%------------------------------------------------------------------------------
% File     : ITP113+4 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : realax.ax [Gau19]
%          : HL4113+4.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  145 (  91 unt;   0 def)
%            Number of atoms       :  241 ( 109 equ)
%            Maximal formula atoms :    8 (   1 avg)
%            Number of connectives :  112 (  16   ~;   7   |;  25   &)
%                                         (  23 <=>;  41  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   4 avg)
%            Maximal term depth    :   14 (   2 avg)
%            Number of predicates  :    2 (   1 usr;   0 prp; 1-2 aty)
%            Number of functors    :   90 (  90 usr;  49 con; 0-3 aty)
%            Number of variables   :  304 ( 289   !;  15   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(reserved_2Eho_2Eboolext,axiom,
    ! [V0_2E0,V1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,V0_2E0))
      <=> p(s(tyop_2Emin_2Ebool,V1_2E0)) )
     => s(tyop_2Emin_2Ebool,V0_2E0) = s(tyop_2Emin_2Ebool,V1_2E0) ) ).

fof(reserved_2Eho_2Etruth,axiom,
    p(s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)) ).

fof(reserved_2Eho_2Enotfalse,axiom,
    ~ p(s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0)) ).

fof(reserved_2Eho_2Ebool__cases__ax,axiom,
    ! [V0t_2E0] :
      ( s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2ET_2E0)
      | s(tyop_2Emin_2Ebool,V0t_2E0) = s(tyop_2Emin_2Ebool,c_2Ebool_2EF_2E0) ) ).

fof(reserved_2Eho_2Ei__thm,axiom,
    ! [A_27a,V0x_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27a),combin_i_2E0),s(A_27a,V0x_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Ek__thm,axiom,
    ! [A_27a,A_27b,V0x_2E0,V1y_2E0] : s(A_27a,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27a),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27a)),combin_k_2E0),s(A_27a,V0x_2E0))),s(A_27b,V1y_2E0))) = s(A_27a,V0x_2E0) ).

fof(reserved_2Eho_2Es__thm,axiom,
    ! [A_27a,A_27b,A_27c,V0f_2E0,V1g_2E0,V2x_2E0] : s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27c),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,A_27b),tyop_2Emin_2Efun(A_27a,A_27c))),combin_s_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0))),s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0))),s(A_27a,V2x_2E0))) = s(A_27c,app_2E2(s(tyop_2Emin_2Efun(A_27b,A_27c),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27b,A_27c)),V0f_2E0),s(A_27a,V2x_2E0))),s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0))))) ).

fof(reserved_2Elogic_2E_2F_5C,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_2F_5C_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        & p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_5C_2F,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_5C_2F_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
        | p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_7E,axiom,
    ! [V0_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_7E_2E1(s(tyop_2Emin_2Ebool,V0_2E0))))
    <=> ~ p(s(tyop_2Emin_2Ebool,V0_2E0)) ) ).

fof(reserved_2Elogic_2E_3D_3D_3E,axiom,
    ! [V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_3D_3E_2E2(s(tyop_2Emin_2Ebool,V0_2E0),s(tyop_2Emin_2Ebool,V1_2E0))))
    <=> ( p(s(tyop_2Emin_2Ebool,V0_2E0))
       => p(s(tyop_2Emin_2Ebool,V1_2E0)) ) ) ).

fof(reserved_2Elogic_2E_3D,axiom,
    ! [A_27a,V0_2E0,V1_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Emin_2E_3D_2E2(s(A_27a,V0_2E0),s(A_27a,V1_2E0))))
    <=> s(A_27a,V0_2E0) = s(A_27a,V1_2E0) ) ).

fof(reserved_2Equant_2E_21,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

fof(reserved_2Equant_2E_3F,axiom,
    ! [A_27a,V0f_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0))))
    <=> ? [V1x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),V0f_2E0),s(A_27a,V1x_2E0)))) ) ).

fof(arityeq2_2Ec_2Ebool_2E_2F_5C_2E2,axiom,
    ! [X0_2E0,X1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
        & p(s(tyop_2Emin_2Ebool,X1_2E0)) )
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_2F_5C_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).

fof(arityeq2_2Ec_2Ebool_2E_5C_2F_2E2,axiom,
    ! [X0_2E0,X1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
        | p(s(tyop_2Emin_2Ebool,X1_2E0)) )
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Ebool_2E_5C_2F_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).

