountersatisfiable:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) mcountersatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False))))) of role definition named mcountersatisfiable
% A new definition: (((eq ((fofType->Prop)->Prop)) mcountersatisfiable) (fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False)))))
% Defined: mcountersatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False))))
% FOF formula (<kernel.Constant object at 0x1280128>, <kernel.DependentProduct object at 0x1280098>) of role type named minvalid_type
% Using role type
% Declaring minvalid:((fofType->Prop)->Prop)
% FOF formula (((eq ((fofType->Prop)->Prop)) minvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False)))) of role definition named minvalid
% A new definition: (((eq ((fofType->Prop)->Prop)) minvalid) (fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False))))
% Defined: minvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False)))
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/LCL013^5.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x1291518>, <kernel.DependentProduct object at 0x12917a0>) of role type named rel_s4_type
% Using role type
% Declaring rel_s4:(fofType->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1291b48>, <kernel.DependentProduct object at 0x1291d88>) of role type named mbox_s4_type
% Using role type
% Declaring mbox_s4:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mbox_s4) (fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V))))) of role definition named mbox_s4
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mbox_s4) (fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V)))))
% Defined: mbox_s4:=(fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V))))
% FOF formula (<kernel.Constant object at 0x1291d88>, <kernel.DependentProduct object at 0x1291908>) of role type named mdia_s4_type
% Using role type
% Declaring mdia_s4:((fofType->Prop)->(fofType->Prop))
% FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) mdia_s4) (fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi))))) of role definition named mdia_s4
% A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) mdia_s4) (fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi)))))
% Defined: mdia_s4:=(fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi))))
% FOF formula (mreflexive rel_s4) of role axiom named a1
% A new axiom: (mreflexive rel_s4)
% FOF formula (mtransitive rel_s4) of role axiom named a2
% A new axiom: (mtransitive rel_s4)
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/LCL015^1.ax, trying next directory
% FOF formula (forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W))) of role axiom named cumulative_ax
% A new axiom: (forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W)))
% FOF formula (<kernel.Constant object at 0x12730e0>, <kernel.DependentProduct object at 0x12918c0>) of role type named inductive_type
% Using role type
% Declaring inductive:(mu->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1273200>, <kernel.DependentProduct object at 0x1291c20>) of role type named subclass_type
% Using role type
% Declaring subclass:(mu->(mu->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x1273200>, <kernel.DependentProduct object at 0x1291710>) of role type named disjoint_type
% Using role type
% Declaring disjoint:(mu->(mu->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x12918c0>, <kernel.DependentProduct object at 0x1291680>) of role type named function_type
% Using role type
% Declaring function:(mu->(fofType->Prop))
% FOF formula (<kernel.Constant object at 0x1291c20>, <kernel.DependentProduct object at 0x12919e0>) of role type named member_type
% Using role type
% Declaring member:(mu->(mu->(fofType->Prop)))
% FOF formula (<kernel.Constant object at 0x1291710>, <kernel.DependentProduct object at 0x12914d0>) of role type named unordered_pair_type
% Using role type
% Declaring unordered_pair:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((unordered_pair V2) V1)) V)) of role axiom named existence_of_unordered_pair_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((unordered_pair V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x1291200>, <kernel.Constant object at 0x1291dd0>) of role type named element_relation_type
% Using role type
% Declaring element_relation:mu
% FOF formula (forall (V:fofType), ((exists_in_world element_relation) V)) of role axiom named existence_of_element_relation_ax
% A new axiom: (forall (V:fofType), ((exists_in_world element_relation) V))
% FOF formula (<kernel.Constant object at 0x1291368>, <kernel.DependentProduct object at 0x12912d8>) of role type named complement_type
% Using role type
% Declaring complement:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (complement V1)) V)) of role axiom named existence_of_complement_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (complement V1)) V))
% FOF formula (<kernel.Constant object at 0x12918c0>, <kernel.DependentProduct object at 0x12911b8>) of role type named intersection_type
% Using role type
% Declaring intersection:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((intersection V2) V1)) V)) of role axiom named existence_of_intersection_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((intersection V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x1291cb0>, <kernel.DependentProduct object at 0x12911b8>) of role type named rotate_type
% Using role type
% Declaring rotate:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (rotate V1)) V)) of role axiom named existence_of_rotate_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (rotate V1)) V))
% FOF formula (<kernel.Constant object at 0x1291128>, <kernel.DependentProduct object at 0x12918c0>) of role type named union_type
% Using role type
% Declaring union:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((union V2) V1)) V)) of role axiom named existence_of_union_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((union V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12910e0>, <kernel.DependentProduct object at 0x1291128>) of role type named successor_type
% Using role type
% Declaring successor:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (successor V1)) V)) of role axiom named existence_of_successor_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (successor V1)) V))
% FOF formula (<kernel.Constant object at 0x1291c68>, <kernel.DependentProduct object at 0x12910e0>) of role type named flip_type
% Using role type
% Declaring flip:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (flip V1)) V)) of role axiom named existence_of_flip_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (flip V1)) V))
% FOF formula (<kernel.Constant object at 0x1291128>, <kernel.DependentProduct object at 0x1291998>) of role type named domain_of_type
% Using role type
% Declaring domain_of:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (domain_of V1)) V)) of role axiom named existence_of_domain_of_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (domain_of V1)) V))
% FOF formula (<kernel.Constant object at 0x12910e0>, <kernel.DependentProduct object at 0x1291878>) of role type named restrict_type
% Using role type
% Declaring restrict:(mu->(mu->(mu->mu)))
% FOF formula (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((restrict V3) V2) V1)) V)) of role axiom named existence_of_restrict_ax
% A new axiom: (forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((restrict V3) V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x1291998>, <kernel.DependentProduct object at 0x1291ef0>) of role type named range_of_type
% Using role type
% Declaring range_of:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (range_of V1)) V)) of role axiom named existence_of_range_of_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (range_of V1)) V))
% FOF formula (<kernel.Constant object at 0x12914d0>, <kernel.Constant object at 0x1291d40>) of role type named successor_relation_type
