%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : LCL865^1 : TPTP v6.1.0. Bugfixed v5.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n090.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 13:26:25 EDT 2014

% Result   : Theorem 97.59s
% Output   : Proof 97.59s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : LCL865^1 : TPTP v6.1.0. Bugfixed v5.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n090.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 11:09:16 CDT 2014
% % CPUTime  : 97.59 
% 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/LCL013^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x190ec20>, <kernel.Type object at 0x190efc8>) of role type named mu_type
% Using role type
% Declaring mu:Type
% FOF formula (<kernel.Constant object at 0x190e908>, <kernel.DependentProduct object at 0x190e7a0>) of role type named meq_ind_type
% Using role type
% Declaring meq_ind:(mu->(mu->(fofType->Prop)))
% FOF formula (((eq (mu->(mu->(fofType->Prop)))) meq_ind) (fun (X:mu) (Y:mu) (W:fofType)=> (((eq mu) X) Y))) of role definition named meq_ind
% A new definition: (((eq (mu->(mu->(fofType->Prop)))) meq_ind) (fun (X:mu) (Y:mu) (W:fofType)=> (((eq mu) X) Y)))
% Defined: meq_ind:=(fun (X:mu) (Y:mu) (W:fofType)=> (((eq mu) X) Y))
% FOF formula (<kernel.Constant object at 0x190e758>, <kernel.DependentProduct object at 0x190e9e0>) 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 0x190ebd8>, <kernel.DependentProduct object at 0x1ce99e0>) 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 0x190ebd8>, <kernel.DependentProduct object at 0x1ce9998>) 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 0x190e9e0>, <kernel.DependentProduct object at 0x1ce99e0>) 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 0x1ce99e0>, <kernel.DependentProduct object at 0x1ce9170>) 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 0x1ce9170>, <kernel.DependentProduct object at 0x1ce9998>) 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 0x1ce9998>, <kernel.DependentProduct object at 0x1ce9ab8>) 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 0x1ce9ab8>, <kernel.DependentProduct object at 0x1ce9128>) 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 0x1ce94d0>, <kernel.DependentProduct object at 0x1ce9998>) 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), ((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), ((Phi X) W))))
% Defined: mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), ((Phi X) W)))
% FOF formula (<kernel.Constant object at 0x1ce9998>, <kernel.DependentProduct object at 0x1ce9050>) 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 0x17d65a8>, <kernel.DependentProduct object at 0x1ce9e60>) 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 0x1ce9e60>, <kernel.DependentProduct object at 0x1ce9998>) 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 0x1ce9710>, <kernel.DependentProduct object at 0x1ce93b0>) 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 0x1ce9f80>, <kernel.DependentProduct object at 0x1cea170>) 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 0x1ce9710>, <kernel.DependentProduct object at 0x1ce9e60>) 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 0x1ce9098>, <kernel.DependentProduct object at 0x1cea518>) 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 0x1ce9f80>, <kernel.DependentProduct object at 0x1cea368>) 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 0x1cea368>, <kernel.DependentProduct object at 0x1cea248>) 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 0x1cea170>, <kernel.DependentProduct object at 0x1cea248>) 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 0x1cea1b8>, <kernel.DependentProduct object at 0x1cea368>) 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 0x1cea248>, <kernel.DependentProduct object at 0x18f3878>) 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 0x1cea170>, <kernel.DependentProduct object at 0x18f36c8>) 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 0x1cea170>, <kernel.DependentProduct object at 0x18f3518>) 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 0x1cea1b8>, <kernel.DependentProduct object at 0x18f3ab8>) 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 0x18f3ab8>, <kernel.DependentProduct object at 0x18f3a28>) 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 0x18f3a28>, <kernel.DependentProduct object at 0x18f3ea8>) 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 0x18f38c0>, <kernel.DependentProduct object at 0x18f3d40>) 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 0x18f3a28>, <kernel.DependentProduct object at 0x18f39e0>) 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)))
% FOF formula (<kernel.Constant object at 0x18f3d40>, <kernel.DependentProduct object at 0x18f3440>) 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 0x18f39e0>, <kernel.DependentProduct object at 0x18f3dd0>) 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 (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) of role conjecture named conj
% Conjecture to prove = (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))):Prop
% Parameter mu_DUMMY:mu.
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))))']
% Parameter mu:Type.
