TPTP Problem File: ITP021_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP021_5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : HOL4 set theory export of thm_2Eextreal_2Emax__le.p, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : thm_2Eextreal_2Emax__le.p [Gau20]
% : HL410001_5.p [TPAP]
% Status : Theorem
% Rating : 0.89 v8.2.0, 1.00 v7.5.0
% Syntax : Number of formulae : 26759 (6280 unt;6772 typ; 0 def)
% Number of atoms : 450638 (21487 equ)
% Maximal formula atoms : 5763 ( 16 avg)
% Number of connectives : 71951 (2527 ~;1056 |;25864 &)
% (6165 <=>;36339 =>; 0 <=; 0 <~>)
% Maximal formula depth : 361 ( 6 avg)
% Maximal term depth : 132 ( 2 avg)
% Number of FOOLs : 361227 (361227 fml; 0 var)
% Number of types : 149 ( 147 usr)
% Number of type conns : 12517 (5412 >;7105 *; 0 +; 0 <<)
% Number of predicates : 86 ( 83 usr; 37 prp; 0-3 aty)
% Number of functors : 6623 (6623 usr;1213 con; 0-11 aty)
% Number of variables : 80810 (66911 !;13899 ?;80810 :)
% SPC : TF0_THM_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001_2.ax').
include('Axioms/ITP001/ITP002_5.ax').
include('Axioms/ITP001/ITP003_5.ax').
include('Axioms/ITP001/ITP004_5.ax').
include('Axioms/ITP001/ITP007_5.ax').
include('Axioms/ITP001/ITP006_5.ax').
include('Axioms/ITP001/ITP005_5.ax').
include('Axioms/ITP001/ITP008_5.ax').
include('Axioms/ITP001/ITP009_5.ax').
include('Axioms/ITP001/ITP010_5.ax').
include('Axioms/ITP001/ITP012_5.ax').
include('Axioms/ITP001/ITP011_5.ax').
include('Axioms/ITP001/ITP013_5.ax').
include('Axioms/ITP001/ITP014_5.ax').
include('Axioms/ITP001/ITP015_5.ax').
include('Axioms/ITP001/ITP017_5.ax').
include('Axioms/ITP001/ITP016_5.ax').
include('Axioms/ITP001/ITP019_5.ax').
include('Axioms/ITP001/ITP018_5.ax').
include('Axioms/ITP001/ITP021_5.ax').
include('Axioms/ITP001/ITP022_5.ax').
include('Axioms/ITP001/ITP020_5.ax').
include('Axioms/ITP001/ITP024_5.ax').
include('Axioms/ITP001/ITP023_5.ax').
include('Axioms/ITP001/ITP025_5.ax').
include('Axioms/ITP001/ITP026_5.ax').
include('Axioms/ITP001/ITP027_5.ax').
include('Axioms/ITP001/ITP028_5.ax').
include('Axioms/ITP001/ITP031_5.ax').
include('Axioms/ITP001/ITP029_5.ax').
include('Axioms/ITP001/ITP033_5.ax').
include('Axioms/ITP001/ITP030_5.ax').
include('Axioms/ITP001/ITP032_5.ax').
include('Axioms/ITP001/ITP038_5.ax').
include('Axioms/ITP001/ITP035_5.ax').
include('Axioms/ITP001/ITP034_5.ax').
include('Axioms/ITP001/ITP036_5.ax').
include('Axioms/ITP001/ITP037_5.ax').
include('Axioms/ITP001/ITP039_5.ax').
include('Axioms/ITP001/ITP041_5.ax').
include('Axioms/ITP001/ITP042_5.ax').
include('Axioms/ITP001/ITP040_5.ax').
include('Axioms/ITP001/ITP044_5.ax').
include('Axioms/ITP001/ITP051_5.ax').
include('Axioms/ITP001/ITP045_5.ax').
include('Axioms/ITP001/ITP056_5.ax').
include('Axioms/ITP001/ITP046_5.ax').
include('Axioms/ITP001/ITP043_5.ax').
include('Axioms/ITP001/ITP052_5.ax').
include('Axioms/ITP001/ITP057_5.ax').
include('Axioms/ITP001/ITP048_5.ax').
include('Axioms/ITP001/ITP047_5.ax').
include('Axioms/ITP001/ITP055_5.ax').
include('Axioms/ITP001/ITP053_5.ax').
include('Axioms/ITP001/ITP054_5.ax').
include('Axioms/ITP001/ITP058_5.ax').
include('Axioms/ITP001/ITP049_5.ax').
include('Axioms/ITP001/ITP050_5.ax').
include('Axioms/ITP001/ITP061_5.ax').
include('Axioms/ITP001/ITP069_5.ax').
include('Axioms/ITP001/ITP062_5.ax').
include('Axioms/ITP001/ITP068_5.ax').
include('Axioms/ITP001/ITP078_5.ax').
include('Axioms/ITP001/ITP064_5.ax').
include('Axioms/ITP001/ITP060_5.ax').
include('Axioms/ITP001/ITP067_5.ax').
include('Axioms/ITP001/ITP075_5.ax').
include('Axioms/ITP001/ITP074_5.ax').
include('Axioms/ITP001/ITP063_5.ax').
include('Axioms/ITP001/ITP059_5.ax').
include('Axioms/ITP001/ITP065_5.ax').
include('Axioms/ITP001/ITP076_5.ax').
include('Axioms/ITP001/ITP066_5.ax').
include('Axioms/ITP001/ITP077_5.ax').
include('Axioms/ITP001/ITP070_5.ax').
include('Axioms/ITP001/ITP071_5.ax').
include('Axioms/ITP001/ITP072_5.ax').
include('Axioms/ITP001/ITP073_5.ax').
include('Axioms/ITP001/ITP080_5.ax').
include('Axioms/ITP001/ITP083_5.ax').
include('Axioms/ITP001/ITP088_5.ax').
include('Axioms/ITP001/ITP082_5.ax').
include('Axioms/ITP001/ITP081_5.ax').
include('Axioms/ITP001/ITP079_5.ax').
include('Axioms/ITP001/ITP087_5.ax').
include('Axioms/ITP001/ITP085_5.ax').
include('Axioms/ITP001/ITP086_5.ax').
include('Axioms/ITP001/ITP090_5.ax').
include('Axioms/ITP001/ITP089_5.ax').
include('Axioms/ITP001/ITP084_5.ax').
include('Axioms/ITP001/ITP104_5.ax').
include('Axioms/ITP001/ITP097_5.ax').
include('Axioms/ITP001/ITP094_5.ax').
include('Axioms/ITP001/ITP095_5.ax').
include('Axioms/ITP001/ITP093_5.ax').
include('Axioms/ITP001/ITP102_5.ax').
include('Axioms/ITP001/ITP091_5.ax').
include('Axioms/ITP001/ITP098_5.ax').
include('Axioms/ITP001/ITP096_5.ax').
include('Axioms/ITP001/ITP100_5.ax').
include('Axioms/ITP001/ITP092_5.ax').
include('Axioms/ITP001/ITP105_5.ax').
include('Axioms/ITP001/ITP101_5.ax').
include('Axioms/ITP001/ITP103_5.ax').
include('Axioms/ITP001/ITP099_5.ax').
include('Axioms/ITP001/ITP111_5.ax').
include('Axioms/ITP001/ITP108_5.ax').
include('Axioms/ITP001/ITP106_5.ax').
include('Axioms/ITP001/ITP107_5.ax').
include('Axioms/ITP001/ITP114_5.ax').
include('Axioms/ITP001/ITP110_5.ax').
include('Axioms/ITP001/ITP109_5.ax').
include('Axioms/ITP001/ITP112_5.ax').
include('Axioms/ITP001/ITP113_5.ax').
include('Axioms/ITP001/ITP117_5.ax').
include('Axioms/ITP001/ITP116_5.ax').
include('Axioms/ITP001/ITP118_5.ax').
include('Axioms/ITP001/ITP120_5.ax').
include('Axioms/ITP001/ITP119_5.ax').
include('Axioms/ITP001/ITP121_5.ax').
include('Axioms/ITP001/ITP123_5.ax').
include('Axioms/ITP001/ITP124_5.ax').
include('Axioms/ITP001/ITP126_5.ax').
include('Axioms/ITP001/ITP127_5.ax').
include('Axioms/ITP001/ITP129_5.ax').
include('Axioms/ITP001/ITP128_5.ax').
include('Axioms/ITP001/ITP130_5.ax').
include('Axioms/ITP001/ITP131_5.ax').
include('Axioms/ITP001/ITP132_5.ax').
include('Axioms/ITP001/ITP134_5.ax').
include('Axioms/ITP001/ITP133_5.ax').
include('Axioms/ITP001/ITP135_5.ax').
include('Axioms/ITP001/ITP136_5.ax').
include('Axioms/ITP001/ITP138_5.ax').
include('Axioms/ITP001/ITP137_5.ax').
include('Axioms/ITP001/ITP140_5.ax').
include('Axioms/ITP001/ITP141_5.ax').
include('Axioms/ITP001/ITP139_5.ax').
include('Axioms/ITP001/ITP142_5.ax').
include('Axioms/ITP001/ITP145_5.ax').
include('Axioms/ITP001/ITP144_5.ax').