% Using role type
% Declaring successor_relation:mu
% FOF formula (forall (V:fofType), ((exists_in_world successor_relation) V)) of role axiom named existence_of_successor_relation_ax
% A new axiom: (forall (V:fofType), ((exists_in_world successor_relation) V))
% FOF formula (<kernel.Constant object at 0x12917a0>, <kernel.DependentProduct object at 0x1291878>) of role type named power_class_type
% Using role type
% Declaring power_class:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (power_class V1)) V)) of role axiom named existence_of_power_class_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (power_class V1)) V))
% FOF formula (<kernel.Constant object at 0x1291878>, <kernel.Constant object at 0x1291d88>) of role type named identity_relation_type
% Using role type
% Declaring identity_relation:mu
% FOF formula (forall (V:fofType), ((exists_in_world identity_relation) V)) of role axiom named existence_of_identity_relation_ax
% A new axiom: (forall (V:fofType), ((exists_in_world identity_relation) V))
% FOF formula (<kernel.Constant object at 0x1291d88>, <kernel.DependentProduct object at 0x12e7998>) of role type named inverse_type
% Using role type
% Declaring inverse:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (inverse V1)) V)) of role axiom named existence_of_inverse_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (inverse V1)) V))
% FOF formula (<kernel.Constant object at 0x1291d88>, <kernel.DependentProduct object at 0x12e7f80>) of role type named compose_type
% Using role type
% Declaring compose:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((compose V2) V1)) V)) of role axiom named existence_of_compose_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((compose V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12915f0>, <kernel.DependentProduct object at 0x12e7f80>) of role type named cross_product_type
% Using role type
% Declaring cross_product:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((cross_product V2) V1)) V)) of role axiom named existence_of_cross_product_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((cross_product V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12915f0>, <kernel.DependentProduct object at 0x12e7b48>) of role type named singleton_type
% Using role type
% Declaring singleton:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (singleton V1)) V)) of role axiom named existence_of_singleton_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (singleton V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7998>, <kernel.DependentProduct object at 0x12e7e18>) of role type named image_type
% Using role type
% Declaring image:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((image V2) V1)) V)) of role axiom named existence_of_image_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((image V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12e74d0>, <kernel.DependentProduct object at 0x12e7998>) of role type named sum_class_type
% Using role type
% Declaring sum_class:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (sum_class V1)) V)) of role axiom named existence_of_sum_class_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (sum_class V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7170>, <kernel.DependentProduct object at 0x12e72d8>) of role type named apply_type
% Using role type
% Declaring apply:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((apply V2) V1)) V)) of role axiom named existence_of_apply_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((apply V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7170>, <kernel.Constant object at 0x12e79e0>) of role type named null_class_type
% Using role type
% Declaring null_class:mu
% FOF formula (forall (V:fofType), ((exists_in_world null_class) V)) of role axiom named existence_of_null_class_ax
% A new axiom: (forall (V:fofType), ((exists_in_world null_class) V))
% FOF formula (<kernel.Constant object at 0x12e77a0>, <kernel.DependentProduct object at 0x12e7e18>) of role type named second_type
% Using role type
% Declaring second:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (second V1)) V)) of role axiom named existence_of_second_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (second V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7e18>, <kernel.DependentProduct object at 0x12923b0>) of role type named first_type
% Using role type
% Declaring first:(mu->mu)
% FOF formula (forall (V:fofType) (V1:mu), ((exists_in_world (first V1)) V)) of role axiom named existence_of_first_ax
% A new axiom: (forall (V:fofType) (V1:mu), ((exists_in_world (first V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7e18>, <kernel.DependentProduct object at 0x12e7098>) of role type named ordered_pair_type
% Using role type
% Declaring ordered_pair:(mu->(mu->mu))
% FOF formula (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((ordered_pair V2) V1)) V)) of role axiom named existence_of_ordered_pair_ax
% A new axiom: (forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((ordered_pair V2) V1)) V))
% FOF formula (<kernel.Constant object at 0x12e7e18>, <kernel.Constant object at 0x1292c20>) of role type named universal_class_type
% Using role type
% Declaring universal_class:mu
% FOF formula (forall (V:fofType), ((exists_in_world universal_class) V)) of role axiom named existence_of_universal_class_ax
% A new axiom: (forall (V:fofType), ((exists_in_world universal_class) V))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X)))))) of role axiom named reflexivity
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X))))))))))) of role axiom named symmetry
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z)))))))))))))) of role axiom named transitivity
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply A) C)) ((apply B) C))))))))))))))) of role axiom named apply_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply A) C)) ((apply B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply C) A)) ((apply C) B))))))))))))))) of role axiom named apply_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply C) A)) ((apply C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (complement A)) (complement B)))))))))))) of role axiom named complement_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (complement A)) (complement B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose A) C)) ((compose B) C))))))))))))))) of role axiom named compose_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose A) C)) ((compose B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose C) A)) ((compose C) B))))))))))))))) of role axiom named compose_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose C) A)) ((compose C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product A) C)) ((cross_product B) C))))))))))))))) of role axiom named cross_product_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product A) C)) ((cross_product B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product C) A)) ((cross_product C) B))))))))))))))) of role axiom named cross_product_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product C) A)) ((cross_product C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (domain_of A)) (domain_of B)))))))))))) of role axiom named domain_of_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (domain_of A)) (domain_of B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (first A)) (first B)))))))))))) of role axiom named first_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (first A)) (first B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (flip A)) (flip B)))))))))))) of role axiom named flip_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (flip A)) (flip B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image A) C)) ((image B) C))))))))))))))) of role axiom named