% Parameter fofType:Type.
% Definition meq_ind:=(fun (X:mu) (Y:mu) (W:fofType)=> (((eq mu) X) Y)):(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 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 mforall_ind:=(fun (Phi:(mu->(fofType->Prop))) (W:fofType)=> (forall (X:mu), ((Phi X) W))):((mu->(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 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 mtrue:=(fun (W:fofType)=> True):(fofType->Prop).
% Definition mfalse:=(mnot mtrue):(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 mdia:=(fun (R:(fofType->(fofType->Prop))) (Phi:(fofType->Prop))=> (mnot ((mbox R) (mnot Phi)))):((fofType->(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 minvalid:=(fun (Phi:(fofType->Prop))=> (forall (W:fofType), ((Phi W)->False))):((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).
% Trying to prove (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))))
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1) as proof of ((mreflexive R)->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of (((and (mserial R)) (msymmetric R))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1) as proof of ((mreflexive R)->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of (((and (mserial R)) (msymmetric R))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of (((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1) as proof of ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1) as proof of ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found (fun (x1:(meuclidean R))=> x1) as proof of (meuclidean R)
% Found (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of ((meuclidean R)->(meuclidean R))
% Found (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1) as proof of (((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->(meuclidean R)))
% Found (and_rect00 (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found ((and_rect0 (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found (((fun (P:Type) (x0:(((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:((and (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (msymmetric R))) (x1:(meuclidean R))=> x1)) as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x00:=(x0 S):((R S) S)
% Found (x0 S) as proof of ((R S) x2)
% Found (x0 S) as proof of ((R S) x2)
% Found (x0 S) as proof of ((R S) x2)
% Found (ex_intro000 (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((ex_intro00 S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (((ex_intro0 (fun (T:fofType)=> ((R S) T))) S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((meuclidean R)->((ex fofType) (fun (T:fofType)=> ((R S) T))))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((mreflexive R)->((meuclidean R)->((ex fofType) (fun (T:fofType)=> ((R S) T)))))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((and_rect0 ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))))) as proof of (mserial R)
% Found x2:((R S) T)
% Instantiate: S0:=S:fofType
% Found x2 as proof of ((R S0) T)
% Found x00:=(x0 S0):((R S0) S0)
% Found (x0 S0) as proof of ((R S0) S)
% Found (x0 S0) as proof of ((R S0) S)
% Found ((conj30 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found (((conj3 ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found (x1000 ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0))) as proof of ((R T) S)
% Found ((x100 S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (((fun (S0:fofType)=> ((x10 S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (fun (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((R T) S)
% Found (fun (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (((R S) T)->((R T) S))
% Found (fun (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (forall (T:fofType), (((R S) T)->((R T) S)))
% Found (fun (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (msymmetric R)
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((meuclidean R)->(msymmetric R))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((mreflexive R)->((meuclidean R)->(msymmetric R)))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found ((and_rect0 (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found ((conj20 (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found (((conj2 (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((conj10 ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (((conj1 (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)))) as proof of (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x00:=(x0 S):((R S) S)
% Found (x0 S) as proof of ((R S) x2)
% Found (x0 S) as proof of ((R S) x2)
% Found (x0 S) as proof of ((R S) x2)
% Found (ex_intro000 (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((ex_intro00 S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (((ex_intro0 (fun (T:fofType)=> ((R S) T))) S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((meuclidean R)->((ex fofType) (fun (T:fofType)=> ((R S) T))))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of ((mreflexive R)->((meuclidean R)->((ex fofType) (fun (T:fofType)=> ((R S) T)))))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found ((and_rect0 ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))))) as proof of ((ex fofType) (fun (T:fofType)=> ((R S) T)))
% Found (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))))) as proof of (mserial R)
% Found x2:((R S) T)
% Instantiate: S0:=S:fofType
% Found x2 as proof of ((R S0) T)
% Found x00:=(x0 S0):((R S0) S0)
% Found (x0 S0) as proof of ((R S0) S)