fof(arityeq1_2Ec_2Ebool_2E_7E_2E1,axiom,
    ! [X0_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,X0_2E0))
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),c_2Ebool_2E_7E_2E0),s(tyop_2Emin_2Ebool,X0_2E0)))) ) ).

fof(arityeq2_2Ec_2Emin_2E_3D_3D_3E_2E2,axiom,
    ! [X0_2E0,X1_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,X0_2E0))
       => p(s(tyop_2Emin_2Ebool,X1_2E0)) )
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_3D_3E_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Emin_2Ebool,X1_2E0)))) ) ).

fof(arityeq2_2Ec_2Emin_2E_3D_2E2_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0,X1_2E0] :
      ( s(A_27a,X0_2E0) = s(A_27a,X1_2E0)
    <=> p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool)),c_2Emin_2E_3D_2E0),s(A_27a,X0_2E0))),s(A_27a,X1_2E0)))) ) ).

fof(arityeq1_2Ec_2Ebool_2E_21_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_21_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_21_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Ebool_2E_3F_2E1_2Emono_2EA_27a,axiom,
    ! [A_27a,X0_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2E_3F_2E1(s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),c_2Ebool_2E_3F_2E0),s(tyop_2Emin_2Efun(A_27a,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Ehreal_2Ehreal,axiom,
    ! [X0_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),tyop_2Ehreal_2Ehreal),c_2Emin_2E_40_2E0),s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Emin_2E_40_2E1_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29,axiom,
    ! [X0_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Emin_2E_40_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__REP__CLASS_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),c_2Erealax_2Ereal__REP__CLASS_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ehreal__of__real_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__real_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ehreal__of__real_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ehreal__of__treal_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Ehreal_2Ehreal),c_2Erealax_2Ehreal__of__treal_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__REP_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Ereal__REP_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Etreal__of__hreal_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,X0_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Etreal__of__hreal_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))) ).

fof(arityeq2_2Ec_2Epair_2E_2C_2E2_2Emono_2Etyop_2Ehreal_2Ehreal_20mono_2Etyop_2Ehreal_2Ehreal,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,X0_2E0),s(tyop_2Ehreal_2Ehreal,X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))),c_2Epair_2E_2C_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))),s(tyop_2Ehreal_2Ehreal,X1_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Einv_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Einv_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__ABS_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__ABS_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Etreal__inv_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Etreal__inv_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Etreal__neg_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Etreal__neg_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Etreal__add_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))),c_2Erealax_2Etreal__add_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Etreal__mul_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))),c_2Erealax_2Etreal__mul_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__ABS__CLASS_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__ABS__CLASS_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Etreal__eq_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),c_2Erealax_2Etreal__eq_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__neg_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__neg_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))) ).

fof(arityeq1_2Ec_2Erealax_2Ereal__of__hreal_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,X0_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__of__hreal_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))) ).

fof(arityeq2_2Ec_2Ehreal_2Ehreal__add_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,X0_2E0),s(tyop_2Ehreal_2Ehreal,X1_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Ehreal_2Ehreal__add_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))),s(tyop_2Ehreal_2Ehreal,X1_2E0))) ).

fof(arityeq2_2Ec_2Ehreal_2Ehreal__mul_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,X0_2E0),s(tyop_2Ehreal_2Ehreal,X1_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Ehreal_2Ehreal__mul_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))),s(tyop_2Ehreal_2Ehreal,X1_2E0))) ).

fof(arityeq2_2Ec_2Ebool_2ETYPE__DEFINITION_2E2_2Emono_2Etyop_2Emin_2Efun_28tyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29_2Ctyop_2Emin_2Ebool_29_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Ebool)),c_2Ebool_2ETYPE__DEFINITION_2E0),s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),X1_2E0))) ).

fof(arityeq2_2Ec_2Ehreal_2Ehreal__lt_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,X0_2E0),s(tyop_2Ehreal_2Ehreal,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),c_2Ehreal_2Ehreal__lt_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))),s(tyop_2Ehreal_2Ehreal,X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Ereal__add_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Erealax_2Ereal__add_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Ereal__mul_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Erealax_2Ereal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Erealax_2Ereal)),c_2Erealax_2Ereal__mul_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Ereal__lt_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,X0_2E0),s(tyop_2Erealax_2Ereal,X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool)),c_2Erealax_2Ereal__lt_2E0),s(tyop_2Erealax_2Ereal,X0_2E0))),s(tyop_2Erealax_2Ereal,X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Etreal__eq_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),c_2Erealax_2Etreal__eq_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) ).