%------------------------------------------------------------------------------
tff(tp_ty_2Eextreal_2Eextreal,type,
ty_2Eextreal_2Eextreal: del ).
tff(stp_ty_2Eextreal_2Eextreal,type,
tp__ty_2Eextreal_2Eextreal: $tType ).
tff(stp_inj_ty_2Eextreal_2Eextreal,type,
inj__ty_2Eextreal_2Eextreal: tp__ty_2Eextreal_2Eextreal > $i ).
tff(stp_surj_ty_2Eextreal_2Eextreal,type,
surj__ty_2Eextreal_2Eextreal: $i > tp__ty_2Eextreal_2Eextreal ).
tff(stp_inj_surj_ty_2Eextreal_2Eextreal,axiom,
! [X: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(inj__ty_2Eextreal_2Eextreal(X)) = X ).
tff(stp_inj_mem_ty_2Eextreal_2Eextreal,axiom,
! [X: tp__ty_2Eextreal_2Eextreal] : mem(inj__ty_2Eextreal_2Eextreal(X),ty_2Eextreal_2Eextreal) ).
tff(stp_iso_mem_ty_2Eextreal_2Eextreal,axiom,
! [X: $i] :
( mem(X,ty_2Eextreal_2Eextreal)
=> ( X = inj__ty_2Eextreal_2Eextreal(surj__ty_2Eextreal_2Eextreal(X)) ) ) ).
tff(tp_c_2Eextreal_2EEXTREAL__SUM__IMAGE,type,
c_2Eextreal_2EEXTREAL__SUM__IMAGE: del > $i ).
tff(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))) ).
tff(tp_c_2Eextreal_2ENegInf,type,
c_2Eextreal_2ENegInf: $i ).
tff(mem_c_2Eextreal_2ENegInf,axiom,
mem(c_2Eextreal_2ENegInf,ty_2Eextreal_2Eextreal) ).
tff(stp_fo_c_2Eextreal_2ENegInf,type,
fo__c_2Eextreal_2ENegInf: tp__ty_2Eextreal_2Eextreal ).
tff(stp_eq_fo_c_2Eextreal_2ENegInf,axiom,
inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf) = c_2Eextreal_2ENegInf ).
tff(tp_c_2Eextreal_2ENormal,type,
c_2Eextreal_2ENormal: $i ).
tff(mem_c_2Eextreal_2ENormal,axiom,
mem(c_2Eextreal_2ENormal,arr(ty_2Erealax_2Ereal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2ENormal,type,
fo__c_2Eextreal_2ENormal: tp__ty_2Erealax_2Ereal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2EPosInf,type,
c_2Eextreal_2EPosInf: $i ).
tff(mem_c_2Eextreal_2EPosInf,axiom,
mem(c_2Eextreal_2EPosInf,ty_2Eextreal_2Eextreal) ).
tff(stp_fo_c_2Eextreal_2EPosInf,type,
fo__c_2Eextreal_2EPosInf: tp__ty_2Eextreal_2Eextreal ).
tff(stp_eq_fo_c_2Eextreal_2EPosInf,axiom,
inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf) = c_2Eextreal_2EPosInf ).
tff(tp_c_2Eextreal_2EQ__set,type,
c_2Eextreal_2EQ__set: $i ).
tff(mem_c_2Eextreal_2EQ__set,axiom,
mem(c_2Eextreal_2EQ__set,arr(ty_2Eextreal_2Eextreal,bool)) ).
tff(stp_fo_c_2Eextreal_2EQ__set,type,
fo__c_2Eextreal_2EQ__set: tp__ty_2Eextreal_2Eextreal > tp__o ).
tff(stp_eq_fo_c_2Eextreal_2EQ__set,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal] : inj__o(fo__c_2Eextreal_2EQ__set(X0)) = ap(c_2Eextreal_2EQ__set,inj__ty_2Eextreal_2Eextreal(X0)) ).
tff(tp_c_2Eextreal_2Eceiling,type,
c_2Eextreal_2Eceiling: $i ).
tff(mem_c_2Eextreal_2Eceiling,axiom,
mem(c_2Eextreal_2Eceiling,arr(ty_2Eextreal_2Eextreal,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Eextreal_2Eceiling,type,
fo__c_2Eextreal_2Eceiling: tp__ty_2Eextreal_2Eextreal > tp__ty_2Enum_2Enum ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eext__mono__decreasing,type,
c_2Eextreal_2Eext__mono__decreasing: $i ).
tff(mem_c_2Eextreal_2Eext__mono__decreasing,axiom,
mem(c_2Eextreal_2Eext__mono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),bool)) ).
tff(tp_c_2Eextreal_2Eext__mono__increasing,type,
c_2Eextreal_2Eext__mono__increasing: $i ).
tff(mem_c_2Eextreal_2Eext__mono__increasing,axiom,
mem(c_2Eextreal_2Eext__mono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),bool)) ).
tff(tp_c_2Eextreal_2Eext__suminf,type,
c_2Eextreal_2Eext__suminf: $i ).
tff(mem_c_2Eextreal_2Eext__suminf,axiom,
mem(c_2Eextreal_2Eext__suminf,arr(arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),ty_2Eextreal_2Eextreal)) ).
tff(tp_c_2Eextreal_2Eextreal__CASE,type,
c_2Eextreal_2Eextreal__CASE: del > $i ).
tff(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))))) ).
tff(tp_c_2Eextreal_2Eextreal__abs,type,
c_2Eextreal_2Eextreal__abs: $i ).
tff(mem_c_2Eextreal_2Eextreal__abs,axiom,
mem(c_2Eextreal_2Eextreal__abs,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__abs,type,
fo__c_2Eextreal_2Eextreal__abs: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__add,type,
c_2Eextreal_2Eextreal__add: $i ).
tff(mem_c_2Eextreal_2Eextreal__add,axiom,
mem(c_2Eextreal_2Eextreal__add,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__add,type,
fo__c_2Eextreal_2Eextreal__add: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__ainv,type,
c_2Eextreal_2Eextreal__ainv: $i ).
tff(mem_c_2Eextreal_2Eextreal__ainv,axiom,
mem(c_2Eextreal_2Eextreal__ainv,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__ainv,type,
fo__c_2Eextreal_2Eextreal__ainv: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__div,type,
c_2Eextreal_2Eextreal__div: $i ).
tff(mem_c_2Eextreal_2Eextreal__div,axiom,
mem(c_2Eextreal_2Eextreal__div,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__div,type,
fo__c_2Eextreal_2Eextreal__div: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__exp,type,
c_2Eextreal_2Eextreal__exp: $i ).
tff(mem_c_2Eextreal_2Eextreal__exp,axiom,
mem(c_2Eextreal_2Eextreal__exp,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__exp,type,
fo__c_2Eextreal_2Eextreal__exp: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__inf,type,
c_2Eextreal_2Eextreal__inf: $i ).
tff(mem_c_2Eextreal_2Eextreal__inf,axiom,
mem(c_2Eextreal_2Eextreal__inf,arr(arr(ty_2Eextreal_2Eextreal,bool),ty_2Eextreal_2Eextreal)) ).
tff(tp_c_2Eextreal_2Eextreal__inv,type,
c_2Eextreal_2Eextreal__inv: $i ).
tff(mem_c_2Eextreal_2Eextreal__inv,axiom,
mem(c_2Eextreal_2Eextreal__inv,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__inv,type,
fo__c_2Eextreal_2Eextreal__inv: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__le,type,
c_2Eextreal_2Eextreal__le: $i ).
tff(mem_c_2Eextreal_2Eextreal__le,axiom,
mem(c_2Eextreal_2Eextreal__le,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__le,type,
fo__c_2Eextreal_2Eextreal__le: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__o ).
tff(stp_eq_fo_c_2Eextreal_2Eextreal__le,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal,X1: tp__ty_2Eextreal_2Eextreal] : inj__o(fo__c_2Eextreal_2Eextreal__le(X0,X1)) = ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(X0)),inj__ty_2Eextreal_2Eextreal(X1)) ).
tff(tp_c_2Eextreal_2Eextreal__lg,type,
c_2Eextreal_2Eextreal__lg: $i ).
tff(mem_c_2Eextreal_2Eextreal__lg,axiom,
mem(c_2Eextreal_2Eextreal__lg,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__lg,type,
fo__c_2Eextreal_2Eextreal__lg: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__logr,type,
c_2Eextreal_2Eextreal__logr: $i ).
tff(mem_c_2Eextreal_2Eextreal__logr,axiom,
mem(c_2Eextreal_2Eextreal__logr,arr(ty_2Erealax_2Ereal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__logr,type,
fo__c_2Eextreal_2Eextreal__logr: ( tp__ty_2Erealax_2Ereal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__lt,type,
c_2Eextreal_2Eextreal__lt: $i ).
tff(mem_c_2Eextreal_2Eextreal__lt,axiom,
mem(c_2Eextreal_2Eextreal__lt,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__lt,type,
fo__c_2Eextreal_2Eextreal__lt: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__o ).