image_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image A) C)) ((image B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image C) A)) ((image C) B))))))))))))))) of role axiom named image_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image C) A)) ((image C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection A) C)) ((intersection B) C))))))))))))))) of role axiom named intersection_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection A) C)) ((intersection B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection C) A)) ((intersection C) B))))))))))))))) of role axiom named intersection_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection C) A)) ((intersection C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (inverse A)) (inverse B)))))))))))) of role axiom named inverse_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (inverse A)) (inverse B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair A) C)) ((ordered_pair B) C))))))))))))))) of role axiom named ordered_pair_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair A) C)) ((ordered_pair B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair C) A)) ((ordered_pair C) B))))))))))))))) of role axiom named ordered_pair_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair C) A)) ((ordered_pair C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (power_class A)) (power_class B)))))))))))) of role axiom named power_class_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (power_class A)) (power_class B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (range_of A)) (range_of B)))))))))))) of role axiom named range_of_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (range_of A)) (range_of B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict A) C) D)) (((restrict B) C) D)))))))))))))))))) of role axiom named restrict_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict A) C) D)) (((restrict B) C) D))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) A) D)) (((restrict C) B) D)))))))))))))))))) of role axiom named restrict_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) A) D)) (((restrict C) B) D))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) D) A)) (((restrict C) D) B)))))))))))))))))) of role axiom named restrict_substitution_3
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) D) A)) (((restrict C) D) B))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (rotate A)) (rotate B)))))))))))) of role axiom named rotate_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (rotate A)) (rotate B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (second A)) (second B)))))))))))) of role axiom named second_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (second A)) (second B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (singleton A)) (singleton B)))))))))))) of role axiom named singleton_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (singleton A)) (singleton B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (successor A)) (successor B)))))))))))) of role axiom named successor_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (successor A)) (successor B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (sum_class A)) (sum_class B)))))))))))) of role axiom named sum_class_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (sum_class A)) (sum_class B))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union A) C)) ((union B) C))))))))))))))) of role axiom named union_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union A) C)) ((union B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union C) A)) ((union C) B))))))))))))))) of role axiom named union_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union C) A)) ((union C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair A) C)) ((unordered_pair B) C))))))))))))))) of role axiom named unordered_pair_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair A) C)) ((unordered_pair B) C)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair C) A)) ((unordered_pair C) B))))))))))))))) of role axiom named unordered_pair_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair C) A)) ((unordered_pair C) B)))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint A) C)))) (mbox_s4 ((disjoint B) C)))))))))))))) of role axiom named disjoint_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint A) C)))) (mbox_s4 ((disjoint B) C))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint C) A)))) (mbox_s4 ((disjoint C) B)))))))))))))) of role axiom named disjoint_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint C) A)))) (mbox_s4 ((disjoint C) B))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (function A)))) (mbox_s4 (function B))))))))))) of role axiom named function_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (function A)))) (mbox_s4 (function B)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (inductive A)))) (mbox_s4 (inductive B))))))))))) of role axiom named inductive_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (inductive A)))) (mbox_s4 (inductive B)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member A) C)))) (mbox_s4 ((member B) C)))))))))))))) of role axiom named member_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member A) C)))) (mbox_s4 ((member B) C))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member C) A)))) (mbox_s4 ((member C) B)))))))))))))) of role axiom named member_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member C) A)))) (mbox_s4 ((member C) B))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass A) C)))) (mbox_s4 ((subclass B) C)))))))))))))) of role axiom named subclass_substitution_1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass A) C)))) (mbox_s4 ((subclass B) C))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass C) A)))) (mbox_s4 ((subclass C) B)))))))))))))) of role axiom named subclass_substitution_2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass C) A)))) (mbox_s4 ((subclass C) B))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((subclass X) Y))) (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))) (mbox_s4 ((subclass X) Y)))))))))))) of role axiom named subclass_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((subclass X) Y))) (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))) (mbox_s4 ((subclass X) Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass X) universal_class)))))) of role axiom named class_elements_are_sets
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass X) universal_class))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))) (mbox_s4 ((qmltpeq X) Y)))))))))))) of role axiom named extensionality
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))) (mbox_s4 ((qmltpeq X) Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) ((unordered_pair X) Y)))) ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))) (mbox_s4 ((member U) ((unordered_pair X) Y)))))))))))))))) of role axiom named unordered_pair_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) ((unordered_pair X) Y)))) ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))) (mbox_s4 ((member U) ((unordered_pair X) Y))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((member ((unordered_pair X) Y)) universal_class))))))))) of role axiom named unordered_pair
