%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : LCL863^1 : TPTP v6.1.0. Bugfixed v5.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n097.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 145.13s % Output : Proof 145.13s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : LCL863^1 : TPTP v6.1.0. Bugfixed v5.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n097.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 11:09:06 CDT 2014 % % CPUTime : 145.13 % 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 0x19c77e8>, <kernel.Type object at 0x19c7830>) of role type named mu_type % Using role type % Declaring mu:Type % FOF formula (<kernel.Constant object at 0x19c7a28>, <kernel.DependentProduct object at 0x19c77e8>) 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 0x1458e18>, <kernel.DependentProduct object at 0x19c7830>) 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 0x19c7758>, <kernel.DependentProduct object at 0x176ffc8>) 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 0x19c7a28>, <kernel.DependentProduct object at 0x176f368>) 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 0x19c7a28>, <kernel.DependentProduct object at 0x176fb00>) 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 0x19c7a28>, <kernel.DependentProduct object at 0x176fcf8>) 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 0x176fcf8>, <kernel.DependentProduct object at 0x176fea8>) 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 0x176fea8>, <kernel.DependentProduct object at 0x176f3f8>) 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 0x176f3f8>, <kernel.DependentProduct object at 0x176fb00>) 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 0x176ffc8>, <kernel.DependentProduct object at 0x176fea8>) 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 0x176fea8>, <kernel.DependentProduct object at 0x176f440>) 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 0x176f440>, <kernel.DependentProduct object at 0x176f170>) 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 0x176f170>, <kernel.DependentProduct object at 0x176f830>) 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 0x176f6c8>, <kernel.DependentProduct object at 0x176f098>) 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 0x176f440>, <kernel.DependentProduct object at 0x1770908>) 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 0x176f6c8>, <kernel.DependentProduct object at 0x176f170>) 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 0x176f638>, <kernel.DependentProduct object at 0x17709e0>) 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 0x176f440>, <kernel.DependentProduct object at 0x1770950>) 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 0x1770950>, <kernel.DependentProduct object at 0x1770440>) 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 0x1770908>, <kernel.DependentProduct object at 0x1770440>) 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 0x17707a0>, <kernel.DependentProduct object at 0x1770950>) 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 0x1770440>, <kernel.DependentProduct object at 0x1572878>) 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 0x1770908>, <kernel.DependentProduct object at 0x15726c8>) 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 0x1770908>, <kernel.DependentProduct object at 0x1572518>) 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 0x17707a0>, <kernel.DependentProduct object at 0x1572ab8>) 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 0x1572ab8>, <kernel.DependentProduct object at 0x1572a28>) 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 0x1572a28>, <kernel.DependentProduct object at 0x1572ea8>) 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 0x15728c0>, <kernel.DependentProduct object at 0x1572d40>) 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 0x1572a28>, <kernel.DependentProduct object at 0x15729e0>) 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 0x1572d40>, <kernel.DependentProduct object at 0x1572440>) 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 0x15729e0>, <kernel.DependentProduct object at 0x1572dd0>) 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)) (mtransitive R))) (msymmetric R)))) of role conjecture named conj % Conjecture to prove = (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric 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)) (mtransitive R))) (msymmetric 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)) (mtransitive R))) (msymmetric 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 (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 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 (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 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 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 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 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 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x2:((R T) U) % Instantiate: S0:=T:fofType % Found x2 as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x30:=(x3 S0):((R S0) S0) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 (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 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 x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (S:fofType)=> ((((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)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))) % Found (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (mserial R) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % 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 (S:fofType)=> ((((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)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))) % Found (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (mserial R) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x2:((R T) U) % Instantiate: S0:=T:fofType % Found x2 as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x30:=(x3 S0):((R S0) S0) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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 (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 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 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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 ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(msymmetric R))) % Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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 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 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 (x1:(meuclidean R)) (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 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((meuclidean R)->(mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) % Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T))))))) % Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 ((and_rect0 (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 (((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)) (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 (((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)) (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % 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 (S:fofType)=> ((((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)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))) % Found (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (mserial R) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % 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 (S:fofType)=> ((((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)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (forall (S:fofType), ((ex fofType) (fun (T:fofType)=> ((R S) T)))) % Found (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S))) as proof of (mserial R) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found x6 as proof of ((R T0) U) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found x6 as proof of ((R T0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 ((conj20 x2) (x0 S0)) as proof of ((and ((R S0) T)) ((R S0) S)) % Found (((conj2 ((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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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 ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(msymmetric R))) % Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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:((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->P)))=> (((((and_rect (mvalid (fun (x1:fofType)=> ((R x1) x1)))) (meuclidean R)) P) x0) x)) (msymmetric R)) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (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 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 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 (x1:(meuclidean R)) (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 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((meuclidean R)->(mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) % Found (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (W:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R W) T))) W) (x0 W))) as proof of ((mvalid (fun (x1:fofType)=> ((R x1) x1)))->((meuclidean R)->(mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T))))))) % Found (and_rect00 (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 ((and_rect0 (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 (((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)) (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 (((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)) (mvalid (fun (x1:fofType)=> ((ex fofType) (fun (T:fofType)=> ((R x1) T)))))) (fun (x0:(mvalid (fun (x1:fofType)=> ((R x1) x1)))) (x1:(meuclidean R)) (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 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found x6 as proof of ((R T0) U) % 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 (x0 S0) as proof of ((R S0) S) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S0):((R S0) S0) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found (x0 S0) as proof of ((R S0) U) % Found x2:((R T) U) % Instantiate: S0:=T:fofType % Found x2 as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found x6 as