tff(stp_eq_fo_c_2Eextreal_2Eextreal__lt,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal,X1: tp__ty_2Eextreal_2Eextreal] : inj__o(fo__c_2Eextreal_2Eextreal__lt(X0,X1)) = ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(X0)),inj__ty_2Eextreal_2Eextreal(X1)) ).
tff(tp_c_2Eextreal_2Eextreal__max,type,
c_2Eextreal_2Eextreal__max: $i ).
tff(mem_c_2Eextreal_2Eextreal__max,axiom,
mem(c_2Eextreal_2Eextreal__max,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__max,type,
fo__c_2Eextreal_2Eextreal__max: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__min,type,
c_2Eextreal_2Eextreal__min: $i ).
tff(mem_c_2Eextreal_2Eextreal__min,axiom,
mem(c_2Eextreal_2Eextreal__min,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__min,type,
fo__c_2Eextreal_2Eextreal__min: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__mul,type,
c_2Eextreal_2Eextreal__mul: $i ).
tff(mem_c_2Eextreal_2Eextreal__mul,axiom,
mem(c_2Eextreal_2Eextreal__mul,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__mul,type,
fo__c_2Eextreal_2Eextreal__mul: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__of__num,type,
c_2Eextreal_2Eextreal__of__num: $i ).
tff(mem_c_2Eextreal_2Eextreal__of__num,axiom,
mem(c_2Eextreal_2Eextreal__of__num,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__of__num,type,
fo__c_2Eextreal_2Eextreal__of__num: tp__ty_2Enum_2Enum > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__pow,type,
c_2Eextreal_2Eextreal__pow: $i ).
tff(mem_c_2Eextreal_2Eextreal__pow,axiom,
mem(c_2Eextreal_2Eextreal__pow,arr(ty_2Eextreal_2Eextreal,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__pow,type,
fo__c_2Eextreal_2Eextreal__pow: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Enum_2Enum ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__size,type,
c_2Eextreal_2Eextreal__size: $i ).
tff(mem_c_2Eextreal_2Eextreal__size,axiom,
mem(c_2Eextreal_2Eextreal__size,arr(ty_2Eextreal_2Eextreal,ty_2Enum_2Enum)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__size,type,
fo__c_2Eextreal_2Eextreal__size: tp__ty_2Eextreal_2Eextreal > tp__ty_2Enum_2Enum ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__sqrt,type,
c_2Eextreal_2Eextreal__sqrt: $i ).
tff(mem_c_2Eextreal_2Eextreal__sqrt,axiom,
mem(c_2Eextreal_2Eextreal__sqrt,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal)) ).
tff(stp_fo_c_2Eextreal_2Eextreal__sqrt,type,
fo__c_2Eextreal_2Eextreal__sqrt: tp__ty_2Eextreal_2Eextreal > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__sub,type,
c_2Eextreal_2Eextreal__sub: $i ).
tff(mem_c_2Eextreal_2Eextreal__sub,axiom,
mem(c_2Eextreal_2Eextreal__sub,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal))) ).
tff(stp_fo_c_2Eextreal_2Eextreal__sub,type,
fo__c_2Eextreal_2Eextreal__sub: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__ty_2Eextreal_2Eextreal ).
tff(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)) ).
tff(tp_c_2Eextreal_2Eextreal__sup,type,
c_2Eextreal_2Eextreal__sup: $i ).
tff(mem_c_2Eextreal_2Eextreal__sup,axiom,
mem(c_2Eextreal_2Eextreal__sup,arr(arr(ty_2Eextreal_2Eextreal,bool),ty_2Eextreal_2Eextreal)) ).
tff(tp_c_2Eextreal_2Emono__decreasing,type,
c_2Eextreal_2Emono__decreasing: $i ).
tff(mem_c_2Eextreal_2Emono__decreasing,axiom,
mem(c_2Eextreal_2Emono__decreasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eextreal_2Emono__increasing,type,
c_2Eextreal_2Emono__increasing: $i ).
tff(mem_c_2Eextreal_2Emono__increasing,axiom,
mem(c_2Eextreal_2Emono__increasing,arr(arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal),bool)) ).
tff(tp_c_2Eextreal_2Eopen__interval,type,
c_2Eextreal_2Eopen__interval: $i ).
tff(mem_c_2Eextreal_2Eopen__interval,axiom,
mem(c_2Eextreal_2Eopen__interval,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,arr(ty_2Eextreal_2Eextreal,bool)))) ).
tff(stp_fo_c_2Eextreal_2Eopen__interval,type,
fo__c_2Eextreal_2Eopen__interval: ( tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal * tp__ty_2Eextreal_2Eextreal ) > tp__o ).
tff(stp_eq_fo_c_2Eextreal_2Eopen__interval,axiom,
! [X0: tp__ty_2Eextreal_2Eextreal,X1: tp__ty_2Eextreal_2Eextreal,X2: tp__ty_2Eextreal_2Eextreal] : inj__o(fo__c_2Eextreal_2Eopen__interval(X0,X1,X2)) = ap(ap(ap(c_2Eextreal_2Eopen__interval,inj__ty_2Eextreal_2Eextreal(X0)),inj__ty_2Eextreal_2Eextreal(X1)),inj__ty_2Eextreal_2Eextreal(X2)) ).
tff(tp_c_2Eextreal_2Eopen__intervals__set,type,
c_2Eextreal_2Eopen__intervals__set: $i ).
tff(mem_c_2Eextreal_2Eopen__intervals__set,axiom,
mem(c_2Eextreal_2Eopen__intervals__set,arr(arr(ty_2Eextreal_2Eextreal,bool),bool)) ).
tff(tp_c_2Eextreal_2Erational__intervals,type,
c_2Eextreal_2Erational__intervals: $i ).
tff(mem_c_2Eextreal_2Erational__intervals,axiom,
mem(c_2Eextreal_2Erational__intervals,arr(arr(ty_2Eextreal_2Eextreal,bool),bool)) ).
tff(tp_c_2Eextreal_2Ereal,type,
c_2Eextreal_2Ereal: $i ).
tff(mem_c_2Eextreal_2Ereal,axiom,
mem(c_2Eextreal_2Ereal,arr(ty_2Eextreal_2Eextreal,ty_2Erealax_2Ereal)) ).
tff(stp_fo_c_2Eextreal_2Ereal,type,
fo__c_2Eextreal_2Ereal: tp__ty_2Eextreal_2Eextreal > tp__ty_2Erealax_2Ereal ).
tff(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)) ).
tff(lamtp_f3341,type,
f3341: $i ).
tff(lameq_f3341,axiom,
! [V7a: tp__ty_2Erealax_2Ereal] : ap(f3341,inj__ty_2Erealax_2Ereal(V7a)) = 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)))),inj__ty_2Erealax_2Ereal(V7a)),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Erealax_2Ereal))) ).
tff(lamtp_f3342,type,
f3342: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal > $i ).
tff(lameq_f3342,axiom,
! [V3a0: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal,V6a: tp__ty_2Erealax_2Ereal] : ap(f3342(V3a0),inj__ty_2Erealax_2Ereal(V6a)) = ap(ap(c_2Emin_2E_3D(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),inj__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal(V3a0)),ap(f3341,inj__ty_2Erealax_2Ereal(V6a))) ).
tff(lamtp_f3343,type,
f3343: $i > $i ).
tff(lameq_f3343,axiom,
! [V2_27extreal_27: $i] :
( mem(V2_27extreal_27,arr(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal),bool))
=> ! [V3a0: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal] : ap(f3343(V2_27extreal_27),inj__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal(V3a0)) = 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)),inj__c_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)),k(ty_2Enum_2Enum,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)),inj__c_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)),k(ty_2Enum_2Enum,c_2Eind__type_2EBOTTOM(ty_2Erealax_2Ereal))))),ap(c_2Ebool_2E_3F(ty_2Erealax_2Ereal),f3342(V3a0))))),ap(V2_27extreal_27,inj__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal(V3a0))) ) ).
tff(lamtp_f3344,type,
f3344: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal > $i ).
tff(lameq_f3344,axiom,
! [V1a0: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal,V2_27extreal_27: $i] : ap(f3344(V1a0),V2_27extreal_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal)),f3343(V2_27extreal_27))),ap(V2_27extreal_27,inj__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal(V1a0))) ).
tff(lamtp_f3345,type,
f3345: $i ).
tff(lameq_f3345,axiom,
! [V1a0: tp__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal] : ap(f3345,inj__c_ty_2Eind__type_2Erecspace_ty_2Erealax_2Ereal(V1a0)) = ap(c_2Ebool_2E_21(arr(ty_2Eind__type_2Erecspace(ty_2Erealax_2Ereal),bool)),f3344(V1a0)) ).
tff(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),f3345),V0rep)) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)))) ) ).
tff(conj_thm_2Eextreal_2Edatatype__extreal,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Eextreal__distinct,axiom,
( ( 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))) ) ).
tff(conj_thm_2Eextreal_2Eextreal__nchotomy,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Eextreal__Axiom,axiom,
! [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)) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__induction,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__case__cong,axiom,
! [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) ) ) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__case__eq,axiom,
! [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 ) ) ) ) ) ) ) ) ).
