include('Axioms/iterate^2.ax').
thf(tp_ty_2Eextreal_2Eextreal,type,( ty_2Eextreal_2Eextreal:del )).
thf(stp_ty_2Eextreal_2Eextreal,type,(tp__ty_2Eextreal_2Eextreal : $tType)).
thf(stp_inj_ty_2Eextreal_2Eextreal,type,(inj__ty_2Eextreal_2Eextreal : (tp__ty_2Eextreal_2Eextreal > $i))).
thf(stp_surj_ty_2Eextreal_2Eextreal,type,(surj__ty_2Eextreal_2Eextreal : ($i > tp__ty_2Eextreal_2Eextreal))).
thf(stp_inj_surj_ty_2Eextreal_2Eextreal,axiom,( ! [X:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (inj__ty_2Eextreal_2Eextreal @ X)) = X))).
thf(stp_inj_mem_ty_2Eextreal_2Eextreal,axiom,( ! [X:tp__ty_2Eextreal_2Eextreal] : (mem @ (inj__ty_2Eextreal_2Eextreal @ X) @ ty_2Eextreal_2Eextreal))).
thf(stp_iso_mem_ty_2Eextreal_2Eextreal,axiom,( ! [X:$i] : ((mem @ X @ ty_2Eextreal_2Eextreal) => (X = (inj__ty_2Eextreal_2Eextreal @ (surj__ty_2Eextreal_2Eextreal @ X)))))).
thf(tp_c_2Eextreal_2EEXTREAL__SUM__IMAGE,type,(c_2Eextreal_2EEXTREAL__SUM__IMAGE:(del>$i))).
thf(mem_c_2Eextreal_2EEXTREAL__SUM__IMAGE,axiom,( ! [A_27a:del] : (mem@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@(arr@(arr@A_27a@ty_2Eextreal_2Eextreal)@(arr@(arr@A_27a@bool)@ty_2Eextreal_2Eextreal))))).
thf(tp_c_2Eextreal_2ENegInf,type,(c_2Eextreal_2ENegInf:$i)).
thf(mem_c_2Eextreal_2ENegInf,axiom,(mem@c_2Eextreal_2ENegInf@ty_2Eextreal_2Eextreal)).
thf(stp_fo_c_2Eextreal_2ENegInf,type,(fo__c_2Eextreal_2ENegInf:tp__ty_2Eextreal_2Eextreal)).
thf(stp_eq_fo_c_2Eextreal_2ENegInf,axiom,((inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf)).
thf(tp_c_2Eextreal_2ENormal,type,(c_2Eextreal_2ENormal:$i)).
thf(mem_c_2Eextreal_2ENormal,axiom,(mem@c_2Eextreal_2ENormal@(arr@ty_2Erealax_2Ereal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2ENormal,type,(fo__c_2Eextreal_2ENormal:(tp__ty_2Erealax_2Ereal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2ENormal,axiom,( ! [X0:tp__ty_2Erealax_2Ereal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2ENormal @ X0)) = (ap @ c_2Eextreal_2ENormal @ (inj__ty_2Erealax_2Ereal @ X0))))).
thf(tp_c_2Eextreal_2EPosInf,type,(c_2Eextreal_2EPosInf:$i)).
thf(mem_c_2Eextreal_2EPosInf,axiom,(mem@c_2Eextreal_2EPosInf@ty_2Eextreal_2Eextreal)).
thf(stp_fo_c_2Eextreal_2EPosInf,type,(fo__c_2Eextreal_2EPosInf:tp__ty_2Eextreal_2Eextreal)).
thf(stp_eq_fo_c_2Eextreal_2EPosInf,axiom,((inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf)).
thf(tp_c_2Eextreal_2EQ__set,type,(c_2Eextreal_2EQ__set:$i)).
thf(mem_c_2Eextreal_2EQ__set,axiom,(mem@c_2Eextreal_2EQ__set@(arr@ty_2Eextreal_2Eextreal@bool))).
thf(tp_c_2Eextreal_2Eceiling,type,(c_2Eextreal_2Eceiling:$i)).
thf(mem_c_2Eextreal_2Eceiling,axiom,(mem@c_2Eextreal_2Eceiling@(arr@ty_2Eextreal_2Eextreal@ty_2Enum_2Enum))).
thf(stp_fo_c_2Eextreal_2Eceiling,type,(fo__c_2Eextreal_2Eceiling:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Enum_2Enum))).
thf(stp_eq_fo_c_2Eextreal_2Eceiling,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Enum_2Enum @ (fo__c_2Eextreal_2Eceiling @ X0)) = (ap @ c_2Eextreal_2Eceiling @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eext__mono__decreasing,type,(c_2Eextreal_2Eext__mono__decreasing:$i)).
thf(mem_c_2Eextreal_2Eext__mono__decreasing,axiom,(mem@c_2Eextreal_2Eext__mono__decreasing@(arr@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)@bool))).
thf(tp_c_2Eextreal_2Eext__mono__increasing,type,(c_2Eextreal_2Eext__mono__increasing:$i)).
thf(mem_c_2Eextreal_2Eext__mono__increasing,axiom,(mem@c_2Eextreal_2Eext__mono__increasing@(arr@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)@bool))).
thf(tp_c_2Eextreal_2Eext__suminf,type,(c_2Eextreal_2Eext__suminf:$i)).
thf(mem_c_2Eextreal_2Eext__suminf,axiom,(mem@c_2Eextreal_2Eext__suminf@(arr@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)@ty_2Eextreal_2Eextreal))).
thf(tp_c_2Eextreal_2Eextreal__CASE,type,(c_2Eextreal_2Eextreal__CASE:(del>$i))).
thf(mem_c_2Eextreal_2Eextreal__CASE,axiom,( ! [A_27a:del] : (mem@(c_2Eextreal_2Eextreal__CASE@A_27a)@(arr@ty_2Eextreal_2Eextreal@(arr@A_27a@(arr@A_27a@(arr@(arr@ty_2Erealax_2Ereal@A_27a)@A_27a))))))).
thf(tp_c_2Eextreal_2Eextreal__abs,type,(c_2Eextreal_2Eextreal__abs:$i)).
thf(mem_c_2Eextreal_2Eextreal__abs,axiom,(mem@c_2Eextreal_2Eextreal__abs@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__abs,type,(fo__c_2Eextreal_2Eextreal__abs:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__abs,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__abs @ X0)) = (ap @ c_2Eextreal_2Eextreal__abs @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__add,type,(c_2Eextreal_2Eextreal__add:$i)).
thf(mem_c_2Eextreal_2Eextreal__add,axiom,(mem@c_2Eextreal_2Eextreal__add@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__add,type,(fo__c_2Eextreal_2Eextreal__add:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__add,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__add @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__add @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__ainv,type,(c_2Eextreal_2Eextreal__ainv:$i)).
thf(mem_c_2Eextreal_2Eextreal__ainv,axiom,(mem@c_2Eextreal_2Eextreal__ainv@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__ainv,type,(fo__c_2Eextreal_2Eextreal__ainv:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__ainv,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__ainv @ X0)) = (ap @ c_2Eextreal_2Eextreal__ainv @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__div,type,(c_2Eextreal_2Eextreal__div:$i)).
thf(mem_c_2Eextreal_2Eextreal__div,axiom,(mem@c_2Eextreal_2Eextreal__div@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__div,type,(fo__c_2Eextreal_2Eextreal__div:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__div,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__div @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__div @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__exp,type,(c_2Eextreal_2Eextreal__exp:$i)).
thf(mem_c_2Eextreal_2Eextreal__exp,axiom,(mem@c_2Eextreal_2Eextreal__exp@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__exp,type,(fo__c_2Eextreal_2Eextreal__exp:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__exp,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__exp @ X0)) = (ap @ c_2Eextreal_2Eextreal__exp @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__inf,type,(c_2Eextreal_2Eextreal__inf:$i)).
thf(mem_c_2Eextreal_2Eextreal__inf,axiom,(mem@c_2Eextreal_2Eextreal__inf@(arr@(arr@ty_2Eextreal_2Eextreal@bool)@ty_2Eextreal_2Eextreal))).
thf(tp_c_2Eextreal_2Eextreal__inv,type,(c_2Eextreal_2Eextreal__inv:$i)).
thf(mem_c_2Eextreal_2Eextreal__inv,axiom,(mem@c_2Eextreal_2Eextreal__inv@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__inv,type,(fo__c_2Eextreal_2Eextreal__inv:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__inv,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__inv @ X0)) = (ap @ c_2Eextreal_2Eextreal__inv @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__le,type,(c_2Eextreal_2Eextreal__le:$i)).