% Found (x0 S0) as proof of ((R S0) S)
% Found ((conj30 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found (((conj3 ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S))
% Found (x1000 ((((conj ((R S0) T)) ((R S0) S)) x2) (x0 S0))) as proof of ((R T) S)
% Found ((x100 S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (((fun (S0:fofType)=> ((x10 S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))) as proof of ((R T) S)
% Found (fun (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((R T) S)
% Found (fun (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (((R S) T)->((R T) S))
% Found (fun (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (forall (T:fofType), (((R S) T)->((R T) S)))
% Found (fun (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of (msymmetric R)
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((meuclidean R)->(msymmetric R))
% Found (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))) as proof of ((mreflexive R)->((meuclidean R)->(msymmetric R)))
% Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found ((and_rect0 (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))) as proof of (msymmetric R)
% Found ((conj20 (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found (((conj2 (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S)))))) as proof of ((and (mserial R)) (msymmetric R))
% Found ((conj10 ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (((conj1 (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)))) as proof of ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))
% Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1)))) as proof of (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x00:=(x0 W):((R W) W)
% Found (x0 W) as proof of ((R W) x1)
% Found (x0 W) as proof of ((R W) x1)
% Found (x0 W) as proof of ((R W) x1)
% Found (ex_intro000 (x0 W)) as proof of ((fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))) W)
% Found ((ex_intro00 W) (x0 W)) as proof of ((fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))) W)
% Found (((ex_intro0 (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))) W)
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x00:=(x0 W):((R W) W)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (ex_intro000 (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((ex_intro00 W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (((ex_intro0 (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of (mvalid (fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))))
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x1:(meuclidean R)
% Found x1 as proof of (meuclidean R)
% Found x5:((R S) x4)
% Instantiate: S:=S0:fofType;S1:=x4:fofType
% Found x5 as proof of ((R S0) S1)
% Found (x300 x5) as proof of ((R S1) S0)
% Found ((x30 S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found x3000:=(x300 x5):((R S1) S0)
% Found (x300 x5) as proof of ((R S1) S0)
% Found ((x30 S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found ((conj20 (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found (((conj2 ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found (x1000 ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5))) as proof of ((R S0) S0)
% Found ((x100 x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found (((fun (S1:fofType)=> ((x10 S1) S0)) x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S0)) x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found x00:=(x0 W):((R W) W)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (ex_intro000 (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((ex_intro00 W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (((ex_intro0 (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of (mvalid (fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))))
% Found x5:((R S) x4)
% Instantiate: S:=S0:fofType;S1:=x4:fofType
% Found x5 as proof of ((R S0) S1)
% Found (x300 x5) as proof of ((R S1) S0)
% Found ((x30 S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found x3000:=(x300 x5):((R S1) S0)
% Found (x300 x5) as proof of ((R S1) S0)
% Found ((x30 S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found (((x3 S0) S1) x5) as proof of ((R S1) S0)
% Found ((conj20 (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found (((conj2 ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5)) as proof of ((and ((R S1) S0)) ((R S1) S0))
% Found (x1000 ((((conj ((R S1) S0)) ((R S1) S0)) (((x3 S0) S1) x5)) (((x3 S0) S1) x5))) as proof of ((R S0) S0)
% Found ((x100 x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found (((fun (S1:fofType)=> ((x10 S1) S0)) x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S0)) x4) ((((conj ((R x4) S0)) ((R x4) S0)) (((x3 S0) x4) x5)) (((x3 S0) x4) x5))) as proof of ((R S0) S0)
% Found x00:=(x0 W):((R W) W)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (x0 W) as proof of ((R W) x2)
% Found (ex_intro000 (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((ex_intro00 W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (((ex_intro0 (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W)) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))) W)
% Found (fun (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of (mvalid (fun (x10:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x10) T)))))
% Found x5:((R S0) x4)
% Instantiate: S1:=x4:fofType;S0:=S:fofType
% Found x5 as proof of ((R S) S1)
% Found (x300 x5) as proof of ((R S1) S)
% Found ((x30 S1) x5) as proof of ((R S1) S)
% Found (((x3 S) S1) x5) as proof of ((R S1) S)