fof(arityeq2_2Ec_2Erealax_2Etreal__lt_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),c_2Erealax_2Etreal__lt_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))) ).

fof(arityeq3_2Ec_2Ebool_2ECOND_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29,axiom,
    ! [X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,X0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X2_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal))),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Ebool,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)))),c_2Ebool_2ECOND_2E0),s(tyop_2Emin_2Ebool,X0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),X2_2E0))) ).

fof(arityeq1_2Ec_2Ehreal_2Ehreal__inv_2E1,axiom,
    ! [X0_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__inv_2E1(s(tyop_2Ehreal_2Ehreal,X0_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ehreal_2Ehreal__inv_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))) ).

fof(arityeq2_2Ec_2Ehreal_2Ehreal__sub_2E2,axiom,
    ! [X0_2E0,X1_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__sub_2E2(s(tyop_2Ehreal_2Ehreal,X0_2E0),s(tyop_2Ehreal_2Ehreal,X1_2E0))) = s(tyop_2Ehreal_2Ehreal,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Ehreal_2Ehreal__sub_2E0),s(tyop_2Ehreal_2Ehreal,X0_2E0))),s(tyop_2Ehreal_2Ehreal,X1_2E0))) ).

fof(arityeq3_2Ec_2Equotient_2EQUOTIENT_2E3_2Emono_2Etyop_2Epair_2Eprod_28tyop_2Ehreal_2Ehreal_2Ctyop_2Ehreal_2Ehreal_29_20mono_2Etyop_2Erealax_2Ereal,axiom,
    ! [X0_2E0,X1_2E0,X2_2E0] : s(tyop_2Emin_2Ebool,c_2Equotient_2EQUOTIENT_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),X0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),X1_2E0),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),X2_2E0))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),tyop_2Emin_2Ebool))),c_2Equotient_2EQUOTIENT_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),X0_2E0))),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),X1_2E0))),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),X2_2E0))) ).

fof(thm_2Erealax_2Etreal__0,axiom,
    s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))) ).

fof(thm_2Erealax_2Etreal__1,axiom,
    s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__1_2E0) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))) ).

fof(thm_2Erealax_2Etreal__neg,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))) ).

fof(thm_2Erealax_2Etreal__add,axiom,
    ! [V0x1_2E0,V1y1_2E0,V2x2_2E0,V3y2_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V2x2_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V1y1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))) ).

fof(thm_2Erealax_2Etreal__mul,axiom,
    ! [V0x1_2E0,V1y1_2E0,V2x2_2E0,V3y2_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V2x2_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V1y1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V1y1_2E0),s(tyop_2Ehreal_2Ehreal,V2x2_2E0))))))) ).

fof(thm_2Erealax_2Etreal__lt,axiom,
    ! [V0x1_2E0,V1y1_2E0,V2x2_2E0,V3y2_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))))) ).

fof(thm_2Erealax_2Etreal__inv,axiom,
    ! [F0_2E0] :
      ( ! [V0x_2E0,V1y_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),F0_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))
        <=> s(tyop_2Ehreal_2Ehreal,V0x_2E0) = s(tyop_2Ehreal_2Ehreal,V1y_2E0) )
     => ! [V0x_2E0,V1y_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),F0_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))),s(tyop_2Ehreal_2Ehreal,V1y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Ebool_2ECOND_2E3(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__inv_2E1(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__sub_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__inv_2E1(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__sub_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))))))))) ) ).

fof(thm_2Erealax_2Etreal__eq,axiom,
    ! [V0x1_2E0,V1y1_2E0,V2x2_2E0,V3y2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))))))
    <=> s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0))) = s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V2x2_2E0),s(tyop_2Ehreal_2Ehreal,V1y1_2E0))) ) ).

fof(thm_2Erealax_2Etreal__of__hreal,axiom,
    ! [V0x_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0x_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__1_2E0))) ).

fof(thm_2Erealax_2Ehreal__of__treal,axiom,
    ! [F0_2E0] :
      ( ! [V0x_2E0,V1y_2E0,V2d_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))),s(tyop_2Ehreal_2Ehreal,V1y_2E0))),s(tyop_2Ehreal_2Ehreal,V2d_2E0))))
        <=> s(tyop_2Ehreal_2Ehreal,V0x_2E0) = s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V2d_2E0))) )
     => ! [V0x_2E0,V1y_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Epair_2E_2C_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))) = s(tyop_2Ehreal_2Ehreal,c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool)),app_2E2(s(tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Efun(tyop_2Ehreal_2Ehreal,tyop_2Emin_2Ebool))),F0_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0))),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))) ) ).