thf(mem_c_2Eextreal_2Eextreal__le,axiom,(mem@c_2Eextreal_2Eextreal__le@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool)))).
thf(tp_c_2Eextreal_2Eextreal__lg,type,(c_2Eextreal_2Eextreal__lg:$i)).
thf(mem_c_2Eextreal_2Eextreal__lg,axiom,(mem@c_2Eextreal_2Eextreal__lg@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__lg,type,(fo__c_2Eextreal_2Eextreal__lg:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__lg,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__lg @ X0)) = (ap @ c_2Eextreal_2Eextreal__lg @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__logr,type,(c_2Eextreal_2Eextreal__logr:$i)).
thf(mem_c_2Eextreal_2Eextreal__logr,axiom,(mem@c_2Eextreal_2Eextreal__logr@(arr@ty_2Erealax_2Ereal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__logr,type,(fo__c_2Eextreal_2Eextreal__logr:(tp__ty_2Erealax_2Ereal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__logr,axiom,( ! [X0:tp__ty_2Erealax_2Ereal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__logr @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__logr @ (inj__ty_2Erealax_2Ereal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__lt,type,(c_2Eextreal_2Eextreal__lt:$i)).
thf(mem_c_2Eextreal_2Eextreal__lt,axiom,(mem@c_2Eextreal_2Eextreal__lt@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool)))).
thf(tp_c_2Eextreal_2Eextreal__max,type,(c_2Eextreal_2Eextreal__max:$i)).
thf(mem_c_2Eextreal_2Eextreal__max,axiom,(mem@c_2Eextreal_2Eextreal__max@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__max,type,(fo__c_2Eextreal_2Eextreal__max:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__max,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__max @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__max @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__min,type,(c_2Eextreal_2Eextreal__min:$i)).
thf(mem_c_2Eextreal_2Eextreal__min,axiom,(mem@c_2Eextreal_2Eextreal__min@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__min,type,(fo__c_2Eextreal_2Eextreal__min:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__min,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__min @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__min @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__mul,type,(c_2Eextreal_2Eextreal__mul:$i)).
thf(mem_c_2Eextreal_2Eextreal__mul,axiom,(mem@c_2Eextreal_2Eextreal__mul@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__mul,type,(fo__c_2Eextreal_2Eextreal__mul:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__mul,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__mul @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__mul @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__of__num,type,(c_2Eextreal_2Eextreal__of__num:$i)).
thf(mem_c_2Eextreal_2Eextreal__of__num,axiom,(mem@c_2Eextreal_2Eextreal__of__num@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__of__num,type,(fo__c_2Eextreal_2Eextreal__of__num:(tp__ty_2Enum_2Enum > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__of__num,axiom,( ! [X0:tp__ty_2Enum_2Enum] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__of__num @ X0)) = (ap @ c_2Eextreal_2Eextreal__of__num @ (inj__ty_2Enum_2Enum @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__pow,type,(c_2Eextreal_2Eextreal__pow:$i)).
thf(mem_c_2Eextreal_2Eextreal__pow,axiom,(mem@c_2Eextreal_2Eextreal__pow@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__pow,type,(fo__c_2Eextreal_2Eextreal__pow:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Enum_2Enum > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__pow,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Enum_2Enum] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__pow @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__pow @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Enum_2Enum @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__size,type,(c_2Eextreal_2Eextreal__size:$i)).
thf(mem_c_2Eextreal_2Eextreal__size,axiom,(mem@c_2Eextreal_2Eextreal__size@(arr@ty_2Eextreal_2Eextreal@ty_2Enum_2Enum))).
thf(stp_fo_c_2Eextreal_2Eextreal__size,type,(fo__c_2Eextreal_2Eextreal__size:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Enum_2Enum))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__size,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Enum_2Enum @ (fo__c_2Eextreal_2Eextreal__size @ X0)) = (ap @ c_2Eextreal_2Eextreal__size @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__sqrt,type,(c_2Eextreal_2Eextreal__sqrt:$i)).
thf(mem_c_2Eextreal_2Eextreal__sqrt,axiom,(mem@c_2Eextreal_2Eextreal__sqrt@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal))).
thf(stp_fo_c_2Eextreal_2Eextreal__sqrt,type,(fo__c_2Eextreal_2Eextreal__sqrt:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__sqrt,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ (fo__c_2Eextreal_2Eextreal__sqrt @ X0)) = (ap @ c_2Eextreal_2Eextreal__sqrt @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(tp_c_2Eextreal_2Eextreal__sub,type,(c_2Eextreal_2Eextreal__sub:$i)).
thf(mem_c_2Eextreal_2Eextreal__sub,axiom,(mem@c_2Eextreal_2Eextreal__sub@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)))).
thf(stp_fo_c_2Eextreal_2Eextreal__sub,type,(fo__c_2Eextreal_2Eextreal__sub:(tp__ty_2Eextreal_2Eextreal > (tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal)))).
thf(stp_eq_fo_c_2Eextreal_2Eextreal__sub,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ( ! [X1:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Eextreal_2Eextreal @ ((fo__c_2Eextreal_2Eextreal__sub @ X0) @ X1)) = (ap @ (ap @ c_2Eextreal_2Eextreal__sub @ (inj__ty_2Eextreal_2Eextreal @ X0)) @ (inj__ty_2Eextreal_2Eextreal @ X1)))))).
thf(tp_c_2Eextreal_2Eextreal__sup,type,(c_2Eextreal_2Eextreal__sup:$i)).
thf(mem_c_2Eextreal_2Eextreal__sup,axiom,(mem@c_2Eextreal_2Eextreal__sup@(arr@(arr@ty_2Eextreal_2Eextreal@bool)@ty_2Eextreal_2Eextreal))).
thf(tp_c_2Eextreal_2Emono__decreasing,type,(c_2Eextreal_2Emono__decreasing:$i)).
thf(mem_c_2Eextreal_2Emono__decreasing,axiom,(mem@c_2Eextreal_2Emono__decreasing@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@bool))).
thf(tp_c_2Eextreal_2Emono__increasing,type,(c_2Eextreal_2Emono__increasing:$i)).
thf(mem_c_2Eextreal_2Emono__increasing,axiom,(mem@c_2Eextreal_2Emono__increasing@(arr@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@bool))).
thf(tp_c_2Eextreal_2Eopen__interval,type,(c_2Eextreal_2Eopen__interval:$i)).
thf(mem_c_2Eextreal_2Eopen__interval,axiom,(mem@c_2Eextreal_2Eopen__interval@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))))).
thf(tp_c_2Eextreal_2Eopen__intervals__set,type,(c_2Eextreal_2Eopen__intervals__set:$i)).
thf(mem_c_2Eextreal_2Eopen__intervals__set,axiom,(mem@c_2Eextreal_2Eopen__intervals__set@(arr@(arr@ty_2Eextreal_2Eextreal@bool)@bool))).
thf(tp_c_2Eextreal_2Erational__intervals,type,(c_2Eextreal_2Erational__intervals:$i)).
thf(mem_c_2Eextreal_2Erational__intervals,axiom,(mem@c_2Eextreal_2Erational__intervals@(arr@(arr@ty_2Eextreal_2Eextreal@bool)@bool))).
thf(tp_c_2Eextreal_2Ereal,type,(c_2Eextreal_2Ereal:$i)).
thf(mem_c_2Eextreal_2Ereal,axiom,(mem@c_2Eextreal_2Ereal@(arr@ty_2Eextreal_2Eextreal@ty_2Erealax_2Ereal))).
thf(stp_fo_c_2Eextreal_2Ereal,type,(fo__c_2Eextreal_2Ereal:(tp__ty_2Eextreal_2Eextreal > tp__ty_2Erealax_2Ereal))).
thf(stp_eq_fo_c_2Eextreal_2Ereal,axiom,( ! [X0:tp__ty_2Eextreal_2Eextreal] : ((inj__ty_2Erealax_2Ereal @ (fo__c_2Eextreal_2Ereal @ X0)) = (ap @ c_2Eextreal_2Ereal @ (inj__ty_2Eextreal_2Eextreal @ X0))))).