tff(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)))) ).
tff(lamtp_f3346,type,
f3346: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3346,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1r: tp__ty_2Erealax_2Ereal] : ap(f3346(V0x),inj__ty_2Erealax_2Ereal(V1r)) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V0x)),ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1r))) ).
tff(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),f3346(V0x)))) ).
tff(conj_thm_2Eextreal_2Ereal__normal,axiom,
! [V0x: tp__ty_2Erealax_2Ereal] : surj__ty_2Erealax_2Ereal(ap(c_2Eextreal_2Ereal,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V0x)))) = V0x ).
tff(conj_thm_2Eextreal_2Enormal__real,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__add__ind,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__add__def,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__sub__ind,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__sub__def,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__le__ind,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__le__def,axiom,
! [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 ) ) ).
tff(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))) ) ).
tff(conj_thm_2Eextreal_2Eextreal__mul__ind,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__mul__def,axiom,
! [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)))) ) ) ).
tff(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)))) ) ).
tff(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)))) ) ).
tff(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)))) ).
tff(lamtp_f3347,type,
f3347: $i ).
tff(lameq_f3347,axiom,
! [V0R: $i] : ap(f3347,V0R) = ap(c_2Erelation_2EWF(ty_2Eextreal_2Eextreal),V0R) ).
tff(lamtp_f3348,type,
f3348: $i ).
tff(lameq_f3348,axiom,
! [V3x: tp__ty_2Erealax_2Ereal] : ap(f3348,inj__ty_2Erealax_2Ereal(V3x)) = ap(c_2Ecombin_2EI(ty_2Eextreal_2Eextreal),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Eabs,inj__ty_2Erealax_2Ereal(V3x)))) ).
tff(lamtp_f3349,type,
f3349: $i ).
tff(lameq_f3349,axiom,
! [V2a: tp__ty_2Eextreal_2Eextreal] : ap(f3349,inj__ty_2Eextreal_2Eextreal(V2a)) = ap(ap(ap(ap(c_2Eextreal_2Eextreal__CASE(ty_2Eextreal_2Eextreal),inj__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))),f3348) ).
tff(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))),f3347)),k(arr(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),f3349)) ).
tff(conj_thm_2Eextreal_2Eextreal__abs__ind,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__abs__def,axiom,
! [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 ) ) ).
tff(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 ) ).
tff(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))) ).
tff(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)))) ) ) ).
tff(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)))) ) ).
tff(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 ) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__LMUL__0__NEG,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__RMUL__0__NEG,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__LMUL__NEG__0__NEG,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__RMUL__NEG__0__NEG,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LT__RDIV__EQ__NEG,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2EREAL__LE__RDIV__EQ__NEG,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2EPOW__POS__EVEN,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2EPOW__NEG__ODD,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2ELOGR__MONO__LE,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2ELOGR__MONO__LE__IMP,axiom,
! [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)))) ) ).
tff(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)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Emono__increasing__suc,axiom,
! [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))))) ) ) ).
tff(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)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Emono__decreasing__suc,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Emono__increasing__converges__to__sup,axiom,
! [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)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__cases,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Eextreal__eq__zero,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__not__infty,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Enum__not__infty,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__11,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Emul__rzero,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Emul__lzero,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Emul__rone,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Emul__lone,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Eentire,axiom,
! [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))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__lt__eq,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__refl,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V0x))) ).
tff(conj_thm_2Eextreal_2Elt__refl,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : ~ p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V0x))) ).
tff(conj_thm_2Eextreal_2Ele__infty,axiom,
( ! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Elt__infty,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__imp__le,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(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))) ) ).
tff(conj_thm_2Eextreal_2Elt__imp__ne,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)))
=> ( V0x != V1y ) ) ).
tff(conj_thm_2Eextreal_2Ele__trans,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Elt__trans,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Elet__trans,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__antisym,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Elt__antisym,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__lt,inj__ty_2Eextreal_2Eextreal(V1y)),inj__ty_2Eextreal_2Eextreal(V0x))) ) ).
tff(conj_thm_2Eextreal_2Elte__trans,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__total,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Elt__total,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__01,axiom,
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)))))) ).
tff(conj_thm_2Eextreal_2Elt__01,axiom,
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)))))) ).
tff(conj_thm_2Eextreal_2Ene__01,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Ele__02,axiom,
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)))))) ).
tff(conj_thm_2Eextreal_2Elt__02,axiom,
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)))))) ).
tff(conj_thm_2Eextreal_2Ene__02,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Ele__num,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Elt__le,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(V0x)),inj__ty_2Eextreal_2Eextreal(V1y)))
& ( V0x != V1y ) ) ) ).
tff(conj_thm_2Eextreal_2Ele__lt,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Ele__neg,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Elt__neg,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__add,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__add,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elet__add,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elte__add,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__add2,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__add2,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elet__add2,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elet__add2__alt,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__addr,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__addr__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__ladd,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__radd,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__radd__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__ladd__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__ladd,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__radd,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__addl,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__lneg,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__mul,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elet__mul,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elte__mul,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__mul__neg,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Emul__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__mul,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__mul__neg,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Emul__lt,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Emul__let,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Emul__lte,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__rmul__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Ele__lmul__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__rmul,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__lmul,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__mul2,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Eabs__pos,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Eabs__refl,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Eabs__bounds,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eabs__bounds__lt,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__comm,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Eadd__assoc,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Eadd__not__infty,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__rzero,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Eadd__lzero,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Eadd__infty,axiom,
( ! [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 ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__EQ__LADD,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Esub__rzero,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Esub__lzero,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Esub__le__imp,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esub__le__imp2,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__sub__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elt__sub__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Esub__lt__imp,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esub__lt__imp2,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esub__zero__lt,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__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)))) ) ).
tff(conj_thm_2Eextreal_2Esub__zero__lt2,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esub__lt__zero,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__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)))) ) ).
tff(conj_thm_2Eextreal_2Esub__lt__zero2,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esub__zero__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Esub__le__zero,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__sub__eq,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__sub__eq2,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__le__eq,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__le__eq2,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__le__switch,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__le__switch2,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__sub,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__add2,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eadd__sub,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eadd__sub2,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Esub__add,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__sub__add,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Esub__0,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eneg__neg,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Eneg__0,axiom,
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))) ).
tff(conj_thm_2Eextreal_2Eneg__eq0,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eeq__neg,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eneg__minus1,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Esub__rneg,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Esub__lneg,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eneg__sub,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__not__infty,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Ele__lsub__imp,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Eeq__sub__ladd__normal,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Eeq__sub__radd,axiom,
! [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))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eeq__sub__ladd,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Eeq__sub__switch,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eeq__add__sub__switch,axiom,
! [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))) ) ) ) ).
tff(conj_thm_2Eextreal_2Esub__refl,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Emul__comm,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Emul__assoc,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Emul__not__infty,axiom,
( ! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Emul__not__infty2,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__ldistrib__pos,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__ldistrib__neg,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__ldistrib__normal,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__ldistrib__normal2,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__rdistrib__normal,axiom,
! [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))))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__rdistrib__normal2,axiom,
! [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))))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__ldistrib,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__rdistrib,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Emul__lneg,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Emul__rneg,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Eneg__mul2,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Eadd2__sub2,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__ldistrib,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Esub__rdistrib,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2Eextreal__div__eq,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Einv__one,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Einv__1over,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Ediv__one,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Einv__pos,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Erinv__uniq,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elinv__uniq,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__rdiv,axiom,
! [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))))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__ldiv,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__rdiv,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__ldiv,axiom,
! [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))))) ) ) ).
tff(conj_thm_2Eextreal_2Elt__rdiv__neg,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ediv__add,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__inv,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__0,axiom,
! [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))))) ).
tff(conj_thm_2Eextreal_2Epow__1,axiom,
! [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 ).
tff(conj_thm_2Eextreal_2Epow__2,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2Epow__zero,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Epow__zero__imp,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__pow2,axiom,
! [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)))))) ).
tff(conj_thm_2Eextreal_2Epow__pos__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__pos__lt,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__lt,axiom,
! [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))))) ) ).
tff(conj_thm_2Eextreal_2Epow__lt2,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__le__mono,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow__pos__even,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Epow__neg__odd,axiom,
! [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))) ) ) ).
tff(conj_thm_2Eextreal_2Eadd__pow2,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Epow__add,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Epow__mul,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2Epow__minus1,axiom,
! [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))))) ).
tff(conj_thm_2Eextreal_2Epow__not__infty,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Esqrt__pos__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Esqrt__pos__lt,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Epow2__sqrt,axiom,
! [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 ) ) ).
tff(conj_thm_2Eextreal_2Esqrt__pow2,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Esqrt__mono__le,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2Elogr__not__infty,axiom,
! [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 ) ) ) ).
tff(conj_thm_2Eextreal_2Ehalf__between,axiom,
( 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)))))) ) ).
tff(conj_thm_2Eextreal_2Ethirds__between,axiom,
( 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)))))) ) ).
tff(conj_thm_2Eextreal_2Ehalf__cancel,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Ethird__cancel,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Efourth__cancel,axiom,
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))))) ).