proof of ((R T0) U) % Found x30:=(x3 S0):((R S0) S0) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found (x3 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found x6 as proof of ((R T0) U) % Found x6:((R S) U) % Instantiate: T0:=S:fofType % Found (fun (x6:((R S) U))=> x6) as proof of ((R T0) U) % Found (fun (x5:((R S) T)) (x6:((R S) U))=> x6) as proof of (((R S) U)->((R T0) U)) % Found (fun (x5:((R S) T)) (x6:((R S) U))=> x6) as proof of (((R S) T)->(((R S) U)->((R T0) U))) % Found (and_rect20 (fun (x5:((R S) T)) (x6:((R S) U))=> x6)) as proof of ((R T0) U) % Found ((and_rect2 ((R T0) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> x6)) as proof of ((R T0) U) % Found (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T0) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> x6)) as proof of ((R T0) U) % Found (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T0) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> x6)) as proof of ((R T0) U) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found (x1 S0) as proof of ((R S0) S) % Found x10:=(x1 S0):((R S0) S0) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found (x1 S0) as proof of ((R S0) U) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % Found x1000:=(x100 x5):((R T) T0) % Found (x100 x5) as proof of ((R T) T0) % Found ((x10 T) x5) as proof of ((R T) T0) % Found (((x1 T0) T) x5) as proof of ((R T) T0) % Found (((x1 T0) T) x5) as proof of ((R T) T0) % Found ((conj20 (((x1 T0) T) x5)) x6) as proof of ((and ((R T) T0)) ((R T0) U)) % Found (((conj2 ((R T0) U)) (((x1 T0) T) x5)) x6) as proof of ((and ((R T) T0)) ((R T0) U)) % Found ((((conj ((R T) T0)) ((R T0) U)) (((x1 T0) T) x5)) x6) as proof of ((and ((R T) T0)) ((R T0) U)) % Found ((((conj ((R T) T0)) ((R T0) U)) (((x1 T0) T) x5)) x6) as proof of ((and ((R T) T0)) ((R T0) U)) % Found (x3000 ((((conj ((R T) T0)) ((R T0) U)) (((x1 T0) T) x5)) x6)) as proof of ((R T) U) % Found ((x300 S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)) as proof of ((R T) U) % Found (((fun (T0:fofType)=> ((x30 T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)) as proof of ((R T) U) % Found (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)) as proof of ((R T) U) % Found (fun (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))) as proof of ((R T) U) % Found (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))) as proof of (((R S) U)->((R T) U)) % Found (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))) as proof of (((R S) T)->(((R S) U)->((R T) U))) % Found (and_rect20 (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))) as proof of ((R T) U) % Found ((and_rect2 ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))) as proof of ((R T) U) % Found (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))) as proof of ((R T) U) % Found (fun (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))) as proof of ((R T) U) % Found (fun (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))) as proof of (((and ((R S) T)) ((R S) U))->((R T) U)) % Found (fun (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))) as proof of (forall (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))) % Found (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))) as proof of (forall (T:fofType) (U:fofType), (((and ((R S) T)) ((R S) U))->((R T) U))) % Found (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))) as proof of (meuclidean R) % Found x5:((R S) x4) % Instantiate: S:=S0:fofType;T:=x4:fofType % Found x5 as proof of ((R S0) T) % Found x5:((R S0) x4) % Instantiate: S0:=S:fofType;T:=x4:fofType % Found x5 as proof of ((R S) T) % Found x5:((R S0) x4) % Instantiate: S0:=S:fofType;T:=x4:fofType % Found x5 as proof of ((R S) T) % Found x1000:=(x100 x5):((R T) S0) % Found (x100 x5) as proof of ((R T) S0) % Found ((x10 T) x5) as proof of ((R T) S0) % Found (((x1 S0) T) x5) as proof of ((R T) S0) % Found (((x1 S0) T) x5) as proof of ((R T) S0) % Found ((conj20 x5) (((x1 S0) T) x5)) as proof of ((and ((R S0) T)) ((R T) S0)) % Found (((conj2 ((R T) S0)) x5) (((x1 S0) T) x5)) as proof of ((and ((R S0) T)) ((R T) S0)) % Found ((((conj ((R S0) T)) ((R T) S0)) x5) (((x1 S0) T) x5)) as proof of ((and ((R S0) T)) ((R T) S0)) % Found ((((conj ((R S0) T)) ((R T) S0)) x5) (((x1 S0) T) x5)) as proof of ((and ((R S0) T)) ((R T) S0)) % Found (x3000 ((((conj ((R S0) T)) ((R T) S0)) x5) (((x1 S0) T) x5))) as proof of ((R S0) S0) % Found ((x300 x4) ((((conj ((R S0) x4)) ((R x4) S0)) x5) (((x1 S0) x4) x5))) as proof of ((R S0) S0) % Found (((fun (T:fofType)=> ((x30 T) S0)) x4) ((((conj ((R S0) x4)) ((R x4) S0)) x5) (((x1 S0) x4) x5))) as proof of ((R S0) S0) % Found (((fun (T:fofType)=> (((x3 S0) T) S0)) x4) ((((conj ((R S0) x4)) ((R x4) S0)) x5) (((x1 S0) x4) x5))) as proof of ((R S0) S0) % Found x1000:=(x100 x5):((R T) S) % Found (x100 x5) as proof of ((R T) S) % Found ((x10 T) x5) as proof of ((R T) S) % Found (((x1 S) T) x5) as proof of ((R T) S) % Found (((x1 S) T) x5) as proof of ((R T) S) % Found ((conj20 x5) (((x1 S) T) x5)) as proof of ((and ((R S) T)) ((R T) S)) % Found (((conj2 ((R T) S)) x5) (((x1 S) T) x5)) as proof of ((and ((R S) T)) ((R T) S)) % Found ((((conj ((R S) T)) ((R T) S)) x5) (((x1 S) T) x5)) as proof of ((and ((R S) T)) ((R T) S)) % Found ((((conj ((R S) T)) ((R T) S)) x5) (((x1 S) T) x5)) as proof of ((and ((R S) T)) ((R T) S)) % Found (x3000 ((((conj ((R S) T)) ((R T) S)) x5) (((x1 S) T) x5))) as proof of ((R S) S) % Found ((x300 x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))) as proof of ((R S) S) % Found (((fun (T:fofType)=> ((x30 T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))) as proof of ((R S) S) % Found (((fun (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))) as proof of ((R S) S) % Found (fun (x5:((R S0) x4))=> (((fun (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5)))) as proof of ((R S) S) % Found (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5)))) as proof of (((R S0) x4)->((R S) S)) % Found (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))) as proof of ((R S) S) % Found ((ex_ind0 ((R S) S)) (fun (x4:fofType) (x5:((R S0) x4))=> (((fun (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))) 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))) 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))) as proof of ((mtransitive R)->((and (mreflexive R)) (meuclidean R))) % Found (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))) as proof of ((mserial R)->((mtransitive R)->((and (mreflexive R)) (meuclidean R)))) % Found (and_rect10 (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found ((and_rect1 ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))) as proof of ((msymmetric R)->((and (mreflexive R)) (meuclidean R))) % Found (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))) as proof of (((and (mserial R)) (mtransitive R))->((msymmetric R)->((and (mreflexive R)) (meuclidean R)))) % Found (and_rect00 (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found ((and_rect0 ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))) as proof of ((and (mreflexive R)) (meuclidean R)) % Found (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))) as proof of (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R))) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S1):((R S1) S1) % Found (x0 S1) as proof of ((R S1) S) % Found (x0 S1) as proof of ((R S1) S) % Found (x0 S1) as proof of ((R S1) S) % Found x00:=(x0 S1):((R S1) S1) % Found (x0 S1) as proof of ((R S1) S0) % Found (x0 S1) as proof of ((R S1) S0) % Found (x0 S1) as proof of ((R S1) S0) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x4:((R T) U) % Instantiate: S0:=T:fofType % Found x4 as proof of ((R S0) U) % 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 (x0 S0) as proof of ((R S0) S) % Found x00:=(x0 S00):((R S00) S00) % Found (x0 S00) as proof of ((R S00) S) % Found (x0 S00) as proof of ((R S00) S) % Found (x0 S00) as proof of ((R S00) S) % Found x00:=(x0 S00):((R S00) S00) % Found (x0 S00) as proof of ((R S00) U) % Found (x0 S00) as proof of ((R S00) U) % Found (x0 S00) as proof of ((R S00) U) % Found x3:((R S) T) % Instantiate: S1:=S:fofType % Found x3 as proof of ((R S1) S0) % Found x00:=(x0 S1):((R S1) S1) % Found (x0 S1) as proof of ((R S1) S) % Found (x0 S1) as proof of ((R S1) S) % Found ((conj40 x3) (x0 S1)) as proof of ((and ((R S1) S0)) ((R S1) S)) % Found (((conj4 ((R S1) S)) x3) (x0 S1)) as proof of ((and ((R S1) S0)) ((R S1) S)) % Found ((((conj ((R S1) S0)) ((R S1) S)) x3) (x0 S1)) as proof of ((and ((R S1) S0)) ((R S1) S)) % Found ((((conj ((R S1) S0)) ((R S1) S)) x3) (x0 S1)) as proof of ((and ((R S1) S0)) ((R S1) S)) % Found (x1100 ((((conj ((R S1) S0)) ((R S1) S)) x3) (x0 S1))) as proof of ((R S0) S) % Found ((x110 S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S))) as proof of ((R S0) S) % Found (((fun (S1:fofType)=> ((x11 S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S))) as proof of ((R S0) S) % Found (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S))) as proof of ((R S0) S) % Found (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S))) as proof of ((R S0) S) % Found ((conj30 (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S)))) x4) as proof of ((and ((R S0) S)) ((R S0) U)) % Found (((conj3 ((R S0) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S)))) x4) as proof of ((and ((R S0) S)) ((R S0) U)) % Found ((((conj ((R S0) S)) ((R S0) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S)))) x4) as proof of ((and ((R S0) S)) ((R S0) U)) % Found ((((conj ((R S0) S)) ((R S0) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S)))) x4) as proof of ((and ((R S0) S)) ((R S0) U)) % Found (x1000 ((((conj ((R S0) S)) ((R S0) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) S0)) S1) S)) S) ((((conj ((R S) S0)) ((R S) S)) x3) (x0 S)))) x4)) as proof of ((R S) U) % Found ((x100 T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)) as proof of ((R S) U) % Found (((fun (S0:fofType)=> ((x10 S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)) as proof of ((R S) U) % Found (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)) as proof of ((R S) U) % Found (fun (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))) as proof of ((R S) U) % Found (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))) as proof of (((R T) U)->((R S) U)) % Found (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))) as proof of (((R S) T)->(((R T) U)->((R S) U))) % Found (and_rect10 (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))) as proof of ((R S) U) % Found ((and_rect1 ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))) as proof of ((R S) U) % Found (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))) as proof of ((R S) U) % Found (fun (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))) as proof of ((R S) U) % Found (fun (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))) as proof of (((and ((R S) T)) ((R T) U))->((R S) U)) % Found (fun (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))) as proof of (forall (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))) % Found (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))) as proof of (forall (T:fofType) (U:fofType), (((and ((R S) T)) ((R T) U))->((R S) U))) % Found (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))) as proof of (mtransitive R) % Found ((conj20 (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))))) as proof of ((and (mserial R)) (mtransitive R)) % Found (((conj2 (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))))) as proof of ((and (mserial R)) (mtransitive R)) % Found ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))))) as proof of ((and (mserial R)) (mtransitive R)) % Found ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4)))))) as proof of ((and (mserial R)) (mtransitive R)) % Found ((conj10 ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (((conj1 (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (fun (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((meuclidean R)->((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((mreflexive R)->((meuclidean R)->((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) % Found (and_rect00 (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found ((and_rect0 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)) % Found (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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 (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found ((conj00 (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found (((conj0 (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6)))))))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))))) as proof of ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) % Found (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))))) as proof of (forall (R:(fofType->(fofType->Prop))), ((iff ((and (mreflexive R)) (meuclidean R))) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) % Got proof (fun (R:(fofType->(fofType->Prop)))=> ((((conj (((and (mreflexive R)) (meuclidean R))->((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))) (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))))) % Time elapsed = 143.577407s % node=14031 cost=2767.000000 depth=49 % :::::::::::::::::::::: % % 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)) (mtransitive R))) (msymmetric R)))) (((and ((and (mserial R)) (mtransitive R))) (msymmetric R))->((and (mreflexive R)) (meuclidean R)))) (fun (x:((and (mreflexive R)) (meuclidean R)))=> (((fun (P:Type) (x0:((mreflexive R)->((meuclidean R)->P)))=> (((((and_rect (mreflexive R)) (meuclidean R)) P) x0) x)) ((and ((and (mserial R)) (mtransitive R))) (msymmetric R))) (fun (x0:(mreflexive R)) (x1:(meuclidean R))=> ((((conj ((and (mserial R)) (mtransitive R))) (msymmetric R)) ((((conj (mserial R)) (mtransitive R)) (fun (S:fofType)=> ((((ex_intro fofType) (fun (T:fofType)=> ((R S) T))) S) (x0 S)))) (fun (S:fofType) (T:fofType) (U:fofType) (x2:((and ((R S) T)) ((R T) U)))=> (((fun (P:Type) (x3:(((R S) T)->(((R T) U)->P)))=> (((((and_rect ((R S) T)) ((R T) U)) P) x3) x2)) ((R S) U)) (fun (x3:((R S) T)) (x4:((R T) U))=> (((fun (S0:fofType)=> (((fun (S0:fofType)=> ((x1 S0) S)) S0) U)) T) ((((conj ((R T) S)) ((R T) U)) (((fun (S1:fofType)=> (((fun (S1:fofType)=> ((x1 S1) T)) S1) S)) S) ((((conj ((R S) T)) ((R S) S)) x3) (x0 S)))) x4))))))) (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))))))))) (fun (x:((and ((and (mserial R)) (mtransitive R))) (msymmetric R)))=> (((fun (P:Type) (x0:(((and (mserial R)) (mtransitive R))->((msymmetric R)->P)))=> (((((and_rect ((and (mserial R)) (mtransitive R))) (msymmetric R)) P) x0) x)) ((and (mreflexive R)) (meuclidean R))) (fun (x0:((and (mserial R)) (mtransitive R))) (x1:(msymmetric R))=> (((fun (P:Type) (x2:((mserial R)->((mtransitive R)->P)))=> (((((and_rect (mserial R)) (mtransitive R)) P) x2) x0)) ((and (mreflexive R)) (meuclidean R))) (fun (x2:(mserial R)) (x3:(mtransitive 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 (T:fofType)=> (((x3 S) T) S)) x4) ((((conj ((R S) x4)) ((R x4) S)) x5) (((x1 S) x4) x5))))))) (fun (S:fofType) (T:fofType) (U:fofType) (x4:((and ((R S) T)) ((R S) U)))=> (((fun (P:Type) (x5:(((R S) T)->(((R S) U)->P)))=> (((((and_rect ((R S) T)) ((R S) U)) P) x5) x4)) ((R T) U)) (fun (x5:((R S) T)) (x6:((R S) U))=> (((fun (T0:fofType)=> (((x3 T) T0) U)) S) ((((conj ((R T) S)) ((R S) U)) (((x1 S) T) x5)) x6))))))))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------