thf(ax_thm_2Eextreal_2Eextreal__TY__DEF,axiom,(? [V0rep:$i] : ((mem@V0rep@(arr@ty_2Eextreal_2Eextreal@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal))) & (p@(ap@(ap@(c_2Ebool_2ETYPE__DEFINITION@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal)@ty_2Eextreal_2Eextreal)@(lam@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal)@(^ [V1a0:$i] : (ap@(c_2Ebool_2E_21@(arr@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal)@bool))@(lam@(arr@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal)@bool)@(^ [V2_27extreal_27:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(c_2Ebool_2E_21@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal))@(lam@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal)@(^ [V3a0:$i] : (ap@(ap@c_2Emin_2E_3D_3D_3E@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@(c_2Emin_2E_3D@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal))@V3a0)@(ap@(ap@(ap@(c_2Eind__type_2ECONSTR@ty_2Erealax_2Ereal)@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))@(c_2Ebool_2EARB@ty_2Erealax_2Ereal))@(lam@ty_2Enum_2Enum@(^ [V4n:$i] : (c_2Eind__type_2EBOTTOM@ty_2Erealax_2Ereal))))))@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@(c_2Emin_2E_3D@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal))@V3a0)@(ap@(ap@(ap@(c_2Eind__type_2ECONSTR@ty_2Erealax_2Ereal)@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(c_2Ebool_2EARB@ty_2Erealax_2Ereal))@(lam@ty_2Enum_2Enum@(^ [V5n:$i] : (c_2Eind__type_2EBOTTOM@ty_2Erealax_2Ereal))))))@(ap@(c_2Ebool_2E_3F@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V6a:$i] : (ap@(ap@(c_2Emin_2E_3D@(ty_2Eind__type_2Erecspace@ty_2Erealax_2Ereal))@V3a0)@(ap@(lam@ty_2Erealax_2Ereal@(^ [V7a:$i] : (ap@(ap@(ap@(c_2Eind__type_2ECONSTR@ty_2Erealax_2Ereal)@(ap@c_2Enum_2ESUC@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@V7a)@(lam@ty_2Enum_2Enum@(^ [V8n:$i] : (c_2Eind__type_2EBOTTOM@ty_2Erealax_2Ereal))))))@V6a))))))))@(ap@V2_27extreal_27@V3a0))))))@(ap@V2_27extreal_27@V1a0))))))))@V0rep))))).
thf(ax_thm_2Eextreal_2Eextreal__case__def,axiom,( ! [A_27a:del]: (( ! [V0v:$i] : ((mem@V0v@A_27a) => ( ! [V1v1:$i] : ((mem@V1v1@A_27a) => ( ! [V2f:$i] : ((mem@V2f@(arr@ty_2Erealax_2Ereal@A_27a)) => ((ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@V0v)@V1v1)@V2f) = V0v))))))) & (( ! [V3v:$i] : ((mem@V3v@A_27a) => ( ! [V4v1:$i] : ((mem@V4v1@A_27a) => ( ! [V5f:$i] : ((mem@V5f@(arr@ty_2Erealax_2Ereal@A_27a)) => ((ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@V3v)@V4v1)@V5f) = V4v1))))))) & ( ! [V6a:tp__ty_2Erealax_2Ereal] : ( ! [V7v:$i] : ((mem@V7v@A_27a) => ( ! [V8v1:$i] : ((mem@V8v1@A_27a) => ( ! [V9f:$i] : ((mem@V9f@(arr@ty_2Erealax_2Ereal@A_27a)) => ((ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6a)))@V7v)@V8v1)@V9f) = (ap@V9f@(inj__ty_2Erealax_2Ereal @ V6a)))))))))))))).
thf(ax_thm_2Eextreal_2Eextreal__size__def,axiom,(((surj__ty_2Enum_2Enum @ (ap@c_2Eextreal_2Eextreal__size@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Enum_2E0) & (((surj__ty_2Enum_2Enum @ (ap@c_2Eextreal_2Eextreal__size@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Enum_2E0) & ( ! [V0a:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Enum_2Enum @ (ap@c_2Eextreal_2Eextreal__size@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0a)))) = (surj__ty_2Enum_2Enum @ (ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))).
thf(conj_thm_2Eextreal_2Edatatype__extreal,lemma,( ! [V0extreal:$i] : ((mem@V0extreal@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@(arr@(arr@ty_2Erealax_2Ereal@ty_2Eextreal_2Eextreal)@bool)))) => (p@(ap@(c_2Ebool_2EDATATYPE@bool)@(ap@(ap@(ap@V0extreal@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@c_2Eextreal_2ENormal)))))).
thf(conj_thm_2Eextreal_2Eextreal__distinct,lemma,((~~ (fo__c_2Eextreal_2ENegInf = fo__c_2Eextreal_2EPosInf)) & (( ! [V0a:tp__ty_2Erealax_2Ereal] : (~~ (fo__c_2Eextreal_2ENegInf = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0a)))))) & ( ! [V1a:tp__ty_2Erealax_2Ereal] : (~~ (fo__c_2Eextreal_2EPosInf = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1a))))))))).
thf(conj_thm_2Eextreal_2Eextreal__nchotomy,lemma,( ! [V0ee:tp__ty_2Eextreal_2Eextreal] : ((V0ee = fo__c_2Eextreal_2ENegInf) | ((V0ee = fo__c_2Eextreal_2EPosInf) | (? [V1r:tp__ty_2Erealax_2Ereal] : (V0ee = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1r))))))))).
thf(conj_thm_2Eextreal_2Eextreal__Axiom,lemma,( ! [A_27a:del]: ( ! [V0f0:$i] : ((mem@V0f0@A_27a) => ( ! [V1f1:$i] : ((mem@V1f1@A_27a) => ( ! [V2f2:$i] : ((mem@V2f2@(arr@ty_2Erealax_2Ereal@A_27a)) => (? [V3fn:$i] : ((mem@V3fn@(arr@ty_2Eextreal_2Eextreal@A_27a)) & (((ap@V3fn@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)) = V0f0) & (((ap@V3fn@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)) = V1f1) & ( ! [V4a:tp__ty_2Erealax_2Ereal] : ((ap@V3fn@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4a))) = (ap@V2f2@(inj__ty_2Erealax_2Ereal @ V4a)))))))))))))))).
thf(conj_thm_2Eextreal_2Eextreal__induction,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@bool)) => (((p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & ((p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & ( ! [V1r:tp__ty_2Erealax_2Ereal] : (p@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1r))))))) => ( ! [V2e:tp__ty_2Eextreal_2Eextreal] : (p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V2e)))))))).
thf(conj_thm_2Eextreal_2Eextreal__case__cong,lemma,( ! [A_27a:del]: ( ! [V0v_27:$i] : ((mem@V0v_27@A_27a) => ( ! [V1v1_27:$i] : ((mem@V1v1_27@A_27a) => ( ! [V2f_27:$i] : ((mem@V2f_27@(arr@ty_2Erealax_2Ereal@A_27a)) => ( ! [V3M:tp__ty_2Eextreal_2Eextreal] : ( ! [V4M_27:tp__ty_2Eextreal_2Eextreal] : ( ! [V5v:$i] : ((mem@V5v@A_27a) => ( ! [V6v1:$i] : ((mem@V6v1@A_27a) => ( ! [V7f:$i] : ((mem@V7f@(arr@ty_2Erealax_2Ereal@A_27a)) => (((V3M = V4M_27) & (((V4M_27 = fo__c_2Eextreal_2ENegInf) => (V5v = V0v_27)) & (((V4M_27 = fo__c_2Eextreal_2EPosInf) => (V6v1 = V1v1_27)) & ( ! [V8a:tp__ty_2Erealax_2Ereal] : ((V4M_27 = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V8a)))) => ((ap@V7f@(inj__ty_2Erealax_2Ereal @ V8a)) = (ap@V2f_27@(inj__ty_2Erealax_2Ereal @ V8a)))))))) => ((ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(inj__ty_2Eextreal_2Eextreal @ V3M))@V5v)@V6v1)@V7f) = (ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(inj__ty_2Eextreal_2Eextreal @ V4M_27))@V0v_27)@V1v1_27)@V2f_27))))))))))))))))))).
thf(conj_thm_2Eextreal_2Eextreal__case__eq,lemma,( ! [A_27a:del]: ( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1v:$i] : ((mem@V1v@A_27a) => ( ! [V2v1:$i] : ((mem@V2v1@A_27a) => ( ! [V3f:$i] : ((mem@V3f@(arr@ty_2Erealax_2Ereal@A_27a)) => ( ! [V4v_27:$i] : ((mem@V4v_27@A_27a) => (((ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@A_27a)@(inj__ty_2Eextreal_2Eextreal @ V0x))@V1v)@V2v1)@V3f) = V4v_27) <=> (((V0x = fo__c_2Eextreal_2ENegInf) & (V1v = V4v_27)) | (((V0x = fo__c_2Eextreal_2EPosInf) & (V2v1 = V4v_27)) | (? [V5r:tp__ty_2Erealax_2Ereal] : ((V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5r)))) & ((ap@V3f@(inj__ty_2Erealax_2Ereal @ V5r)) = V4v_27))))))))))))))))).