tff(conj_thm_2Eextreal_2Equotient__normal,axiom,
! [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))))) ).
tff(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)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eext__mono__increasing__suc,axiom,
! [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))))) ) ) ).
tff(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)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eext__mono__decreasing__suc,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__ARCH,axiom,
! [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)))) ) ) ).
tff(conj_thm_2Eextreal_2ESIMP__REAL__ARCH,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2ESIMP__REAL__ARCH__NEG,axiom,
! [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))) ).
tff(conj_thm_2Eextreal_2ESIMP__EXTREAL__ARCH,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EREAL__ARCH__POW,axiom,
! [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)))) ).
tff(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW,axiom,
! [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)))) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__ARCH__POW__INV,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2EREAL__LE__MUL__EPSILON,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__epsilon,axiom,
! [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))) ) ).
tff(conj_thm_2Eextreal_2Ele__mul__epsilon,axiom,
! [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))) ) ).
tff(lamtp_f3350,type,
f3350: ( del * $i * $i ) > $i ).
tff(lameq_f3350,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2e: $i] :
( mem(V2e,A_27a)
=> ! [V3acc: tp__ty_2Eextreal_2Eextreal] : ap(f3350(A_27a,V0f,V2e),inj__ty_2Eextreal_2Eextreal(V3acc)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,V2e)),inj__ty_2Eextreal_2Eextreal(V3acc)) ) ) ).
tff(lamtp_f3351,type,
f3351: ( del * $i ) > $i ).
tff(lameq_f3351,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2e: $i] : ap(f3351(A_27a,V0f),V2e) = f3350(A_27a,V0f,V2e) ) ).
tff(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),f3351(A_27a,V0f)),V1s),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__THM,axiom,
! [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)))) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__NEG__INF,axiom,
! [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 ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__POS__INF,axiom,
! [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 ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NOT__INFTY,axiom,
! [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 ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SING,axiom,
! [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)) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS,axiom,
! [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))) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SPOS,axiom,
! [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))) ) ) ) ).
tff(lamtp_f3352,type,
f3352: ( del * $i * $i ) > $i ).
tff(lameq_f3352,axiom,
! [A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1P: $i] :
( mem(V1P,arr(A_27a,bool))
=> ! [V4x: $i] : ap(f3352(A_27a,V2f,V1P),V4x) = 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))) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IF__ELIM,axiom,
! [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),f3352(A_27a,V2f,V1P)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V0s)) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__SAME,axiom,
! [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))) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__FINITE__CONST,axiom,
! [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))) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO,axiom,
! [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),k(A_27a,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))) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__0,axiom,
! [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))) ) ) ) ) ).
tff(lamtp_f3353,type,
f3353: ( del * $i * $i ) > $i ).
tff(lameq_f3353,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2x: $i] : ap(f3353(A_27a,V1f,V0s),V2x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V2x),V0s)),ap(V1f,V2x)),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF,axiom,
! [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))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3353(A_27a,V1f,V0s)),V0s)) ) ) ) ) ).
tff(lamtp_f3354,type,
f3354: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3354,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Erealax_2Ereal,V4x: $i] : ap(f3354(A_27a,V1f,V2c),V4x) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V2c))),ap(V1f,V4x)) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL,axiom,
! [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))
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( 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)))
| ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1f,V3x))) ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3354(A_27a,V1f,V2c)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V2c))),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s))) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CMUL2,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2c: tp__ty_2Erealax_2Ereal] :
( ( 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))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V3x)) != fo__c_2Eextreal_2ENegInf ) ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3354(A_27a,V1f,V2c)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V2c))),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ! [V1f_27: $i] :
( mem(V1f_27,arr(A_27a,A_27b))
=> ( p(ap(ap(ap(c_2Epred__set_2EINJ(A_27a,A_27b),V1f_27),V0s),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0s)))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27b,ty_2Eextreal_2Eextreal))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),V2f),ap(ap(c_2Epred__set_2EIMAGE(A_27a,A_27b),V1f_27),V0s))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),ap(ap(c_2Ecombin_2Eo(A_27a,ty_2Eextreal_2Eextreal,A_27b),V2f),V1f_27)),V0s)) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__DISJOINT__UNION,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27: $i] :
( mem(V1s_27,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V1s_27))
& p(ap(ap(c_2Epred__set_2EDISJOINT(A_27a),V0s),V1s_27)) )
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),ap(ap(c_2Epred__set_2EUNION(A_27a),V0s),V1s_27))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f),V1s_27))) ) ) ) ) ) ).
tff(lamtp_f3355,type,
f3355: ( del * $i ) > $i ).
tff(lameq_f3355,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x: $i] : ap(f3355(A_27a,V0s),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),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,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ__CARD,axiom,
! [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),f3355(A_27a,V0s)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s))) ) ) ) ).
tff(lamtp_f3356,type,
f3356: ( del * $i ) > $i ).
tff(lameq_f3356,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1x: $i] : ap(f3356(A_27a,V0s),V1x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V1x),V0s)),ap(c_2Eextreal_2Eextreal__inv,ap(c_2Eextreal_2Eextreal__of__num,ap(c_2Epred__set_2ECARD(A_27a),V0s)))),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INV__CARD__EQ__1,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ( ( ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
& p(ap(c_2Epred__set_2EFINITE(A_27a),V0s)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3356(A_27a,V0s)),V0s)) = 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))))) ) ) ) ).
tff(lamtp_f3357,type,
f3357: ( del * $i ) > $i ).
tff(lameq_f3357,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2p: $i] : ap(f3357(A_27a,V1f),V2p) = ap(c_2Ebool_2E_7E,ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),ap(V1f,V2p)),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0)))) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__NONZERO,axiom,
! [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))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),f3357(A_27a,V1f)))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__INTER__ELIM,axiom,
! [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))
=> ! [V2s_27: $i] :
( mem(V2s_27,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( ~ p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2s_27))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V3x)) = 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),V1f),ap(ap(c_2Epred__set_2EINTER(A_27a),V0s),V2s_27))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ZERO__DIFF,axiom,
! [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))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V3x)) = 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),V1f),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),ap(ap(c_2Epred__set_2EDIFF(A_27a),V0s),V2t))) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO,axiom,
! [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))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1f,V3x)),ap(V2f_27,V3x))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s))) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__MONO__SET,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t: $i] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
& p(ap(c_2Epred__set_2EFINITE(A_27a),V2t))
& p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V2t))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,V3x))) ) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V2t))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EQ,axiom,
! [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))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V1f,V3x)) = surj__ty_2Eextreal_2Eextreal(ap(V2f_27,V3x)) ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s)) ) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__POS__MEM__LE,axiom,
! [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))) ) ) )
=> ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V1s))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,V3x)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ) ).
tff(lamtp_f3358,type,
f3358: ( del * $i * $i ) > $i ).
tff(lameq_f3358,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V3x: $i] : ap(f3358(A_27a,V1f,V2f_27),V3x) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V1f,V3x)),ap(V2f_27,V3x)) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__ADD,axiom,
! [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))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3358(A_27a,V1f,V2f_27)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s))) ) ) ) ) ) ).
tff(lamtp_f3359,type,
f3359: ( del * $i * $i ) > $i ).
tff(lameq_f3359,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V5x: $i] : ap(f3359(A_27a,V1f,V2f_27),V5x) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V1f,V5x)),ap(V2f_27,V5x)) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__SUB,axiom,
! [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))
=> ! [V2f_27: $i] :
( mem(V2f_27,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ( ( ! [V3x: $i] :
( mem(V3x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2f_27,V3x)) != fo__c_2Eextreal_2ENegInf ) ) )
| ! [V4x: $i] :
( mem(V4x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V4x),V0s))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V2f_27,V4x)) != fo__c_2Eextreal_2EPosInf ) ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3359(A_27a,V1f,V2f_27)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__sub,ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)),ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V2f_27),V0s))) ) ) ) ) ) ) ).
tff(lamtp_f3360,type,
f3360: ( del * del * $i * $i ) > $i ).
tff(lameq_f3360,axiom,
! [A_27b: del,A_27a: del,V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Eextreal_2Eextreal)))
=> ! [V1s_27: $i] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V3x: $i] : ap(f3360(A_27b,A_27a,V2f,V1s_27),V3x) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27b),ap(V2f,V3x)),V1s_27) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__EXTREAL__SUM__IMAGE,axiom,
! [A_27a: del,A_27b: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1s_27: $i] :
( mem(V1s_27,arr(A_27b,bool))
=> ! [V2f: $i] :
( mem(V2f,arr(A_27a,arr(A_27b,ty_2Eextreal_2Eextreal)))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V1s_27)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3360(A_27b,A_27a,V2f,V1s_27)),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),f3092(A_27a,A_27b,V2f)),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V0s),V1s_27))) ) ) ) ) ) ).
tff(lamtp_f3361,type,
f3361: ( del * $i ) > $i ).
tff(lameq_f3361,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V2x: $i] : ap(f3361(A_27a,V0f),V2x) = ap(c_2Eextreal_2ENormal,ap(V0f,V2x)) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__NORMAL,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Erealax_2Ereal))
=> ! [V1s: $i] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3361(A_27a,V0f)),V1s)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,ap(ap(c_2Ereal__sigma_2EREAL__SUM__IMAGE(A_27a),V0f),V1s))) ) ) ) ) ).