fof(thm_2Erealax_2Ereal__TY__DEF,axiom,
    ! [F0_2E0] :
      ( ! [V1c_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1c_2E0))))
        <=> ? [V2r_2E0] :
              ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))))
              & s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1c_2E0) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))) ) )
     => ? [V0rep_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ebool_2ETYPE__DEFINITION_2E2(s(tyop_2Emin_2Efun(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),tyop_2Emin_2Ebool),F0_2E0),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),V0rep_2E0)))) ) ).

fof(thm_2Erealax_2Ereal__bijections,axiom,
    ( ! [V0a_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,V0a_2E0))))) = s(tyop_2Erealax_2Ereal,V0a_2E0)
    & ! [V1r_2E0] :
        ( ? [V2r_2E0] :
            ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))))
            & s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1r_2E0) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))) )
      <=> s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1r_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1r_2E0) ) ) ).

fof(thm_2Erealax_2Ereal__REP__def,axiom,
    ! [V0a_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0a_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Emin_2E_40_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,V0a_2E0))))) ).

fof(thm_2Erealax_2Ereal__ABS__def,axiom,
    ! [V0r_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0r_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0r_2E0))))) ).

fof(thm_2Erealax_2Ereal__0,axiom,
    s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0))) ).

fof(thm_2Erealax_2Ereal__1,axiom,
    s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__1_2E0) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__1_2E0))) ).

fof(thm_2Erealax_2Ereal__neg,axiom,
    ! [V0T1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))))))) ).

fof(thm_2Erealax_2Ereal__inv,axiom,
    ! [V0T1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))))))) ).

fof(thm_2Erealax_2Ereal__add,axiom,
    ! [V0T1_2E0,V1T2_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0T1_2E0),s(tyop_2Erealax_2Ereal,V1T2_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V1T2_2E0))))))) ).

fof(thm_2Erealax_2Ereal__mul,axiom,
    ! [V0T1_2E0,V1T2_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0T1_2E0),s(tyop_2Erealax_2Ereal,V1T2_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V1T2_2E0))))))) ).

fof(thm_2Erealax_2Ereal__lt,axiom,
    ! [V0T1_2E0,V1T2_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0T1_2E0),s(tyop_2Erealax_2Ereal,V1T2_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V1T2_2E0))))) ).

fof(thm_2Erealax_2Ereal__of__hreal,axiom,
    ! [V0T1_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0T1_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0T1_2E0))))) ).

fof(thm_2Erealax_2Ehreal__of__real,axiom,
    ! [V0T1_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__real_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))) = s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Ereal__REP_2E1(s(tyop_2Erealax_2Ereal,V0T1_2E0))))) ).

fof(thm_2Erealax_2EHREAL__RDISTRIB,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))),s(tyop_2Ehreal_2Ehreal,V2z_2E0))) = s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V2z_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__mul_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V2z_2E0))))) ).

fof(thm_2Erealax_2EHREAL__EQ__ADDR,axiom,
    ! [V0X_2E0,V1Y_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0X_2E0),s(tyop_2Ehreal_2Ehreal,V1Y_2E0))) != s(tyop_2Ehreal_2Ehreal,V0X_2E0) ).

fof(thm_2Erealax_2EHREAL__EQ__ADDL,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Ehreal_2Ehreal,V0x_2E0) != s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))) ).

fof(thm_2Erealax_2EHREAL__EQ__LADD,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))) = s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V2z_2E0)))
    <=> s(tyop_2Ehreal_2Ehreal,V1y_2E0) = s(tyop_2Ehreal_2Ehreal,V2z_2E0) ) ).

fof(thm_2Erealax_2EHREAL__LT__REFL,axiom,
    ! [V0x_2E0] : ~ p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0)))) ).

fof(thm_2Erealax_2EHREAL__LT__ADDL,axiom,
    ! [V0x_2E0,V1y_2E0] : p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0)))))) ).

fof(thm_2Erealax_2EHREAL__LT__NE,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))
     => s(tyop_2Ehreal_2Ehreal,V0x_2E0) != s(tyop_2Ehreal_2Ehreal,V1y_2E0) ) ).