thf(ax_thm_2Eextreal_2Eextreal__of__num__def,axiom,( ! [V0n:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V0n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V0n))))))).
thf(ax_thm_2Eextreal_2Ereal__def,axiom,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eextreal_2Ereal@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Erealax_2Ereal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Erealax_2Ereal)@(ap@(ap@c_2Ebool_2E_5C_2F@(ap@(ap@(c_2Emin_2E_3D@ty_2Eextreal_2Eextreal)@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))@(ap@(ap@(c_2Emin_2E_3D@ty_2Eextreal_2Eextreal)@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(c_2Emin_2E_40@ty_2Erealax_2Ereal)@(lam@ty_2Erealax_2Ereal@(^ [V1r:$i] : (ap@(ap@(c_2Emin_2E_3D@ty_2Eextreal_2Eextreal)@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2ENormal@V1r)))))))))).
thf(conj_thm_2Eextreal_2Ereal__normal,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Erealax_2Ereal @ (ap@c_2Eextreal_2Ereal@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = V0x))).
thf(conj_thm_2Eextreal_2Enormal__real,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Eextreal_2Ereal@(inj__ty_2Eextreal_2Eextreal @ V0x)))) = V0x)))).
thf(conj_thm_2Eextreal_2Eextreal__add__ind,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2y)))))) & (( ! [V3a:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V3a)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & (( ! [V4v2:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & (( ! [V5v5:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v5))))) & ( ! [V6v3:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))))))))) => ( ! [V7v:tp__ty_2Eextreal_2Eextreal] : ( ! [V8v1:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V7v))@(inj__ty_2Eextreal_2Eextreal @ V8v1))))))))).
thf(conj_thm_2Eextreal_2Eextreal__add__def,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Eextreal_2Eextreal] : ( ! [V3v2:tp__ty_2Erealax_2Ereal] : ( ! [V4v5:tp__ty_2Erealax_2Ereal] : ( ! [V5v3:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Erealax_2Ereal__add@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V2a))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V3v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v5)))) = fo__c_2Eextreal_2ENegInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf)))))))))))))).
thf(conj_thm_2Eextreal_2Eextreal__sub__ind,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2y)))))) & (( ! [V3a:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V3a)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & (( ! [V4v2:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & (( ! [V5v5:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v5))))) & ( ! [V6v3:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))))))))) => ( ! [V7v:tp__ty_2Eextreal_2Eextreal] : ( ! [V8v1:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V7v))@(inj__ty_2Eextreal_2Eextreal @ V8v1))))))))).
thf(conj_thm_2Eextreal_2Eextreal__sub__def,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Eextreal_2Eextreal] : ( ! [V3v2:tp__ty_2Erealax_2Ereal] : ( ! [V4v5:tp__ty_2Erealax_2Ereal] : ( ! [V5v3:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Ereal_2Ereal__sub@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V2a))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2ENegInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V3v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2ENegInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v5)))) = fo__c_2Eextreal_2ENegInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2EPosInf)))))))))))))).
thf(conj_thm_2Eextreal_2Eextreal__le__ind,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : ( ! [V2y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2y)))))) & (( ! [V3a:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ V3a)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & (( ! [V4v2:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & (( ! [V5v3:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))) & ( ! [V6v5:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6v5))))))))))) => ( ! [V7v:tp__ty_2Eextreal_2Eextreal] : ( ! [V8v1:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V7v))@(inj__ty_2Eextreal_2Eextreal @ V8v1))))))))).
thf(conj_thm_2Eextreal_2Eextreal__le__def,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2a:tp__ty_2Eextreal_2Eextreal] : ( ! [V3v2:tp__ty_2Erealax_2Ereal] : ( ! [V4v3:tp__ty_2Erealax_2Ereal] : ( ! [V5v5:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ V2a))) <=> $true) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) <=> $true) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V3v2)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) <=> $true) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) <=> $false) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4v3)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) <=> $false) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5v5)))) <=> $false)))))))))))))).
thf(ax_thm_2Eextreal_2Eextreal__lt__def,axiom,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> (~ (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Eextreal__mul__ind,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))) => (((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & ((p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & (( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))) & (( ! [V2y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2y))))) & (( ! [V3x:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V3x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))) & (( ! [V4y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4y))))) & ( ! [V5x:tp__ty_2Erealax_2Ereal] : ( ! [V6y:tp__ty_2Erealax_2Ereal] : (p@(ap@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V5x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6y)))))))))))))) => ( ! [V7v:tp__ty_2Eextreal_2Eextreal] : ( ! [V8v1:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V7v))@(inj__ty_2Eextreal_2Eextreal @ V8v1))))))))).
thf(conj_thm_2Eextreal_2Eextreal__mul__def,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2ENegInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Erealax_2Ereal)@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))))) & ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))))))))))).
thf(ax_thm_2Eextreal_2Eextreal__inv__def,axiom,(((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) & ( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Erealax_2Einv@(inj__ty_2Erealax_2Ereal @ V0x))))))))).
thf(ax_thm_2Eextreal_2Eextreal__ainv__def,axiom,(((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2EPosInf) & (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2ENegInf) & ( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Erealax_2Ereal__neg@(inj__ty_2Erealax_2Ereal @ V0x))))))))).
thf(ax_thm_2Eextreal_2Eextreal__div__def,axiom,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(ax_thm_2Eextreal_2Eextreal__abs__primitive__def,axiom,(c_2Eextreal_2Eextreal__abs = (ap@(ap@(c_2Erelation_2EWFREC@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)@(ap@(c_2Emin_2E_40@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool)))@(lam@(arr@ty_2Eextreal_2Eextreal@(arr@ty_2Eextreal_2Eextreal@bool))@(^ [V0R:$i] : (ap@(c_2Erelation_2EWF@ty_2Eextreal_2Eextreal)@V0R)))))@(lam@(arr@ty_2Eextreal_2Eextreal@ty_2Eextreal_2Eextreal)@(^ [V1extreal__abs:$i] : (lam@ty_2Eextreal_2Eextreal@(^ [V2a:$i] : (ap@(ap@(ap@(ap@(c_2Eextreal_2Eextreal__CASE@ty_2Eextreal_2Eextreal)@V2a)@(ap@(c_2Ecombin_2EI@ty_2Eextreal_2Eextreal)@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))@(ap@(c_2Ecombin_2EI@ty_2Eextreal_2Eextreal)@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))@(lam@ty_2Erealax_2Ereal@(^ [V3x:$i] : (ap@(c_2Ecombin_2EI@ty_2Eextreal_2Eextreal)@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Eabs@V3x))))))))))))).
thf(conj_thm_2Eextreal_2Eextreal__abs__ind,lemma,( ! [V0P:$i] : ((mem@V0P@(arr@ty_2Eextreal_2Eextreal@bool)) => ((( ! [V1x:tp__ty_2Erealax_2Ereal] : (p@(ap@V0P@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x))))) & ((p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) & (p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))) => ( ! [V2v:tp__ty_2Eextreal_2Eextreal] : (p@(ap@V0P@(inj__ty_2Eextreal_2Eextreal @ V2v)))))))).
thf(conj_thm_2Eextreal_2Eextreal__abs__def,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__abs@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Eabs@(inj__ty_2Erealax_2Ereal @ V0x))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2EPosInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf))))).
thf(ax_thm_2Eextreal_2Eextreal__logr__def,axiom,(( ! [V0b:tp__ty_2Erealax_2Ereal] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(inj__ty_2Erealax_2Ereal @ V0b))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Eutil__prob_2Elogr@(inj__ty_2Erealax_2Ereal @ V0b))@(inj__ty_2Erealax_2Ereal @ V1x))))))) & (( ! [V2b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf)) & ( ! [V3b:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(inj__ty_2Erealax_2Ereal @ V3b))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf))))).