tff(lamtp_f3362,type,
f3362: ( del * $i * $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3362,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V2z: tp__ty_2Eextreal_2Eextreal,V3x: $i] : ap(f3362(A_27a,V1f,V0s,V2z),V3x) = ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Ebool_2EIN(A_27a),V3x),V0s)),ap(V1f,V3x)),inj__ty_2Eextreal_2Eextreal(V2z)) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__IN__IF__ALT,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1f: $i] :
( mem(V1f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V2z: tp__ty_2Eextreal_2Eextreal] :
( p(ap(c_2Epred__set_2EFINITE(A_27a),V0s))
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),V1f),V0s)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(A_27a),f3362(A_27a,V1f,V0s,V2z)),V0s)) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__CROSS__SYM,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Epair_2Eprod(A_27a,A_27b),ty_2Eextreal_2Eextreal))
=> ! [V1s1: $i] :
( mem(V1s1,arr(A_27a,bool))
=> ! [V2s2: $i] :
( mem(V2s2,arr(A_27b,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(A_27a),V1s1))
& p(ap(c_2Epred__set_2EFINITE(A_27b),V2s2)) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27a,A_27b)),ap(c_2Epair_2EUNCURRY(A_27a,A_27b,ty_2Eextreal_2Eextreal),f3103(A_27b,A_27a,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27a,A_27b),V1s1),V2s2))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Epair_2Eprod(A_27b,A_27a)),ap(c_2Epair_2EUNCURRY(A_27b,A_27a,ty_2Eextreal_2Eextreal),f3105(A_27a,A_27b,V0f))),ap(ap(c_2Epred__set_2ECROSS(A_27b,A_27a),V2s2),V1s1))) ) ) ) ) ) ).
tff(conj_thm_2Eextreal_2EEXTREAL__SUM__IMAGE__COUNT,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,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(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))) )
& ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,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(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,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__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,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(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,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(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,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__add,ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO)))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT1,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))),ap(V0f,ap(c_2Earithmetic_2ENUMERAL,ap(c_2Earithmetic_2EBIT2,ap(c_2Earithmetic_2EBIT1,inj__ty_2Enum_2Enum(fo__c_2Earithmetic_2EZERO))))))) ) ) ) ).
tff(lamtp_f3363,type,
f3363: ( $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3363,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal,V2y: tp__ty_2Eextreal_2Eextreal] : ap(f3363(V0p,V1x),inj__ty_2Eextreal_2Eextreal(V2y)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V0p,inj__ty_2Eextreal_2Eextreal(V2y))),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ).
tff(lamtp_f3364,type,
f3364: $i > $i ).
tff(lameq_f3364,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] : ap(f3364(V0p),inj__ty_2Eextreal_2Eextreal(V1x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3363(V0p,V1x))),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf))) ) ).
tff(lamtp_f3365,type,
f3365: $i > $i ).
tff(lameq_f3365,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V3x: tp__ty_2Eextreal_2Eextreal] : ap(f3365(V0p),inj__ty_2Eextreal_2Eextreal(V3x)) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(V0p,inj__ty_2Eextreal_2Eextreal(V3x))),ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V3x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf))) ) ).
tff(lamtp_f3366,type,
f3366: $i > $i ).
tff(lameq_f3366,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V4r: tp__ty_2Erealax_2Ereal] : ap(f3366(V0p),inj__ty_2Erealax_2Ereal(V4r)) = ap(V0p,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V4r))) ) ).
tff(ax_thm_2Eextreal_2Eextreal__sup__def,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,V0p)) = surj__ty_2Eextreal_2Eextreal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3364(V0p))),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf)),ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(c_2Ebool_2E_21(ty_2Eextreal_2Eextreal),f3365(V0p))),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf)),ap(c_2Eextreal_2ENormal,ap(c_2Ereal_2Esup,f3366(V0p)))))) ) ) ).
tff(ax_thm_2Eextreal_2Eextreal__inf__def,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,V0p)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__ainv,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Eextreal_2Eextreal,ty_2Eextreal_2Eextreal),c_2Eextreal_2Eextreal__ainv),V0p)))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__sup__imp,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),ap(c_2Eextreal_2Eextreal__sup,V0p))) ) ) ).
tff(conj_thm_2Eextreal_2Esup__le,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,V0p)),inj__ty_2Eextreal_2Eextreal(V1x)))
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2y)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ) ) ).
tff(conj_thm_2Eextreal_2Ele__sup,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),ap(c_2Eextreal_2Eextreal__sup,V0p)))
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( ! [V3z: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V3z)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V3z)),inj__ty_2Eextreal_2Eextreal(V2y))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__eq,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,V0p)) = V1x )
<=> ( ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2y)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V1x))) )
& ! [V3y: tp__ty_2Eextreal_2Eextreal] :
( ! [V4z: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V4z)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V4z)),inj__ty_2Eextreal_2Eextreal(V3y))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V3y))) ) ) ) ) ).
tff(lamtp_f3367,type,
f3367: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3367,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : ap(f3367(V0x),inj__ty_2Eextreal_2Eextreal(V1y)) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V1y)),inj__ty_2Eextreal_2Eextreal(V0x)) ).
tff(conj_thm_2Eextreal_2Esup__const,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,f3367(V0x))) = V0x ).
tff(conj_thm_2Eextreal_2Esup__const__alt,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z: tp__ty_2Eextreal_2Eextreal] :
( ( ? [V2x: tp__ty_2Eextreal_2Eextreal] : p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2x)))
& ! [V3x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V3x)))
=> ( V3x = V1z ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,V0p)) = V1z ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__const__over__set,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1k: tp__ty_2Eextreal_2Eextreal] :
( ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),k(A_27a,inj__ty_2Eextreal_2Eextreal(V1k))),V0s))) = V1k ) ) ) ).
tff(lamtp_f3368,type,
f3368: tp__ty_2Eextreal_2Eextreal > $i ).
tff(lameq_f3368,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1n: tp__ty_2Enum_2Enum] : ap(f3368(V0x),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Emin_2E_3D(ty_2Eextreal_2Eextreal),inj__ty_2Eextreal_2Eextreal(V0x)),ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(V1n))) ).
tff(lamtp_f3369,type,
f3369: $i ).
tff(lameq_f3369,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : ap(f3369,inj__ty_2Eextreal_2Eextreal(V0x)) = ap(c_2Ebool_2E_3F(ty_2Enum_2Enum),f3368(V0x)) ).
tff(conj_thm_2Eextreal_2Esup__num,axiom,
surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,f3369)) = fo__c_2Eextreal_2EPosInf ).
tff(conj_thm_2Eextreal_2Esup__le__sup__imp,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1q: $i] :
( mem(V1q,arr(ty_2Eextreal_2Eextreal,bool))
=> ( ! [V2x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2x)))
=> ? [V3y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V1q,inj__ty_2Eextreal_2Eextreal(V3y)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V3y))) ) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,V0p)),ap(c_2Eextreal_2Eextreal__sup,V1q))) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__mono,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1q: $i] :
( mem(V1q,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0p,inj__ty_2Enum_2Enum(V2n))),ap(V1q,inj__ty_2Enum_2Enum(V2n))))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0p),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V1q),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ) ).
tff(lamtp_f3370,type,
f3370: $i > $i ).
tff(lameq_f3370,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ap(f3370(V0f),inj__ty_2Enum_2Enum(V3n)) = ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n))) ) ).
tff(conj_thm_2Eextreal_2Esup__suc,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [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)))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3370(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ) ).
tff(lamtp_f3371,type,
f3371: $i > $i ).
tff(lameq_f3371,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ap(f3371(V0f),inj__ty_2Enum_2Enum(V2n)) = ap(c_2Eextreal_2ENormal,ap(V0f,inj__ty_2Enum_2Enum(V2n))) ) ).
tff(conj_thm_2Eextreal_2Esup__seq,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l: 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(V1l)))
<=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3371(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1l))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__lt__infty,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V0p)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf)))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf))) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__max,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z: tp__ty_2Eextreal_2Eextreal] :
( ( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1z)))
& ! [V2x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2x)),inj__ty_2Eextreal_2Eextreal(V1z))) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,V0p)) = V1z ) ) ) ).
tff(lamtp_f3372,type,
f3372: ( $i * $i ) > $i ).
tff(lameq_f3372,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V6n: tp__ty_2Enum_2Enum] : ap(f3372(V0f,V1g),inj__ty_2Enum_2Enum(V6n)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,inj__ty_2Enum_2Enum(V6n))),ap(V1g,inj__ty_2Enum_2Enum(V6n))) ) ) ).
tff(conj_thm_2Eextreal_2Esup__add__mono,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [V2n: 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(V0f,inj__ty_2Enum_2Enum(V2n))))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,inj__ty_2Enum_2Enum(V3n))),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))))
& ! [V4n: 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(V1g,inj__ty_2Enum_2Enum(V4n))))
& ! [V5n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,inj__ty_2Enum_2Enum(V5n))),ap(V1g,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n))))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3372(V0f,V1g)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V1g),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) ) ) ) ) ).