% Found (((x3 S) S1) x5) as proof of ((R S1) S)
% Found x3000:=(x300 x5):((R S1) S)
% Found (x300 x5) as proof of ((R S1) S)
% Found ((x30 S1) x5) as proof of ((R S1) S)
% Found (((x3 S) S1) x5) as proof of ((R S1) S)
% Found (((x3 S) S1) x5) as proof of ((R S1) S)
% Found ((conj20 (((x3 S) S1) x5)) (((x3 S) S1) x5)) as proof of ((and ((R S1) S)) ((R S1) S))
% Found (((conj2 ((R S1) S)) (((x3 S) S1) x5)) (((x3 S) S1) x5)) as proof of ((and ((R S1) S)) ((R S1) S))
% Found ((((conj ((R S1) S)) ((R S1) S)) (((x3 S) S1) x5)) (((x3 S) S1) x5)) as proof of ((and ((R S1) S)) ((R S1) S))
% Found ((((conj ((R S1) S)) ((R S1) S)) (((x3 S) S1) x5)) (((x3 S) S1) x5)) as proof of ((and ((R S1) S)) ((R S1) S))
% Found (x1000 ((((conj ((R S1) S)) ((R S1) S)) (((x3 S) S1) x5)) (((x3 S) S1) x5))) as proof of ((R S) S)
% Found ((x100 x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))) as proof of ((R S) S)
% Found (((fun (S1:fofType)=> ((x10 S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))) as proof of ((R S) S)
% Found (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))) as proof of ((R S) S)
% Found (fun (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5)))) as proof of ((R S) S)
% Found (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5)))) as proof of (((R S0) x4)->((R S) S))
% Found (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5)))) as proof of (forall (x:fofType), (((R S0) x)->((R S) S)))
% Found (ex_ind00 (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))) as proof of ((R S) S)
% Found ((ex_ind0 ((R S) S)) (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))) as proof of ((R S) S)
% Found (((fun (P:Prop) (x4:(forall (x:fofType), (((R S0) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S0) T))) P) x4) x20)) ((R S) S)) (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))) as proof of ((R S) S)
% Found (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))) as proof of ((R S) S)
% Found (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5)))))) as proof of ((R S) S)
% Found (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5)))))) as proof of (mreflexive R)
% Found ((conj10 (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (((conj1 (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1) as proof of ((and (mreflexive R)) (meuclidean R))
% Found ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)) as proof of ((msymmetric R)->((and (mreflexive R)) (meuclidean R)))
% Found (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)) as proof of ((mserial R)->((msymmetric R)->((and (mreflexive R)) (meuclidean R))))
% Found (and_rect10 (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found ((and_rect1 ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))) as proof of ((meuclidean R)->((and (mreflexive R)) (meuclidean R)))
% Found (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))) as proof of (((and (mserial R)) (msymmetric R))->((meuclidean R)->((and (mreflexive R)) (meuclidean R))))
% Found (and_rect00 (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found ((and_rect0 ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x:fofType), (((R S) x)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))))) as proof of ((and (mreflexive R)) (meuclidean R))
% Found (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))))) as proof of (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))
% Found ((conj00 (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found (((conj0 (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))
% Found (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1)))))))) as proof of (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (msymmetric R))) (meuclidean R))))
% Got proof (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))))))
% Time elapsed = 96.208759s
% node=12262 cost=2473.000000 depth=42
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))) (((and ((and (mserial R)) (msymmetric R))) (meuclidean R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> ((((conj ((and (mserial R)) (msymmetric R))) (meuclidean R)) ((((conj (mserial R)) (msymmetric R)) (fun (S:fofType)=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((ex fofType) (fun (T:fofType)=> ((R S) T)))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R)) (S:fofType) (T:fofType) (x2:((R S) T))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) T)) S0) S)) S) ((((conj ((R S) T)) ((R S) S)) x2) (x0 S))))))) (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) (meuclidean R)) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> x1))))) (fun (x:((and ((and (mserial R)) (msymmetric R))) (meuclidean R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (msymmetric R))->((meuclidean R)->P)))=> (((((and_rect ((and (mserial R)) (msymmetric R))) (meuclidean R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (msymmetric R))) (x1:(meuclidean R))=> (((fun (P:Type) (x2:((mserial R)->((msymmetric R)->P)))=> (((((and_rect (mserial R)) (msymmetric R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(msymmetric R))=> ((((conj (mreflexive R)) (meuclidean R)) (fun (S:fofType)=> (((fun (P:Prop) (x4:(forall (x0:fofType), (((R S) x0)->P)))=> (((((ex_ind fofType) (fun (T:fofType)=> ((R S) T))) P) x4) (x2 S))) ((R S) S)) (fun (x4:fofType) (x5:((R S) x4))=> (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S)) S1) S)) x4) ((((conj ((R x4) S)) ((R x4) S)) (((x3 S) x4) x5)) (((x3 S) x4) x5))))))) x1))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------