thf(ax_thm_2Eextreal_2Eextreal__lg__def,axiom,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__lg@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(ax_thm_2Eextreal_2Eextreal__exp__def,axiom,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__exp@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Etransc_2Eexp@(inj__ty_2Erealax_2Ereal @ V0x)))))) & (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__exp@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__exp@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(ax_thm_2Eextreal_2Eextreal__pow__def,axiom,(( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0a)))@(inj__ty_2Enum_2Enum @ V1n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V0a))@(inj__ty_2Enum_2Enum @ V1n))))))) & (( ! [V2n:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Enum_2Enum @ V2n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V2n))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))) & ( ! [V3n:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Enum_2Enum @ V3n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@(ap@(c_2Emin_2E_3D@ty_2Enum_2Enum)@(inj__ty_2Enum_2Enum @ V3n))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2ENormal@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@c_2Earithmetic_2EEVEN@(inj__ty_2Enum_2Enum @ V3n)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))))))))).
thf(ax_thm_2Eextreal_2Eextreal__sqrt__def,axiom,(( ! [V0x:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__sqrt@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@c_2Etransc_2Esqrt@(inj__ty_2Erealax_2Ereal @ V0x)))))) & ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf))).
thf(conj_thm_2Eextreal_2EREAL__LT__LMUL__0__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__RMUL__0__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1y)))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1y))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1y))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))))))).
thf(conj_thm_2Eextreal_2EREAL__LT__RDIV__EQ__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Erealax_2Ereal @ V0x))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Eextreal_2EREAL__LE__RDIV__EQ__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V1y))@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Erealax_2Ereal @ V0x))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Eextreal_2EPOW__POS__EVEN,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))) <=> (p@(ap@c_2Earithmetic_2EEVEN@(inj__ty_2Enum_2Enum @ V0n)))))))).
thf(conj_thm_2Eextreal_2EPOW__NEG__ODD,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V1x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@(ap@c_2Ereal_2Epow@(inj__ty_2Erealax_2Ereal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (p@(ap@c_2Earithmetic_2EODD@(inj__ty_2Enum_2Enum @ V0n)))))))).
thf(conj_thm_2Eextreal_2ELOGR__MONO__LE,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) & ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V1y))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V2b))))) => ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Eutil__prob_2Elogr@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Eutil__prob_2Elogr@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(conj_thm_2Eextreal_2ELOGR__MONO__LE__IMP,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ( ! [V2b:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) & ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))) & (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Erealax_2Ereal @ V2b))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Eutil__prob_2Elogr@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Eutil__prob_2Elogr@(inj__ty_2Erealax_2Ereal @ V2b))@(inj__ty_2Erealax_2Ereal @ V1y))))))))).
thf(ax_thm_2Eextreal_2Emono__increasing__def,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Eextreal_2Emono__increasing@V0f)) <=> ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Enum_2Enum @ V1m)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V2n))))))))))).
thf(conj_thm_2Eextreal_2Emono__increasing__suc,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Eextreal_2Emono__increasing@V0f)) <=> ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Enum_2Enum @ V1n)))@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n)))))))))).
thf(ax_thm_2Eextreal_2Emono__decreasing__def,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Eextreal_2Emono__decreasing@V0f)) <=> ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(inj__ty_2Enum_2Enum @ V2n)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V1m))))))))))).
thf(conj_thm_2Eextreal_2Emono__decreasing__suc,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ((p@(ap@c_2Eextreal_2Emono__decreasing@V0f)) <=> ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n))))@(ap@V0f@(inj__ty_2Enum_2Enum @ V1n))))))))).
thf(conj_thm_2Eextreal_2Emono__increasing__converges__to__sup,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Erealax_2Ereal)) => ( ! [V1r:tp__ty_2Erealax_2Ereal] : (((p@(ap@c_2Eextreal_2Emono__increasing@V0f)) & (p@(ap@(ap@c_2Eseq_2E_2D_2D_3E@V0f)@(inj__ty_2Erealax_2Ereal @ V1r)))) => (V1r = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Esup@(ap@(ap@(c_2Epred__set_2EIMAGE@ty_2Enum_2Enum@ty_2Erealax_2Ereal)@V0f)@(c_2Epred__set_2EUNIV@ty_2Enum_2Enum)))))))))).
thf(conj_thm_2Eextreal_2Eextreal__cases,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((V0x = fo__c_2Eextreal_2ENegInf) | ((V0x = fo__c_2Eextreal_2EPosInf) | (? [V1r:tp__ty_2Erealax_2Ereal] : (V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1r))))))))).
thf(conj_thm_2Eextreal_2Eextreal__eq__zero,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V0x = (surj__ty_2Erealax_2Ereal @ (ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Eextreal_2Eextreal__not__infty,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ((~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))) = fo__c_2Eextreal_2ENegInf)) & (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))) = fo__c_2Eextreal_2EPosInf))))).
thf(conj_thm_2Eextreal_2Enum__not__infty,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V0n))) = fo__c_2Eextreal_2ENegInf)) & (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V0n))) = fo__c_2Eextreal_2EPosInf))))).
thf(conj_thm_2Eextreal_2Eextreal__11,lemma,( ! [V0a:tp__ty_2Erealax_2Ereal] : ( ! [V1a_27:tp__ty_2Erealax_2Ereal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0a))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1a_27)))) <=> (V0a = V1a_27))))).
thf(conj_thm_2Eextreal_2Emul__rzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))).
thf(conj_thm_2Eextreal_2Emul__lzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))).
thf(conj_thm_2Eextreal_2Emul__rone,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = V0x))).
thf(conj_thm_2Eextreal_2Emul__lone,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = V0x))).
thf(conj_thm_2Eextreal_2Eentire,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> ((V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) | (V1y = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Eextreal_2Eextreal__lt__eq,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) <=> (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))).
thf(conj_thm_2Eextreal_2Ele__refl,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V0x))))).
thf(conj_thm_2Eextreal_2Elt__refl,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (~~ (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Ele__infty,lemma,(( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))))) & (( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) <=> (V1x = fo__c_2Eextreal_2ENegInf))) & ( ! [V2x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V2x))) <=> (V2x = fo__c_2Eextreal_2EPosInf)))))).
thf(conj_thm_2Eextreal_2Elt__infty,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) & ((~~ (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf)))) & ((~~ (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) & (((~~ (V0x = fo__c_2Eextreal_2EPosInf)) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf)))) & ((~~ (V0x = fo__c_2Eextreal_2ENegInf)) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))))))).
thf(conj_thm_2Eextreal_2Elt__imp__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Elt__imp__ne,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) => (~~ (V0x = V1y)))))).
thf(conj_thm_2Eextreal_2Ele__trans,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Elt__trans,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Elet__trans,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Ele__antisym,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) <=> (V0x = V1y))))).
thf(conj_thm_2Eextreal_2Elt__antisym,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (~~ ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Elte__trans,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Ele__total,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) | (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Elt__total,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((V0x = V1y) | ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) | (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Ele__01,lemma,(p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Elt__01,lemma,(p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Ene__01,lemma,(~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(conj_thm_2Eextreal_2Ele__02,lemma,(p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Elt__02,lemma,(p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Ene__02,lemma,(~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(conj_thm_2Eextreal_2Ele__num,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V0n)))))).
thf(conj_thm_2Eextreal_2Elt__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (~~ (V0x = V1y))))))).
thf(conj_thm_2Eextreal_2Ele__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) | (V0x = V1y)))))).
thf(conj_thm_2Eextreal_2Ele__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Elt__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Ele__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elt__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elet__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elte__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__add2,lemma,( ! [V0w:tp__ty_2Eextreal_2Eextreal] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : ( ! [V3z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Eextreal_2Eextreal @ V3z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V2y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))))))).
thf(conj_thm_2Eextreal_2Elt__add2,lemma,( ! [V0w:tp__ty_2Eextreal_2Eextreal] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : ( ! [V3z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Eextreal_2Eextreal @ V3z)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V2y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))))))).
thf(conj_thm_2Eextreal_2Elet__add2,lemma,( ! [V0w:tp__ty_2Eextreal_2Eextreal] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : ( ! [V3z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0w = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0w = fo__c_2Eextreal_2EPosInf)) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V2y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))))))).
thf(conj_thm_2Eextreal_2Elet__add2__alt,lemma,( ! [V0w:tp__ty_2Eextreal_2Eextreal] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : ( ! [V3z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1x = fo__c_2Eextreal_2EPosInf)) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0w))@(inj__ty_2Eextreal_2Eextreal @ V2y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V3z)))))))))).