tff(lamtp_f3373,type,
f3373: ( $i * tp__ty_2Enum_2Enum ) > $i ).
tff(lameq_f3373,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V6n: tp__ty_2Enum_2Enum,V7i: tp__ty_2Enum_2Enum] : ap(f3373(V0f,V6n),inj__ty_2Enum_2Enum(V7i)) = ap(ap(V0f,inj__ty_2Enum_2Enum(V7i)),inj__ty_2Enum_2Enum(V6n)) ) ).
tff(lamtp_f3374,type,
f3374: ( $i * $i ) > $i ).
tff(lameq_f3374,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V1s: $i] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ! [V6n: tp__ty_2Enum_2Enum] : ap(f3374(V0f,V1s),inj__ty_2Enum_2Enum(V6n)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3373(V0f,V6n)),V1s) ) ) ).
tff(lamtp_f3375,type,
f3375: $i > $i ).
tff(lameq_f3375,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V8i: tp__ty_2Enum_2Enum] : ap(f3375(V0f),inj__ty_2Enum_2Enum(V8i)) = ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),ap(V0f,inj__ty_2Enum_2Enum(V8i))),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))) ) ).
tff(conj_thm_2Eextreal_2Esup__sum__mono,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal)))
=> ! [V1s: $i] :
( mem(V1s,arr(ty_2Enum_2Enum,bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(ty_2Enum_2Enum),V1s))
& ! [V2i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V2i)),V1s))
=> ! [V3n: 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(V0f,inj__ty_2Enum_2Enum(V2i)),inj__ty_2Enum_2Enum(V3n)))) )
& ! [V4i: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Ebool_2EIN(ty_2Enum_2Enum),inj__ty_2Enum_2Enum(V4i)),V1s))
=> ! [V5n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(V0f,inj__ty_2Enum_2Enum(V4i)),inj__ty_2Enum_2Enum(V5n))),ap(ap(V0f,inj__ty_2Enum_2Enum(V4i)),ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V5n))))) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3374(V0f,V1s)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),f3375(V0f)),V1s)) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__le__mono,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1z: tp__ty_2Eextreal_2Eextreal] :
( ( ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V0f,inj__ty_2Enum_2Enum(V2n))),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1z)),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum))))) )
=> ? [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1z)),ap(V0f,inj__ty_2Enum_2Enum(V3n)))) ) ) ).
tff(lamtp_f3376,type,
f3376: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3376,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2n: $i] : ap(f3376(A_27a,V0f,V1c),V2n) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(V0f,V2n)) ) ).
tff(conj_thm_2Eextreal_2Esup__cmul,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( 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(V1c)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),f3376(A_27a,V0f,V1c)),c_2Epred__set_2EUNIV(A_27a)))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(A_27a))))) ) ) ) ).
tff(conj_thm_2Eextreal_2Esup__lt,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1y: tp__ty_2Eextreal_2Eextreal] :
( ? [V2x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0P,inj__ty_2Eextreal_2Eextreal(V2x)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1y)),inj__ty_2Eextreal_2Eextreal(V2x))) )
<=> p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(V1y)),ap(c_2Eextreal_2Eextreal__sup,V0P))) ) ) ).
tff(conj_thm_2Eextreal_2Esup__lt__epsilon,axiom,
! [V0P: $i] :
( mem(V0P,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1e: 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(V1e)))
& ? [V2x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0P,inj__ty_2Eextreal_2Eextreal(V2x)))
& ( V2x != fo__c_2Eextreal_2ENegInf ) )
& ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,V0P)) != fo__c_2Eextreal_2EPosInf ) )
=> ? [V3x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0P,inj__ty_2Eextreal_2Eextreal(V3x)))
& p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(c_2Eextreal_2Eextreal__sup,V0P)),ap(ap(c_2Eextreal_2Eextreal__add,inj__ty_2Eextreal_2Eextreal(V3x)),inj__ty_2Eextreal_2Eextreal(V1e)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__le__imp,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__inf,V0p)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__inf,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),ap(c_2Eextreal_2Eextreal__inf,V0p)))
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2y)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__le,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__inf,V0p)),inj__ty_2Eextreal_2Eextreal(V1x)))
<=> ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( ! [V3z: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V3z)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V3z))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__eq,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,V0p)) = V1x )
<=> ( ! [V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2y)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))) )
& ! [V3y: tp__ty_2Eextreal_2Eextreal] :
( ! [V4z: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V4z)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V3y)),inj__ty_2Eextreal_2Eextreal(V4z))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V3y)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__const,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,f3367(V0x))) = V0x ).
tff(conj_thm_2Eextreal_2Einf__const__alt,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z: tp__ty_2Eextreal_2Eextreal] :
( ( ? [V2x: tp__ty_2Eextreal_2Eextreal] : p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2x)))
& ! [V3x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V3x)))
=> ( V3x = V1z ) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,V0p)) = V1z ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__const__over__set,axiom,
! [A_27a: del,V0s: $i] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1k: tp__ty_2Eextreal_2Eextreal] :
( ( V0s != c_2Epred__set_2EEMPTY(A_27a) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),k(A_27a,inj__ty_2Eextreal_2Eextreal(V1k))),V0s))) = V1k ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__suc,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [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)))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3370(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__seq,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V1l: tp__ty_2Erealax_2Ereal] :
( p(ap(c_2Eextreal_2Emono__decreasing,V0f))
=> ( p(ap(ap(c_2Eseq_2E_2D_2D_3E,V0f),inj__ty_2Erealax_2Ereal(V1l)))
<=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3371(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1l))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__lt__infty,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ( p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf)),ap(c_2Eextreal_2Eextreal__inf,V0p)))
=> ! [V1x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__lt,inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf)),inj__ty_2Eextreal_2Eextreal(V1x))) ) ) ) ).
tff(conj_thm_2Eextreal_2Einf__min,axiom,
! [V0p: $i] :
( mem(V0p,arr(ty_2Eextreal_2Eextreal,bool))
=> ! [V1z: tp__ty_2Eextreal_2Eextreal] :
( ( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V1z)))
& ! [V2x: tp__ty_2Eextreal_2Eextreal] :
( p(ap(V0p,inj__ty_2Eextreal_2Eextreal(V2x)))
=> p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1z)),inj__ty_2Eextreal_2Eextreal(V2x))) ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__inf,V0p)) = V1z ) ) ) ).
tff(lamtp_f3377,type,
f3377: ( del * $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3377,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V2n: $i] : ap(f3377(A_27a,V0f,V1c),V2n) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(V0f,V2n)) ) ).
tff(conj_thm_2Eextreal_2Einf__cminus,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(A_27a,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] : surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(c_2Eextreal_2Eextreal__inf,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),V0f),c_2Epred__set_2EUNIV(A_27a))))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(A_27a,ty_2Eextreal_2Eextreal),f3377(A_27a,V0f,V1c)),c_2Epred__set_2EUNIV(A_27a)))) ) ).
tff(lamtp_f3378,type,
f3378: $i > $i ).
tff(lameq_f3378,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1n: tp__ty_2Enum_2Enum] : ap(f3378(V0f),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V1n))) ) ).
tff(ax_thm_2Eextreal_2Eext__suminf__def,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,V0f)) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eextreal__sup,ap(ap(c_2Epred__set_2EIMAGE(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal),f3378(V0f)),c_2Epred__set_2EUNIV(ty_2Enum_2Enum)))) ) ) ).
tff(lamtp_f3379,type,
f3379: ( $i * $i ) > $i ).
tff(lameq_f3379,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ap(f3379(V0f,V1g),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Eextreal_2Eextreal__add,ap(V0f,inj__ty_2Enum_2Enum(V3n))),ap(V1g,inj__ty_2Enum_2Enum(V3n))) ) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__add,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n: 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(V0f,inj__ty_2Enum_2Enum(V2n))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eextreal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V1g,inj__ty_2Enum_2Enum(V2n)))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3379(V0f,V1g))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__add,ap(c_2Eextreal_2Eext__suminf,V0f)),ap(c_2Eextreal_2Eext__suminf,V1g))) ) ) ) ) ).
tff(lamtp_f3380,type,
f3380: ( $i * tp__ty_2Eextreal_2Eextreal ) > $i ).
tff(lameq_f3380,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Eextreal_2Eextreal,V3n: tp__ty_2Enum_2Enum] : ap(f3380(V0f,V1c),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Eextreal_2Eextreal__mul,inj__ty_2Eextreal_2Eextreal(V1c)),ap(V0f,inj__ty_2Enum_2Enum(V3n))) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__cmul,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c: 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(V1c)))
& ! [V2n: 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(V0f,inj__ty_2Enum_2Enum(V2n)))) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3380(V0f,V1c))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,inj__ty_2Eextreal_2Eextreal(V1c)),ap(c_2Eextreal_2Eext__suminf,V0f))) ) ) ) ).
tff(lamtp_f3381,type,
f3381: ( $i * tp__ty_2Erealax_2Ereal ) > $i ).