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((member ((unordered_pair X) Y)) universal_class)))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (singleton X)) ((unordered_pair X) X))))))) of role axiom named singleton_set_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (singleton X)) ((unordered_pair X) X)))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((ordered_pair X) Y)) ((unordered_pair (singleton X)) ((unordered_pair X) (singleton Y)))))))))))) of role axiom named ordered_pair_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((ordered_pair X) Y)) ((unordered_pair (singleton X)) ((unordered_pair X) (singleton Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y)))) ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))) (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y))))))))))))))))))) of role axiom named cross_product_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y)))) ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))) (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y)))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((cross_product X) Y)))) (mbox_s4 ((qmltpeq Z) ((ordered_pair (first Z)) (second Z)))))))))))))))) of role axiom named cross_product
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((cross_product X) Y)))) (mbox_s4 ((qmltpeq Z) ((ordered_pair (first Z)) (second Z))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) element_relation))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))) (mbox_s4 ((member ((ordered_pair X) Y)) element_relation)))))))))))) of role axiom named element_relation_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) element_relation))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))) (mbox_s4 ((member ((ordered_pair X) Y)) element_relation))))))))))))
% FOF formula (mvalid (mbox_s4 ((subclass element_relation) ((cross_product universal_class) universal_class)))) of role axiom named element_relation
% A new axiom: (mvalid (mbox_s4 ((subclass element_relation) ((cross_product universal_class) universal_class))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((intersection X) Y)))) ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((intersection X) Y)))))))))))))))) of role axiom named intersection
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((intersection X) Y)))) ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((intersection X) Y))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (complement X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))) (mbox_s4 ((member Z) (complement X))))))))))))) of role axiom named complement
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (complement X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))) (mbox_s4 ((member Z) (complement X)))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (((restrict XR) X) Y)) ((intersection XR) ((cross_product X) Y)))))))))))))) of role axiom named restrict_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (((restrict XR) X) Y)) ((intersection XR) ((cross_product X) Y))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((member X) null_class)))))))) of role axiom named null_class_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((member X) null_class))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (domain_of X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))) (mbox_s4 ((member Z) (domain_of X))))))))))))) of role axiom named domain_of
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (domain_of X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))) (mbox_s4 ((member Z) (domain_of X)))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X))))))))))))))))))) of role axiom named rotate_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X)))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (rotate X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class))))))) of role axiom named rotate
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (rotate X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X))))))))))))))))))) of role axiom named flip_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X)))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (flip X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class))))))) of role axiom named flip
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (flip X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((union X) Y)))) ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((union X) Y)))))))))))))))) of role axiom named union_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((union X) Y)))) ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((union X) Y))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (successor X)) ((union X) (singleton X)))))))) of role axiom named successor_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (successor X)) ((union X) (singleton X))))))))
% FOF formula (mvalid (mbox_s4 ((subclass successor_relation) ((cross_product universal_class) universal_class)))) of role axiom named successor_relation_defn1
% A new axiom: (mvalid (mbox_s4 ((subclass successor_relation) ((cross_product universal_class) universal_class))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation))) ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))) (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation)))))))))))) of role axiom named successor_relation_defn2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation))) ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))) (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (inverse Y)) (domain_of (flip ((cross_product Y) universal_class))))))))) of role axiom named inverse_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (inverse Y)) (domain_of (flip ((cross_product Y) universal_class)))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((qmltpeq (range_of Z)) (domain_of (inverse Z)))))))) of role axiom named range_of_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((qmltpeq (range_of Z)) (domain_of (inverse Z))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 ((qmltpeq ((image XR) X)) (range_of (((restrict XR) X) universal_class))))))))))) of role axiom named image_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 ((qmltpeq ((image XR) X)) (range_of (((restrict XR) X) universal_class)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 (inductive X))) ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))) (mbox_s4 (inductive X))))))))) of role axiom named inductive_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 (inductive X))) ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))) (mbox_s4 (inductive X)))))))))
% FOF formula (mvalid (mexists_ind (fun (X:mu)=> ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 (inductive X))) (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (inductive Y))) (mbox_s4 ((subclass X) Y)))))))))))) of role axiom named infinity
% A new axiom: (mvalid (mexists_ind (fun (X:mu)=> ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 (inductive X))) (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (inductive Y))) (mbox_s4 ((subclass X) Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (sum_class X)))) (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))))) (mbox_s4 ((mimplies (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))) (mbox_s4 ((member U) (sum_class X))))))))))))) of role axiom named sum_class_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (sum_class X)))) (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))))) (mbox_s4 ((mimplies (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))) (mbox_s4 ((member U) (sum_class X)))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member X) universal_class))) (mbox_s4 ((member (sum_class X)) universal_class)))))))) of role axiom named sum_class
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member X) universal_class))) (mbox_s4 ((member (sum_class X)) universal_class))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (power_class X)))) ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))) (mbox_s4 ((member U) (power_class X))))))))))))) of role axiom named power_class_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (power_class X)))) ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))) (mbox_s4 ((member U) (power_class X)))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member (power_class U)) universal_class)))))))) of role axiom named power_class