thf(conj_thm_2Eextreal_2Ele__addr,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__addr__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__ladd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Ele__radd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Ele__radd__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Ele__ladd__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Elt__ladd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Elt__radd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Elt__addl,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Ele__lneg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__mul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elet__mul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elte__mul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__mul__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Emul__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Elt__mul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elt__mul__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Emul__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Emul__let,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Emul__lte,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Ele__rmul__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Ele__lmul__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Elt__rmul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Elt__lmul,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Elt__mul2,lemma,( ! [V0x1:tp__ty_2Eextreal_2Eextreal] : ( ! [V1x2:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y1:tp__ty_2Eextreal_2Eextreal] : ( ! [V3y2:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x1))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2y1))) & ((~~ (V0x1 = fo__c_2Eextreal_2EPosInf)) & ((~~ (V2y1 = fo__c_2Eextreal_2EPosInf)) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x1))@(inj__ty_2Eextreal_2Eextreal @ V1x2))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V2y1))@(inj__ty_2Eextreal_2Eextreal @ V3y2)))))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x1))@(inj__ty_2Eextreal_2Eextreal @ V2y1)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1x2))@(inj__ty_2Eextreal_2Eextreal @ V3y2)))))))))).
thf(conj_thm_2Eextreal_2Eabs__pos,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Eabs__refl,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ V0x))) = V0x) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Eabs__bounds,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1k:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1k))) <=> ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1k)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1k)))))))).
thf(conj_thm_2Eextreal_2Eabs__bounds__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1k:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__abs@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1k))) <=> ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1k)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1k)))))))).
thf(conj_thm_2Eextreal_2Eadd__comm,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Eadd__assoc,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Eadd__not__infty,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2ENegInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2ENegInf))) & (((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2EPosInf))))))).
thf(conj_thm_2Eextreal_2Eadd__rzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = V0x))).
thf(conj_thm_2Eextreal_2Eadd__lzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = V0x))).
thf(conj_thm_2Eextreal_2Eadd__infty,lemma,(( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))) = fo__c_2Eextreal_2EPosInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2EPosInf))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = fo__c_2Eextreal_2EPosInf))) & ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ((~~ (V1x = fo__c_2Eextreal_2EPosInf)) => (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf) & ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ fo__c_2Eextreal_2ENegInf))@(inj__ty_2Eextreal_2Eextreal @ V1x))) = fo__c_2Eextreal_2ENegInf)))))).
thf(conj_thm_2Eextreal_2EEXTREAL__EQ__LADD,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> (V1y = V2z))))))).
thf(conj_thm_2Eextreal_2Esub__rzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = V0x))).
thf(conj_thm_2Eextreal_2Esub__lzero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Esub__le__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Esub__le__imp2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Ele__sub__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Elt__sub__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Esub__lt__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Esub__lt__imp2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & ((~~ (V2z = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Esub__zero__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Esub__zero__lt2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Esub__lt__zero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Esub__lt__zero2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Esub__zero__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Esub__le__zero,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Eextreal_2Ele__sub__eq,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Ele__sub__eq2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & ((~~ (V2z = fo__c_2Eextreal_2EPosInf)) & ((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2ENegInf))))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Esub__le__eq,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))))).
thf(conj_thm_2Eextreal_2Esub__le__eq2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & ((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2ENegInf))))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))))).
thf(conj_thm_2Eextreal_2Esub__le__switch,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & ((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Esub__le__switch2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Elt__sub,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))))).
thf(conj_thm_2Eextreal_2Esub__add2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))) = V1y))))).
thf(conj_thm_2Eextreal_2Eadd__sub,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = V0x))))).
thf(conj_thm_2Eextreal_2Eadd__sub2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = V0x))))).
thf(conj_thm_2Eextreal_2Esub__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = V0x))))).
thf(conj_thm_2Eextreal_2Eextreal__sub__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Esub__0,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => (V0x = V1y))))).
thf(conj_thm_2Eextreal_2Eneg__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))) = V0x))).
thf(conj_thm_2Eextreal_2Eneg__0,lemma,((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))).
thf(conj_thm_2Eextreal_2Eneg__eq0,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Eextreal_2Eeq__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (V0x = V1y))))).
thf(conj_thm_2Eextreal_2Eneg__minus1,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__ainv@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Esub__rneg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Esub__lneg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2ENegInf))) | ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Eneg__sub,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) | ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Esub__not__infty,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2ENegInf))) & (((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & (~~ (V1y = fo__c_2Eextreal_2ENegInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2EPosInf))))))).
thf(conj_thm_2Eextreal_2Ele__lsub__imp,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Eeq__sub__ladd__normal,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))) <=> ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z)))) = V1y)))))).
thf(conj_thm_2Eextreal_2Eeq__sub__radd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))) => (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = V2z) <=> (V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))))).
thf(conj_thm_2Eextreal_2Eeq__sub__ladd,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))) => ((V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) <=> ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z))) = V1y))))))).
thf(conj_thm_2Eextreal_2Eeq__sub__switch,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) <=> (V1y = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z)))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))))).
thf(conj_thm_2Eextreal_2Eeq__add__sub__switch,lemma,( ! [V0a:tp__ty_2Eextreal_2Eextreal] : ( ! [V1b:tp__ty_2Eextreal_2Eextreal] : ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1b = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1b = fo__c_2Eextreal_2EPosInf)) & ((~~ (V2c = fo__c_2Eextreal_2ENegInf)) & (~~ (V2c = fo__c_2Eextreal_2EPosInf))))) => (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0a))@(inj__ty_2Eextreal_2Eextreal @ V1b))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V2c))@(inj__ty_2Eextreal_2Eextreal @ V3d)))) <=> ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0a))@(inj__ty_2Eextreal_2Eextreal @ V2c))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V3d))@(inj__ty_2Eextreal_2Eextreal @ V1b))))))))))).
thf(conj_thm_2Eextreal_2Esub__refl,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))).
thf(conj_thm_2Eextreal_2Emul__comm,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Emul__assoc,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))).
thf(conj_thm_2Eextreal_2Emul__not__infty,lemma,(( ! [V0c:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0c))) & (~~ (V1y = fo__c_2Eextreal_2ENegInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0c)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2ENegInf))))) & (( ! [V2c:tp__ty_2Erealax_2Ereal] : ( ! [V3y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2c))) & (~~ (V3y = fo__c_2Eextreal_2EPosInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2c)))@(inj__ty_2Eextreal_2Eextreal @ V3y))) = fo__c_2Eextreal_2EPosInf))))) & (( ! [V4c:tp__ty_2Erealax_2Ereal] : ( ! [V5y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V4c))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (~~ (V5y = fo__c_2Eextreal_2ENegInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V4c)))@(inj__ty_2Eextreal_2Eextreal @ V5y))) = fo__c_2Eextreal_2EPosInf))))) & ( ! [V6c:tp__ty_2Erealax_2Ereal] : ( ! [V7y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V6c))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (~~ (V7y = fo__c_2Eextreal_2EPosInf))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V6c)))@(inj__ty_2Eextreal_2Eextreal @ V7y))) = fo__c_2Eextreal_2ENegInf))))))))).
thf(conj_thm_2Eextreal_2Emul__not__infty2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V1y = fo__c_2Eextreal_2EPosInf))))) => ((~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2ENegInf)) & (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = fo__c_2Eextreal_2EPosInf))))))).
thf(conj_thm_2Eextreal_2Eadd__ldistrib__pos,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eadd__ldistrib__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eadd__ldistrib__normal,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))) | ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2ENegInf)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eadd__ldistrib__normal2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eadd__rdistrib__normal,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))) | ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2ENegInf)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))))))))).
thf(conj_thm_2Eextreal_2Eadd__rdistrib__normal2,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))))))))).
thf(conj_thm_2Eextreal_2Eadd__ldistrib,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) | ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eadd__rdistrib,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) | ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))))).
thf(conj_thm_2Eextreal_2Emul__lneg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Emul__rneg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__ainv@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Eneg__mul2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@c_2Eextreal_2Eextreal__ainv@(inj__ty_2Eextreal_2Eextreal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Eadd2__sub2,lemma,( ! [V0a:tp__ty_2Eextreal_2Eextreal] : ( ! [V1b:tp__ty_2Eextreal_2Eextreal] : ( ! [V2c:tp__ty_2Eextreal_2Eextreal] : ( ! [V3d:tp__ty_2Eextreal_2Eextreal] : ((((~~ (V1b = fo__c_2Eextreal_2EPosInf)) & (~~ (V3d = fo__c_2Eextreal_2EPosInf))) | ((~~ (V1b = fo__c_2Eextreal_2ENegInf)) & (~~ (V3d = fo__c_2Eextreal_2ENegInf)))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0a))@(inj__ty_2Eextreal_2Eextreal @ V1b)))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V2c))@(inj__ty_2Eextreal_2Eextreal @ V3d)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0a))@(inj__ty_2Eextreal_2Eextreal @ V2c)))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1b))@(inj__ty_2Eextreal_2Eextreal @ V3d))))))))))).