tff(lameq_f3381,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal,V4n: tp__ty_2Enum_2Enum] : ap(f3381(V0f,V1c),inj__ty_2Enum_2Enum(V4n)) = ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(V0f,inj__ty_2Enum_2Enum(V4n))) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__cmul__alt,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1c: tp__ty_2Erealax_2Ereal] :
( ( 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(V1c)))
& ( ! [V2n: tp__ty_2Enum_2Enum] : surj__ty_2Eextreal_2Eextreal(ap(V0f,inj__ty_2Enum_2Enum(V2n))) != fo__c_2Eextreal_2ENegInf
| ! [V3n: tp__ty_2Enum_2Enum] : surj__ty_2Eextreal_2Eextreal(ap(V0f,inj__ty_2Enum_2Enum(V3n))) != fo__c_2Eextreal_2EPosInf ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3381(V0f,V1c))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__mul,ap(c_2Eextreal_2ENormal,inj__ty_2Erealax_2Ereal(V1c))),ap(c_2Eextreal_2Eext__suminf,V0f))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__lt__infty,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [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(V0f,inj__ty_2Enum_2Enum(V1n))))
& ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,V0f)) != fo__c_2Eextreal_2EPosInf ) )
=> ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Eextreal_2Eextreal__lt,ap(V0f,inj__ty_2Enum_2Enum(V2n))),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf))) ) ) ).
tff(lamtp_f3382,type,
f3382: $i > $i ).
tff(lameq_f3382,axiom,
! [V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V2n: tp__ty_2Enum_2Enum] : ap(f3382(V0r),inj__ty_2Enum_2Enum(V2n)) = ap(c_2Eextreal_2ENormal,ap(V0r,inj__ty_2Enum_2Enum(V2n))) ) ).
tff(lamtp_f3383,type,
f3383: $i > $i ).
tff(lameq_f3383,axiom,
! [V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ! [V3n: tp__ty_2Enum_2Enum] : ap(f3383(V0r),inj__ty_2Enum_2Enum(V3n)) = ap(c_2Eextreal_2ENormal,ap(V0r,inj__ty_2Enum_2Enum(V3n))) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__suminf,axiom,
! [V0r: $i] :
( mem(V0r,arr(ty_2Enum_2Enum,ty_2Erealax_2Ereal))
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ereal_2Ereal__lte,ap(c_2Ereal_2Ereal__of__num,inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),ap(V0r,inj__ty_2Enum_2Enum(V1n))))
& ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3382(V0r))) != fo__c_2Eextreal_2EPosInf ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3383(V0r))) = surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2ENormal,ap(c_2Eseq_2Esuminf,V0r))) ) ) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__mono,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ! [V2n: tp__ty_2Enum_2Enum] :
( ( surj__ty_2Eextreal_2Eextreal(ap(V1g,inj__ty_2Enum_2Enum(V2n))) != fo__c_2Eextreal_2ENegInf )
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,inj__ty_2Enum_2Enum(V2n))),ap(V0f,inj__ty_2Enum_2Enum(V2n)))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(c_2Eextreal_2Eext__suminf,V1g)),ap(c_2Eextreal_2Eext__suminf,V0f))) ) ) ) ).
tff(lamtp_f3384,type,
f3384: ( $i * $i ) > $i ).
tff(lameq_f3384,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V3i: tp__ty_2Enum_2Enum] : ap(f3384(V0f,V1g),inj__ty_2Enum_2Enum(V3i)) = ap(ap(c_2Eextreal_2Eextreal__sub,ap(V0f,inj__ty_2Enum_2Enum(V3i))),ap(V1g,inj__ty_2Enum_2Enum(V3i))) ) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__sub,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ( ( ! [V2n: 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(V1g,inj__ty_2Enum_2Enum(V2n))))
& p(ap(ap(c_2Eextreal_2Eextreal__le,ap(V1g,inj__ty_2Enum_2Enum(V2n))),ap(V0f,inj__ty_2Enum_2Enum(V2n)))) )
& ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,V0f)) != fo__c_2Eextreal_2EPosInf ) )
=> ( surj__ty_2Eextreal_2Eextreal(ap(c_2Eextreal_2Eext__suminf,f3384(V0f,V1g))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__sub,ap(c_2Eextreal_2Eext__suminf,V0f)),ap(c_2Eextreal_2Eext__suminf,V1g))) ) ) ) ) ).
tff(conj_thm_2Eextreal_2Eext__suminf__sum,axiom,
! [V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eextreal_2Eextreal))
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ! [V2n: 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(V0f,inj__ty_2Enum_2Enum(V2n))))
& ! [V3m: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V1n)),inj__ty_2Enum_2Enum(V3m)))
=> ( surj__ty_2Eextreal_2Eextreal(ap(V0f,inj__ty_2Enum_2Enum(V3m))) = 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_2Eext__suminf,V0f)) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2EEXTREAL__SUM__IMAGE(ty_2Enum_2Enum),V0f),ap(c_2Epred__set_2Ecount,inj__ty_2Enum_2Enum(V1n)))) ) ) ) ).
tff(ax_thm_2Eextreal_2Eextreal__min__def,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))) = surj__ty_2Eextreal_2Eextreal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))),inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))) ).
tff(ax_thm_2Eextreal_2Eextreal__max__def,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__max,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))) = surj__ty_2Eextreal_2Eextreal(ap(ap(ap(c_2Ebool_2ECOND(ty_2Eextreal_2Eextreal),ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))),inj__ty_2Eextreal_2Eextreal(V1y)),inj__ty_2Eextreal_2Eextreal(V0x))) ).
tff(conj_thm_2Eextreal_2Emin__le,axiom,
! [V0z: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal,V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))),inj__ty_2Eextreal_2Eextreal(V0z)))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0z)))
| p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V0z))) ) ) ).
tff(conj_thm_2Eextreal_2Emin__le1,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))),inj__ty_2Eextreal_2Eextreal(V0x))) ).
tff(conj_thm_2Eextreal_2Emin__le2,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))),inj__ty_2Eextreal_2Eextreal(V1y))) ).
tff(conj_thm_2Eextreal_2Ele__min,axiom,
! [V0z: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal,V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),inj__ty_2Eextreal_2Eextreal(V1x)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),inj__ty_2Eextreal_2Eextreal(V2y))) ) ) ).
tff(conj_thm_2Eextreal_2Emin__le2__imp,axiom,
! [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,inj__ty_2Eextreal_2Eextreal(V0x1)),inj__ty_2Eextreal_2Eextreal(V2y1)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x2)),inj__ty_2Eextreal_2Eextreal(V3y2))) )
=> p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x1)),inj__ty_2Eextreal_2Eextreal(V1x2))),ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V2y1)),inj__ty_2Eextreal_2Eextreal(V3y2)))) ) ).
tff(conj_thm_2Eextreal_2Emin__refl,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V0x))) = V0x ).
tff(conj_thm_2Eextreal_2Emin__comm,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y))) = surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V1y)),inj__ty_2Eextreal_2Eextreal(V0x))) ).
tff(conj_thm_2Eextreal_2Emin__infty,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal] :
( ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf))) = V0x )
& ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2EPosInf)),inj__ty_2Eextreal_2Eextreal(V0x))) = V0x )
& ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf)),inj__ty_2Eextreal_2Eextreal(V0x))) = fo__c_2Eextreal_2ENegInf )
& ( surj__ty_2Eextreal_2Eextreal(ap(ap(c_2Eextreal_2Eextreal__min,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(fo__c_2Eextreal_2ENegInf))) = fo__c_2Eextreal_2ENegInf ) ) ).
tff(conj_thm_2Eextreal_2Ele__max,axiom,
! [V0z: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal,V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),ap(ap(c_2Eextreal_2Eextreal__max,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),inj__ty_2Eextreal_2Eextreal(V1x)))
| p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0z)),inj__ty_2Eextreal_2Eextreal(V2y))) ) ) ).
tff(conj_thm_2Eextreal_2Ele__max1,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V0x)),ap(ap(c_2Eextreal_2Eextreal__max,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y)))) ).
tff(conj_thm_2Eextreal_2Ele__max2,axiom,
! [V0x: tp__ty_2Eextreal_2Eextreal,V1y: tp__ty_2Eextreal_2Eextreal] : p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1y)),ap(ap(c_2Eextreal_2Eextreal__max,inj__ty_2Eextreal_2Eextreal(V0x)),inj__ty_2Eextreal_2Eextreal(V1y)))) ).
tff(conj_thm_2Eextreal_2Emax__le,conjecture,
! [V0z: tp__ty_2Eextreal_2Eextreal,V1x: tp__ty_2Eextreal_2Eextreal,V2y: tp__ty_2Eextreal_2Eextreal] :
( p(ap(ap(c_2Eextreal_2Eextreal__le,ap(ap(c_2Eextreal_2Eextreal__max,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V2y))),inj__ty_2Eextreal_2Eextreal(V0z)))
<=> ( p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V1x)),inj__ty_2Eextreal_2Eextreal(V0z)))
& p(ap(ap(c_2Eextreal_2Eextreal__le,inj__ty_2Eextreal_2Eextreal(V2y)),inj__ty_2Eextreal_2Eextreal(V0z))) ) ) ).
%------------------------------------------------------------------------------