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member (power_class U)) universal_class))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (YR:mu)=> (mbox_s4 ((subclass ((compose YR) XR)) ((cross_product universal_class) universal_class)))))))))) of role axiom named compose_defn1
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (YR:mu)=> (mbox_s4 ((subclass ((compose YR) XR)) ((cross_product universal_class) universal_class))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (YR:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair U) V)) ((compose YR) XR)))) ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member V) ((image YR) ((image YR) (singleton U))))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member V) ((image YR) ((image YR) (singleton U))))))) (mbox_s4 ((member ((ordered_pair U) V)) ((compose YR) XR))))))))))))))))))) of role axiom named compose_defn2
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (YR:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair U) V)) ((compose YR) XR)))) ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member V) ((image YR) ((image YR) (singleton U))))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((member V) ((image YR) ((image YR) (singleton U))))))) (mbox_s4 ((member ((ordered_pair U) V)) ((compose YR) XR)))))))))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (XF:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 (function XF))) ((mand (mbox_s4 ((subclass XF) ((cross_product universal_class) universal_class)))) (mbox_s4 ((subclass ((compose XF) (inverse XF))) identity_relation)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((subclass XF) ((cross_product universal_class) universal_class)))) (mbox_s4 ((subclass ((compose XF) (inverse XF))) identity_relation)))) (mbox_s4 (function XF))))))))) of role axiom named function_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (XF:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 (function XF))) ((mand (mbox_s4 ((subclass XF) ((cross_product universal_class) universal_class)))) (mbox_s4 ((subclass ((compose XF) (inverse XF))) identity_relation)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((subclass XF) ((cross_product universal_class) universal_class)))) (mbox_s4 ((subclass ((compose XF) (inverse XF))) identity_relation)))) (mbox_s4 (function XF)))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XF:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((member X) universal_class))) (mbox_s4 (function XF)))) (mbox_s4 ((member ((image XF) X)) universal_class))))))))))) of role axiom named replacement
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XF:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((member X) universal_class))) (mbox_s4 (function XF)))) (mbox_s4 ((member ((image XF) X)) universal_class)))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((disjoint X) Y))) (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mnot ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mnot ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y))))))))) (mbox_s4 ((disjoint X) Y)))))))))))) of role axiom named disjoint_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((disjoint X) Y))) (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mnot ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y))))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mnot ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y))))))))) (mbox_s4 ((disjoint X) Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((qmltpeq X) null_class))))) (mexists_ind (fun (U:mu)=> ((mand (mbox_s4 ((member U) universal_class))) ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((disjoint U) X)))))))))))) of role axiom named regularity
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies (mbox_s4 (mnot (mbox_s4 ((qmltpeq X) null_class))))) (mexists_ind (fun (U:mu)=> ((mand (mbox_s4 ((member U) universal_class))) ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((disjoint U) X))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (XF:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((apply XF) Y)) (sum_class ((image XF) (singleton Y)))))))))))) of role axiom named apply_defn
% A new axiom: (mvalid (mbox_s4 (mforall_ind (fun (XF:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((apply XF) Y)) (sum_class ((image XF) (singleton Y))))))))))))
% FOF formula (mvalid (mexists_ind (fun (XF:mu)=> ((mand (mbox_s4 (function XF))) (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member Y) universal_class))) ((mor (mbox_s4 ((qmltpeq Y) null_class))) (mbox_s4 ((member ((apply XF) Y)) Y)))))))))))) of role axiom named choice
% A new axiom: (mvalid (mexists_ind (fun (XF:mu)=> ((mand (mbox_s4 (function XF))) (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member Y) universal_class))) ((mor (mbox_s4 ((qmltpeq Y) null_class))) (mbox_s4 ((member ((apply XF) Y)) Y))))))))))))
% FOF formula (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mand (mbox_s4 ((member V) universal_class))) (mbox_s4 ((qmltpeq X) ((ordered_pair U) V)))))) ((mand (mbox_s4 ((qmltpeq (first X)) U))) (mbox_s4 ((qmltpeq (second X)) V))))))))))))))) of role conjecture named unique_1st_and_2nd_in_pair_of_sets1
% Conjecture to prove = (mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mand (mbox_s4 ((member V) universal_class))) (mbox_s4 ((qmltpeq X) ((ordered_pair U) V)))))) ((mand (mbox_s4 ((qmltpeq (first X)) U))) (mbox_s4 ((qmltpeq (second X)) V))))))))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mand (mbox_s4 ((member V) universal_class))) (mbox_s4 ((qmltpeq X) ((ordered_pair U) V)))))) ((mand (mbox_s4 ((qmltpeq (first X)) U))) (mbox_s4 ((qmltpeq (second X)) V)))))))))))))))']
% Parameter mu:Type.
% Parameter fofType:Type.
% Parameter qmltpeq:(mu->(mu->(fofType->Prop))).
% Definition meq_prop:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (W:fofType)=> (((eq Prop) (X W)) (Y W))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mnot:=(fun (Phi:(fofType->Prop)) (W:fofType)=> ((Phi W)->False)):((fofType->Prop)->(fofType->Prop)).
% Definition mor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop)) (W:fofType)=> ((or (Phi W)) (Psi W))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mbox:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((R W) V)->False)) (Phi V)))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop))).
% Definition mforall_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop))) (W:fofType)=> (forall (P:(fofType->Prop)), ((Phi P) W))):(((fofType->Prop)->(fofType->Prop))->(fofType->Prop)).
% Definition mtrue:=(fun (W:fofType)=> True):(fofType->Prop).
% Definition mfalse:=(mnot mtrue):(fofType->Prop).
% Definition mand:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mor (mnot Phi)) (mnot Psi)))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mimplies:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Phi)) Psi)):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mimplied:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mor (mnot Psi)) Phi)):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mequiv:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> ((mand ((mimplies Phi) Psi)) ((mimplies Psi) Phi))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mxor:=(fun (Phi:(fofType->Prop)) (Psi:(fofType->Prop))=> (mnot ((mequiv Phi) Psi))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))).
% Definition mdia:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi)))):((fofType->(fofType->Prop))->((fofType->Prop)->(fofType->Prop))).
% Parameter exists_in_world:(mu->(fofType->Prop)).
% Axiom nonempty_ax:(forall (V:fofType), ((ex mu) (fun (X:mu)=> ((exists_in_world X) V)))).
% Definition mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), (((exists_in_world X) W)->((Phi X) W)))):((mu->(fofType->Prop))->(fofType->Prop)).
% Definition mexists_ind:=(fun (Phi:(mu->(fofType->Prop)))=> (mnot (mforall_ind (fun (X:mu)=> (mnot (Phi X)))))):((mu->(fofType->Prop))->(fofType->Prop)).