thf(conj_thm_2Eextreal_2Esub__ldistrib,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & ((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Esub__rdistrib,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) & ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) & ((~~ (V2z = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = fo__c_2Eextreal_2EPosInf))))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__sub@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__sub@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))))))))).
thf(conj_thm_2Eextreal_2Eextreal__div__eq,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x)))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V1y)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Ereal_2E_2F@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Einv__one,lemma,((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Einv__1over,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ V0x))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Ediv__one,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = V0x))).
thf(conj_thm_2Eextreal_2Einv__pos,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Erinv__uniq,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) => (V1y = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))).
thf(conj_thm_2Eextreal_2Elinv__uniq,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) => (V0x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Ele__rdiv,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Eextreal_2Eextreal @ V2z))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))))))))).
thf(conj_thm_2Eextreal_2Ele__ldiv,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V0x))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V0x))))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Elt__rdiv,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2z))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Elt__ldiv,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2z))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))@(inj__ty_2Eextreal_2Eextreal @ V1y))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))))))))).
thf(conj_thm_2Eextreal_2Elt__rdiv__neg,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Erealax_2Ereal] : ((p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))@(inj__ty_2Eextreal_2Eextreal @ V0x))) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2ENormal@(inj__ty_2Erealax_2Ereal @ V2z))))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))))).
thf(conj_thm_2Eextreal_2Ediv__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & ((~~ (V1y = fo__c_2Eextreal_2ENegInf)) & (~~ (V2z = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V2z)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2z)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(inj__ty_2Eextreal_2Eextreal @ V2z))))))))).
thf(conj_thm_2Eextreal_2Ele__inv,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__inv@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Epow__0,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(conj_thm_2Eextreal_2Epow__1,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = V0x))).
thf(conj_thm_2Eextreal_2Epow__2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Epow__zero,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V0n)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (V1x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Epow__zero__imp,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => (V1x = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))).
thf(conj_thm_2Eextreal_2Ele__pow2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(conj_thm_2Eextreal_2Epow__pos__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eextreal_2Epow__pos__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eextreal_2Epow__le,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V2y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Enum_2Enum @ V0n))))))))).
thf(conj_thm_2Eextreal_2Epow__lt,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V2y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V0n))))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V2y))@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V0n)))))))))).
thf(conj_thm_2Eextreal_2Epow__lt2,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : (((~~ (V0n = fo__c_2Enum_2E0)) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V1x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V2y))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Enum_2Enum @ V0n))))))))).
thf(conj_thm_2Eextreal_2Epow__le__mono,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2m:tp__ty_2Enum_2Enum] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2m)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V2m))))))))).
thf(conj_thm_2Eextreal_2Epow__pos__even,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))) <=> (p@(ap@c_2Earithmetic_2EEVEN@(inj__ty_2Enum_2Enum @ V0n)))))))).
thf(conj_thm_2Eextreal_2Epow__neg__odd,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) => ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) <=> (p@(ap@c_2Earithmetic_2EODD@(inj__ty_2Enum_2Enum @ V0n)))))))).
thf(conj_thm_2Eextreal_2Eadd__pow2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__add@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Epow__add,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1n:tp__ty_2Enum_2Enum] : ( ! [V2m:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Earithmetic_2E_2B@(inj__ty_2Enum_2Enum @ V1n))@(inj__ty_2Enum_2Enum @ V2m)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V1n)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Enum_2Enum @ V2m))))))))).
thf(conj_thm_2Eextreal_2Epow__mul,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : ( ! [V2y:tp__ty_2Eextreal_2Eextreal] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Eextreal_2Eextreal @ V2y)))@(inj__ty_2Enum_2Enum @ V0n))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n)))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V2y))@(inj__ty_2Enum_2Enum @ V0n))))))))).
thf(conj_thm_2Eextreal_2Epow__minus1,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@c_2Eextreal_2Eextreal__ainv@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Earithmetic_2E_2A@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))@(inj__ty_2Enum_2Enum @ V0n)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))).
thf(conj_thm_2Eextreal_2Epow__not__infty,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1x:tp__ty_2Eextreal_2Eextreal] : (((~~ (V1x = fo__c_2Eextreal_2ENegInf)) & (~~ (V1x = fo__c_2Eextreal_2EPosInf))) => ((~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n))) = fo__c_2Eextreal_2ENegInf)) & (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V1x))@(inj__ty_2Enum_2Enum @ V0n))) = fo__c_2Eextreal_2EPosInf))))))).
thf(conj_thm_2Eextreal_2Esqrt__pos__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Esqrt__pos__lt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ V0x))))))).
thf(conj_thm_2Eextreal_2Epow2__sqrt,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__sqrt@(ap@(ap@c_2Eextreal_2Eextreal__pow@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))) = V0x)))).
thf(conj_thm_2Eextreal_2Esqrt__pow2,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))) = V0x) <=> (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))).
thf(conj_thm_2Eextreal_2Esqrt__mono__le,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y)))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(ap@c_2Eextreal_2Eextreal__sqrt@(inj__ty_2Eextreal_2Eextreal @ V1y)))))))).
thf(conj_thm_2Eextreal_2Elogr__not__infty,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1b:tp__ty_2Erealax_2Ereal] : (((~~ (V0x = fo__c_2Eextreal_2ENegInf)) & (~~ (V0x = fo__c_2Eextreal_2EPosInf))) => ((~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = fo__c_2Eextreal_2ENegInf)) & (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__logr@(inj__ty_2Erealax_2Ereal @ V1b))@(inj__ty_2Eextreal_2Eextreal @ V0x))) = fo__c_2Eextreal_2EPosInf))))))).
thf(conj_thm_2Eextreal_2Ehalf__between,lemma,(((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))))).
thf(conj_thm_2Eextreal_2Ethirds__between,lemma,((((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) & ((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))))).
thf(conj_thm_2Eextreal_2Ehalf__cancel,lemma,((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Ethird__cancel,lemma,((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Efourth__cancel,lemma,((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))).
thf(conj_thm_2Eextreal_2Equotient__normal,lemma,( ! [V0n:tp__ty_2Enum_2Enum] : ( ! [V1m:tp__ty_2Enum_2Enum] : ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__div@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V0n)))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V1m)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V0n)))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1m))))))))).
thf(ax_thm_2Eextreal_2Eext__mono__increasing__def,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)) => ((p@(ap@c_2Eextreal_2Eext__mono__increasing@V0f)) <=> ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@V0f@(inj__ty_2Enum_2Enum @ V1m)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V2n))))))))))).
thf(conj_thm_2Eextreal_2Eext__mono__increasing__suc,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)) => ((p@(ap@c_2Eextreal_2Eext__mono__increasing@V0f)) <=> ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@V0f@(inj__ty_2Enum_2Enum @ V1n)))@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n)))))))))).
thf(ax_thm_2Eextreal_2Eext__mono__decreasing__def,axiom,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)) => ((p@(ap@c_2Eextreal_2Eext__mono__decreasing@V0f)) <=> ( ! [V1m:tp__ty_2Enum_2Enum] : ( ! [V2n:tp__ty_2Enum_2Enum] : ((p@(ap@(ap@c_2Earithmetic_2E_3C_3D@(inj__ty_2Enum_2Enum @ V1m))@(inj__ty_2Enum_2Enum @ V2n))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@V0f@(inj__ty_2Enum_2Enum @ V2n)))@(ap@V0f@(inj__ty_2Enum_2Enum @ V1m))))))))))).
thf(conj_thm_2Eextreal_2Eext__mono__decreasing__suc,lemma,( ! [V0f:$i] : ((mem@V0f@(arr@ty_2Enum_2Enum@ty_2Eextreal_2Eextreal)) => ((p@(ap@c_2Eextreal_2Eext__mono__decreasing@V0f)) <=> ( ! [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@V0f@(ap@c_2Enum_2ESUC@(inj__ty_2Enum_2Enum @ V1n))))@(ap@V0f@(inj__ty_2Enum_2Enum @ V1n))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__ARCH,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0x))) => ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : ((~~ (V1y = fo__c_2Eextreal_2EPosInf)) => (? [V2n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V1y))@(ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V2n)))@(inj__ty_2Eextreal_2Eextreal @ V0x)))))))))).
thf(conj_thm_2Eextreal_2ESIMP__REAL__ARCH,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eextreal_2ESIMP__REAL__ARCH__NEG,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@c_2Erealax_2Ereal__neg@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ V1n))))@(inj__ty_2Erealax_2Ereal @ V0x)))))).
thf(conj_thm_2Eextreal_2ESIMP__EXTREAL__ARCH,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) => (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eextreal_2EREAL__ARCH__POW,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V0x))@(ap@(ap@c_2Ereal_2Epow@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1n))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ((~~ (V0x = fo__c_2Eextreal_2EPosInf)) => (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__pow@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(inj__ty_2Enum_2Enum @ V1n)))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW__INV,lemma,( ! [V0e:tp__ty_2Eextreal_2Eextreal] : ((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V0e))) => (? [V1n:tp__ty_2Enum_2Enum] : (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2ENormal@(ap@(ap@c_2Ereal_2Epow@(ap@(ap@c_2Ereal_2E_2F@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO)))))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT2@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))@(inj__ty_2Enum_2Enum @ V1n))))@(inj__ty_2Eextreal_2Eextreal @ V0e))))))).
thf(conj_thm_2Eextreal_2EREAL__LE__MUL__EPSILON,lemma,( ! [V0x:tp__ty_2Erealax_2Ereal] : ( ! [V1y:tp__ty_2Erealax_2Ereal] : (( ! [V2z:tp__ty_2Erealax_2Ereal] : (((p@(ap@(ap@c_2Erealax_2Ereal__lt@(ap@c_2Ereal_2Ereal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Erealax_2Ereal @ V2z))) & (p@(ap@(ap@c_2Erealax_2Ereal__lt@(inj__ty_2Erealax_2Ereal @ V2z))@(ap@c_2Ereal_2Ereal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(ap@(ap@c_2Erealax_2Ereal__mul@(inj__ty_2Erealax_2Ereal @ V2z))@(inj__ty_2Erealax_2Ereal @ V0x)))@(inj__ty_2Erealax_2Ereal @ V1y))))) => (p@(ap@(ap@c_2Ereal_2Ereal__lte@(inj__ty_2Erealax_2Ereal @ V0x))@(inj__ty_2Erealax_2Ereal @ V1y))))))).
thf(conj_thm_2Eextreal_2Ele__epsilon,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (( ! [V2e:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2e))) & (~~ (V2e = fo__c_2Eextreal_2EPosInf))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(ap@(ap@c_2Eextreal_2Eextreal__add@(inj__ty_2Eextreal_2Eextreal @ V1y))@(inj__ty_2Eextreal_2Eextreal @ V2e)))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(conj_thm_2Eextreal_2Ele__mul__epsilon,lemma,( ! [V0x:tp__ty_2Eextreal_2Eextreal] : ( ! [V1y:tp__ty_2Eextreal_2Eextreal] : (( ! [V2z:tp__ty_2Eextreal_2Eextreal] : (((p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(inj__ty_2Eextreal_2Eextreal @ V2z))) & (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(inj__ty_2Eextreal_2Eextreal @ V2z))@(ap@c_2Eextreal_2Eextreal__of__num@(ap@c_2Earithmetic_2ENUMERAL@(ap@c_2Earithmetic_2EBIT1@(inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO))))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@(ap@c_2Eextreal_2Eextreal__mul@(inj__ty_2Eextreal_2Eextreal @ V2z))@(inj__ty_2Eextreal_2Eextreal @ V0x)))@(inj__ty_2Eextreal_2Eextreal @ V1y))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(inj__ty_2Eextreal_2Eextreal @ V0x))@(inj__ty_2Eextreal_2Eextreal @ V1y))))))).
thf(ax_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DEF,axiom,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(ap@(c_2Epred__set_2EITSET@A_27a@ty_2Eextreal_2Eextreal)@(lam@A_27a@(^ [V2e:$i] : (lam@ty_2Eextreal_2Eextreal@(^ [V3acc:$i] : (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@V0f@V2e))@V3acc))))))@V1s)@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__THM,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => (((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@(c_2Epred__set_2EEMPTY@A_27a))) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))) & ( ! [V1e:$i] : ((mem@V1e@A_27a) => ( ! [V2s:$i] : ((mem@V2s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V2s)) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1e)@V2s))) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__add@(ap@V0f@V1e))@(ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@(ap@(ap@(c_2Epred__set_2EDELETE@A_27a)@V2s)@V1e))))))))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__NEG__INF,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V2x)) = fo__c_2Eextreal_2ENegInf)))))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = fo__c_2Eextreal_2ENegInf))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__POS__INF,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V2x)) = fo__c_2Eextreal_2EPosInf)))))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = fo__c_2Eextreal_2EPosInf))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__INFTY,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => ((((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V2x)) = fo__c_2Eextreal_2ENegInf)))))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = fo__c_2Eextreal_2ENegInf))) & (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V1s)) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V3x)) = fo__c_2Eextreal_2EPosInf)))))) => (~~ ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = fo__c_2Eextreal_2EPosInf)))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SING,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1e:$i] : ((mem@V1e@A_27a) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@(ap@(ap@(c_2Epred__set_2EINSERT@A_27a)@V1e)@(c_2Epred__set_2EEMPTY@A_27a)))) = (surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V1e))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@V2x))))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__le@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SPOS,lemma,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ((~~ (V1s = (c_2Epred__set_2EEMPTY@A_27a))) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@V0f@V2x)))))))) => (p@(ap@(ap@c_2Eextreal_2Eextreal__lt@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))@(ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IF__ELIM,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ( ! [V1P:$i] : ((mem@V1P@(arr@A_27a@bool)) => ( ! [V2f:$i] : ((mem@V2f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) & ( ! [V3x:$i] : ((mem@V3x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3x)@V0s)) => (p@(ap@V1P@V3x)))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@(lam@A_27a@(^ [V4x:$i] : (ap@(ap@(ap@(c_2Ebool_2ECOND@ty_2Eextreal_2Eextreal)@(ap@V1P@V4x))@(ap@V2f@V4x))@(ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))@V0s)) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V2f)@V0s)))))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__SAME,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V2p:$i] : ((mem@V2p@A_27a) => (((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2p)@V0s)) & ( ! [V3q:$i] : ((mem@V3q@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V3q)@V0s)) => ((surj__ty_2Eextreal_2Eextreal @ (ap@V1f@V2p)) = (surj__ty_2Eextreal_2Eextreal @ (ap@V1f@V3q))))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V1f)@V0s)) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@(c_2Epred__set_2ECARD@A_27a)@V0s)))@(ap@V1f@V2p)))))))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__CONST,lemma,( ! [A_27a:del]: ( ! [V0P:$i] : ((mem@V0P@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0P)) => ( ! [V1f:$i] : ((mem@V1f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V2x:tp__ty_2Eextreal_2Eextreal] : (( ! [V3y:$i] : ((mem@V3y@A_27a) => ((surj__ty_2Eextreal_2Eextreal @ (ap@V1f@V3y)) = V2x))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V1f)@V0P)) = (surj__ty_2Eextreal_2Eextreal @ (ap@(ap@c_2Eextreal_2Eextreal__mul@(ap@c_2Eextreal_2Eextreal__of__num@(ap@(c_2Epred__set_2ECARD@A_27a)@V0P)))@(inj__ty_2Eextreal_2Eextreal @ V2x))))))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO,lemma,( ! [A_27a:del]: ( ! [V0s:$i] : ((mem@V0s@(arr@A_27a@bool)) => ((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V0s)) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@(lam@A_27a@(^ [V1x:$i] : (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))@V0s)) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))).
thf(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__0,conjecture,( ! [A_27a:del]: ( ! [V0f:$i] : ((mem@V0f@(arr@A_27a@ty_2Eextreal_2Eextreal)) => ( ! [V1s:$i] : ((mem@V1s@(arr@A_27a@bool)) => (((p@(ap@(c_2Epred__set_2EFINITE@A_27a)@V1s)) & ( ! [V2x:$i] : ((mem@V2x@A_27a) => ((p@(ap@(ap@(c_2Ebool_2EIN@A_27a)@V2x)@V1s)) => ((surj__ty_2Eextreal_2Eextreal @ (ap@V0f@V2x)) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0)))))))) => ((surj__ty_2Eextreal_2Eextreal @ (ap@(ap@(c_2Eextreal_2EEXTREAL__SUM__IMAGE@A_27a)@V0f)@V1s)) = (surj__ty_2Eextreal_2Eextreal @ (ap@c_2Eextreal_2Eextreal__of__num@(inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0))))))))))).