% Definition mexists_prop:=(fun (Phi:((fofType->Prop)->(fofType->Prop)))=> (mnot (mforall_prop (fun (P:(fofType->Prop))=> (mnot (Phi P)))))):(((fofType->Prop)->(fofType->Prop))->(fofType->Prop)).
% Definition mreflexive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((R S) S))):((fofType->(fofType->Prop))->Prop).
% Definition msymmetric:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (((R S) T)->((R T) S)))):((fofType->(fofType->Prop))->Prop).
% Definition mserial:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T))))):((fofType->(fofType->Prop))->Prop).
% Definition mtransitive:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U)))):((fofType->(fofType->Prop))->Prop).
% Definition meuclidean:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U)))):((fofType->(fofType->Prop))->Prop).
% Definition mpartially_functional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->(((eq fofType) T) U)))):((fofType->(fofType->Prop))->Prop).
% Definition mfunctional:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((and ((R S) T)) (forall (U:fofType), (((R S) U)->(((eq fofType) T) U)))))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_dense:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType), (fofType->(((R S) T)->((ex fofType) (fun (U:fofType)=> ((and ((R S) U)) ((R U) T)))))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_connected:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((or ((or ((R T) U)) (((eq fofType) T) U))) ((R U) T))))):((fofType->(fofType->Prop))->Prop).
% Definition mweakly_directed:=(fun (R:(fofType->(fofType->Prop)))=> (forall (S:fofType) (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((ex fofType) (fun (V:fofType)=> ((and ((R T) V)) ((R U) V))))))):((fofType->(fofType->Prop))->Prop).
% Definition mvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), (Phi W))):((fofType->Prop)->Prop).
% Definition msatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> (Phi W)))):((fofType->Prop)->Prop).
% Definition mcountersatisfiable:=(fun (Phi:(fofType->Prop))=> ((ex fofType) (fun (W:fofType)=> ((Phi W)->False)))):((fofType->Prop)->Prop).
% Definition minvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False))):((fofType->Prop)->Prop).
% Parameter rel_s4:(fofType->(fofType->Prop)).
% Definition mbox_s4:=(fun (Phi:(fofType->Prop)) (W:fofType)=> (forall (V:fofType), ((or (((rel_s4 W) V)->False)) (Phi V)))):((fofType->Prop)->(fofType->Prop)).
% Definition mdia_s4:=(fun (Phi:(fofType->Prop))=> (mnot (mbox_s4 (mnot Phi)))):((fofType->Prop)->(fofType->Prop)).
% Axiom a1:(mreflexive rel_s4).
% Axiom a2:(mtransitive rel_s4).
% Axiom cumulative_ax:(forall (X:mu) (V:fofType) (W:fofType), (((and ((exists_in_world X) V)) ((rel_s4 V) W))->((exists_in_world X) W))).
% Parameter inductive:(mu->(fofType->Prop)).
% Parameter subclass:(mu->(mu->(fofType->Prop))).
% Parameter disjoint:(mu->(mu->(fofType->Prop))).
% Parameter function:(mu->(fofType->Prop)).
% Parameter member:(mu->(mu->(fofType->Prop))).
% Parameter unordered_pair:(mu->(mu->mu)).
% Axiom existence_of_unordered_pair_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((unordered_pair V2) V1)) V)).
% Parameter element_relation:mu.
% Axiom existence_of_element_relation_ax:(forall (V:fofType), ((exists_in_world element_relation) V)).
% Parameter complement:(mu->mu).
% Axiom existence_of_complement_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (complement V1)) V)).
% Parameter intersection:(mu->(mu->mu)).
% Axiom existence_of_intersection_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((intersection V2) V1)) V)).
% Parameter rotate:(mu->mu).
% Axiom existence_of_rotate_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (rotate V1)) V)).
% Parameter union:(mu->(mu->mu)).
% Axiom existence_of_union_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((union V2) V1)) V)).
% Parameter successor:(mu->mu).
% Axiom existence_of_successor_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (successor V1)) V)).
% Parameter flip:(mu->mu).
% Axiom existence_of_flip_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (flip V1)) V)).
% Parameter domain_of:(mu->mu).
% Axiom existence_of_domain_of_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (domain_of V1)) V)).
% Parameter restrict:(mu->(mu->(mu->mu))).
% Axiom existence_of_restrict_ax:(forall (V:fofType) (V3:mu) (V2:mu) (V1:mu), ((exists_in_world (((restrict V3) V2) V1)) V)).
% Parameter range_of:(mu->mu).
% Axiom existence_of_range_of_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (range_of V1)) V)).
% Parameter successor_relation:mu.
% Axiom existence_of_successor_relation_ax:(forall (V:fofType), ((exists_in_world successor_relation) V)).
% Parameter power_class:(mu->mu).
% Axiom existence_of_power_class_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (power_class V1)) V)).
% Parameter identity_relation:mu.
% Axiom existence_of_identity_relation_ax:(forall (V:fofType), ((exists_in_world identity_relation) V)).
% Parameter inverse:(mu->mu).
% Axiom existence_of_inverse_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (inverse V1)) V)).
% Parameter compose:(mu->(mu->mu)).
% Axiom existence_of_compose_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((compose V2) V1)) V)).
% Parameter cross_product:(mu->(mu->mu)).
% Axiom existence_of_cross_product_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((cross_product V2) V1)) V)).
% Parameter singleton:(mu->mu).
% Axiom existence_of_singleton_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (singleton V1)) V)).
% Parameter image:(mu->(mu->mu)).
% Axiom existence_of_image_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((image V2) V1)) V)).
% Parameter sum_class:(mu->mu).
% Axiom existence_of_sum_class_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (sum_class V1)) V)).
% Parameter apply:(mu->(mu->mu)).
% Axiom existence_of_apply_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((apply V2) V1)) V)).
% Parameter null_class:mu.
% Axiom existence_of_null_class_ax:(forall (V:fofType), ((exists_in_world null_class) V)).
% Parameter second:(mu->mu).
% Axiom existence_of_second_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (second V1)) V)).
% Parameter first:(mu->mu).
% Axiom existence_of_first_ax:(forall (V:fofType) (V1:mu), ((exists_in_world (first V1)) V)).
% Parameter ordered_pair:(mu->(mu->mu)).
% Axiom existence_of_ordered_pair_ax:(forall (V:fofType) (V2:mu) (V1:mu), ((exists_in_world ((ordered_pair V2) V1)) V)).
% Parameter universal_class:mu.
% Axiom existence_of_universal_class_ax:(forall (V:fofType), ((exists_in_world universal_class) V)).
% Axiom reflexivity:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq X) X)))))).
% Axiom symmetry:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) X))))))))))).
% Axiom transitivity:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq X) Y))) (mbox_s4 ((qmltpeq Y) Z)))) (mbox_s4 ((qmltpeq X) Z)))))))))))))).
% Axiom apply_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply A) C)) ((apply B) C))))))))))))))).
% Axiom apply_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((apply C) A)) ((apply C) B))))))))))))))).
% Axiom complement_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (complement A)) (complement B)))))))))))).
% Axiom compose_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose A) C)) ((compose B) C))))))))))))))).
% Axiom compose_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((compose C) A)) ((compose C) B))))))))))))))).
% Axiom cross_product_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product A) C)) ((cross_product B) C))))))))))))))).
% Axiom cross_product_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((cross_product C) A)) ((cross_product C) B))))))))))))))).
% Axiom domain_of_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (domain_of A)) (domain_of B)))))))))))).
% Axiom first_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (first A)) (first B)))))))))))).
% Axiom flip_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (flip A)) (flip B)))))))))))).
% Axiom image_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image A) C)) ((image B) C))))))))))))))).
% Axiom image_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((image C) A)) ((image C) B))))))))))))))).
% Axiom intersection_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection A) C)) ((intersection B) C))))))))))))))).
% Axiom intersection_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((intersection C) A)) ((intersection C) B))))))))))))))).
% Axiom inverse_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (inverse A)) (inverse B)))))))))))).
% Axiom ordered_pair_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair A) C)) ((ordered_pair B) C))))))))))))))).
% Axiom ordered_pair_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((ordered_pair C) A)) ((ordered_pair C) B))))))))))))))).
% Axiom power_class_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (power_class A)) (power_class B)))))))))))).
% Axiom range_of_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (range_of A)) (range_of B)))))))))))).
% Axiom restrict_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict A) C) D)) (((restrict B) C) D)))))))))))))))))).
% Axiom restrict_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) A) D)) (((restrict C) B) D)))))))))))))))))).
% Axiom restrict_substitution_3:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 (mforall_ind (fun (D:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (((restrict C) D) A)) (((restrict C) D) B)))))))))))))))))).
% Axiom rotate_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (rotate A)) (rotate B)))))))))))).
% Axiom second_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (second A)) (second B)))))))))))).
% Axiom singleton_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (singleton A)) (singleton B)))))))))))).
% Axiom successor_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (successor A)) (successor B)))))))))))).
% Axiom sum_class_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq (sum_class A)) (sum_class B)))))))))))).
% Axiom union_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union A) C)) ((union B) C))))))))))))))).
% Axiom union_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((union C) A)) ((union C) B))))))))))))))).
% Axiom unordered_pair_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair A) C)) ((unordered_pair B) C))))))))))))))).
% Axiom unordered_pair_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((qmltpeq ((unordered_pair C) A)) ((unordered_pair C) B))))))))))))))).
% Axiom disjoint_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint A) C)))) (mbox_s4 ((disjoint B) C)))))))))))))).
% Axiom disjoint_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((disjoint C) A)))) (mbox_s4 ((disjoint C) B)))))))))))))).
% Axiom function_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (function A)))) (mbox_s4 (function B))))))))))).
% Axiom inductive_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 (inductive A)))) (mbox_s4 (inductive B))))))))))).
% Axiom member_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member A) C)))) (mbox_s4 ((member B) C)))))))))))))).
% Axiom member_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((member C) A)))) (mbox_s4 ((member C) B)))))))))))))).
% Axiom subclass_substitution_1:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass A) C)))) (mbox_s4 ((subclass B) C)))))))))))))).
% Axiom subclass_substitution_2:(mvalid (mbox_s4 (mforall_ind (fun (A:mu)=> (mbox_s4 (mforall_ind (fun (B:mu)=> (mbox_s4 (mforall_ind (fun (C:mu)=> (mbox_s4 ((mimplies ((mand (mbox_s4 ((qmltpeq A) B))) (mbox_s4 ((subclass C) A)))) (mbox_s4 ((subclass C) B)))))))))))))).
% Axiom subclass_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((subclass X) Y))) (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))))) (mbox_s4 ((mimplies (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member U) X))) (mbox_s4 ((member U) Y)))))))) (mbox_s4 ((subclass X) Y)))))))))))).
% Axiom class_elements_are_sets:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass X) universal_class)))))).
% Axiom extensionality:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((qmltpeq X) Y))) ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((subclass X) Y))) (mbox_s4 ((subclass Y) X)))) (mbox_s4 ((qmltpeq X) Y)))))))))))).
% Axiom unordered_pair_defn:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) ((unordered_pair X) Y)))) ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) ((mor (mbox_s4 ((qmltpeq U) X))) (mbox_s4 ((qmltpeq U) Y))))) (mbox_s4 ((member U) ((unordered_pair X) Y)))))))))))))))).
% Axiom unordered_pair_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((member ((unordered_pair X) Y)) universal_class))))))))).
% Axiom singleton_set_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (singleton X)) ((unordered_pair X) X))))))).
% Axiom ordered_pair_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq ((ordered_pair X) Y)) ((unordered_pair (singleton X)) ((unordered_pair X) (singleton Y)))))))))))).
% Axiom cross_product_defn:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y)))) ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) X))) (mbox_s4 ((member V) Y)))) (mbox_s4 ((member ((ordered_pair U) V)) ((cross_product X) Y))))))))))))))))))).
% Axiom cross_product_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((cross_product X) Y)))) (mbox_s4 ((qmltpeq Z) ((ordered_pair (first Z)) (second Z)))))))))))))))).
% Axiom element_relation_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) element_relation))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((member X) Y)))) (mbox_s4 ((member ((ordered_pair X) Y)) element_relation)))))))))))).
% Axiom element_relation_TPTP_next:(mvalid (mbox_s4 ((subclass element_relation) ((cross_product universal_class) universal_class)))).
% Axiom intersection_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((intersection X) Y)))) ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((intersection X) Y)))))))))))))))).
% Axiom complement_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (complement X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((member Z) X)))))) (mbox_s4 ((member Z) (complement X))))))))))))).
% Axiom restrict_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (((restrict XR) X) Y)) ((intersection XR) ((cross_product X) Y)))))))))))))).
% Axiom null_class_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mnot (mbox_s4 ((member X) null_class)))))))).
% Axiom domain_of_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) (domain_of X)))) ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member Z) universal_class))) (mbox_s4 (mnot (mbox_s4 ((qmltpeq (((restrict X) (singleton Z)) universal_class)) null_class)))))) (mbox_s4 ((member Z) (domain_of X))))))))))))).
% Axiom rotate_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) W)) U)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (rotate X))))))))))))))))))).
% Axiom rotate_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (rotate X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class))))))).
% Axiom flip_defn:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (V:mu)=> (mbox_s4 (mforall_ind (fun (W:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X)))) ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) ((cross_product ((cross_product universal_class) universal_class)) universal_class)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair V) U)) W)) X)))) (mbox_s4 ((member ((ordered_pair ((ordered_pair U) V)) W)) (flip X))))))))))))))))))).
% Axiom flip_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((subclass (flip X)) ((cross_product ((cross_product universal_class) universal_class)) universal_class))))))).
% Axiom union_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 (mforall_ind (fun (Z:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member Z) ((union X) Y)))) ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))))) (mbox_s4 ((mimplies ((mor (mbox_s4 ((member Z) X))) (mbox_s4 ((member Z) Y)))) (mbox_s4 ((member Z) ((union X) Y)))))))))))))))).
% Axiom successor_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((qmltpeq (successor X)) ((union X) (singleton X)))))))).
% Axiom successor_relation_defn1:(mvalid (mbox_s4 ((subclass successor_relation) ((cross_product universal_class) universal_class)))).
% Axiom successor_relation_defn2:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (Y:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation))) ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 ((member Y) universal_class))) (mbox_s4 ((qmltpeq (successor X)) Y))))) (mbox_s4 ((member ((ordered_pair X) Y)) successor_relation)))))))))))).
% Axiom inverse_defn:(mvalid (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((qmltpeq (inverse Y)) (domain_of (flip ((cross_product Y) universal_class))))))))).
% Axiom range_of_defn:(mvalid (mbox_s4 (mforall_ind (fun (Z:mu)=> (mbox_s4 ((qmltpeq (range_of Z)) (domain_of (inverse Z)))))))).
% Axiom image_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 (mforall_ind (fun (XR:mu)=> (mbox_s4 ((qmltpeq ((image XR) X)) (range_of (((restrict XR) X) universal_class))))))))))).
% Axiom inductive_defn:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 (inductive X))) ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member null_class) X))) (mbox_s4 ((subclass ((image successor_relation) X)) X)))) (mbox_s4 (inductive X))))))))).
% Axiom infinity:(mvalid (mexists_ind (fun (X:mu)=> ((mand (mbox_s4 ((member X) universal_class))) ((mand (mbox_s4 (inductive X))) (mbox_s4 (mforall_ind (fun (Y:mu)=> (mbox_s4 ((mimplies (mbox_s4 (inductive Y))) (mbox_s4 ((subclass X) Y)))))))))))).
% Axiom sum_class_defn:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (sum_class X)))) (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))))) (mbox_s4 ((mimplies (mexists_ind (fun (Y:mu)=> ((mand (mbox_s4 ((member U) Y))) (mbox_s4 ((member Y) X)))))) (mbox_s4 ((member U) (sum_class X))))))))))))).
% Axiom sum_class_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (X:mu)=> (mbox_s4 ((mimplies (mbox_s4 ((member X) universal_class))) (mbox_s4 ((member (sum_class X)) universal_class)))))))).
% Axiom power_class_defn:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (mbox_s4 (mforall_ind (fun (X:mu)=> ((mand (mbox_s4 ((mimplies (mbox_s4 ((member U) (power_class X)))) ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))))) (mbox_s4 ((mimplies ((mand (mbox_s4 ((member U) universal_class))) (mbox_s4 ((subclass U) X)))) (mbox_s4 ((member U) (power_class X))))))))))))).
% Axiom power_class_TPTP_next:(mvalid (mbox_s4 (mforall_ind (fun (U:mu)=> (
% EOF
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