fof(thm_2Erealax_2EHREAL__LT__ADDR,axiom,
    ! [V0x_2E0,V1y_2E0] : ~ p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))),s(tyop_2Ehreal_2Ehreal,V0x_2E0)))) ).

fof(thm_2Erealax_2EHREAL__LT__GT,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))))
     => ~ p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V0x_2E0)))) ) ).

fof(thm_2Erealax_2EHREAL__LT__ADD2,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y1_2E0,V3y2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V2y1_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V1x2_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x1_2E0),s(tyop_2Ehreal_2Ehreal,V1x2_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V2y1_2E0),s(tyop_2Ehreal_2Ehreal,V3y2_2E0)))))) ) ).

fof(thm_2Erealax_2EHREAL__LT__LADD,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V1y_2E0))),s(tyop_2Ehreal_2Ehreal,c_2Ehreal_2Ehreal__add_2E2(s(tyop_2Ehreal_2Ehreal,V0x_2E0),s(tyop_2Ehreal_2Ehreal,V2z_2E0))))) = s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V1y_2E0),s(tyop_2Ehreal_2Ehreal,V2z_2E0))) ).

fof(thm_2Erealax_2ETREAL__EQ__REFL,axiom,
    ! [V0x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0)))) ).

fof(thm_2Erealax_2ETREAL__EQ__SYM,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))) ).

fof(thm_2Erealax_2ETREAL__EQ__TRANS,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0)))) ) ).

fof(thm_2Erealax_2ETREAL__EQ__EQUIV,axiom,
    ! [V0p_2E0,V1q_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0p_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0))))
    <=> s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0p_2E0))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__EQ__AP,axiom,
    ! [V0p_2E0,V1q_2E0] :
      ( s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0p_2E0) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0)
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0p_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1q_2E0)))) ) ).

fof(thm_2Erealax_2ETREAL__10,axiom,
    ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0)))) ).

fof(thm_2Erealax_2ETREAL__ADD__SYM,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))) ).

fof(thm_2Erealax_2ETREAL__MUL__SYM,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))) ).

fof(thm_2Erealax_2ETREAL__ADD__ASSOC,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))) ).

fof(thm_2Erealax_2ETREAL__MUL__ASSOC,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))) ).

fof(thm_2Erealax_2ETREAL__LDISTRIB,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))))) = s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))))) ).

fof(thm_2Erealax_2ETREAL__ADD__LID,axiom,
    ! [V0x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0)))) ).

fof(thm_2Erealax_2ETREAL__MUL__LID,axiom,
    ! [V0x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0)))) ).

fof(thm_2Erealax_2ETREAL__ADD__LINV,axiom,
    ! [V0x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0)))) ).

fof(thm_2Erealax_2ETREAL__INV__0,axiom,
    p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0)))) ).

fof(thm_2Erealax_2ETREAL__MUL__LINV,axiom,
    ! [V0x_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__1_2E0)))) ) ).

fof(thm_2Erealax_2ETREAL__LT__TOTAL,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))))
      | p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))))
      | p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0)))) ) ).

fof(thm_2Erealax_2ETREAL__LT__REFL,axiom,
    ! [V0x_2E0] : ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0)))) ).

fof(thm_2Erealax_2ETREAL__LT__TRANS,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0)))) ) ).

fof(thm_2Erealax_2ETREAL__LT__ADD,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2z_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__LT__MUL,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__BIJ,axiom,
    ( ! [V0h_2E0] : s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0h_2E0))))) = s(tyop_2Ehreal_2Ehreal,V0h_2E0)
    & ! [V1r_2E0] : s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__0_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1r_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1r_2E0))))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1r_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__ISO,axiom,
    ! [V0h_2E0,V1i_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0h_2E0),s(tyop_2Ehreal_2Ehreal,V1i_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0h_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V1i_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__BIJ__WELLDEF,axiom,
    ! [V0h_2E0,V1i_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0h_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1i_2E0))))
     => s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0h_2E0))) = s(tyop_2Ehreal_2Ehreal,c_2Erealax_2Ehreal__of__treal_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1i_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__NEG__WELLDEF,axiom,
    ! [V0x1_2E0,V1x2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__neg_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__ADD__WELLDEFR,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__ADD__WELLDEF,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y1_2E0,V3y2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__add_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__MUL__WELLDEFR,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__MUL__WELLDEF,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y1_2E0,V3y2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__mul_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0)))))) ) ).

fof(thm_2Erealax_2ETREAL__LT__WELLDEFR,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
     => s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__LT__WELLDEFL,axiom,
    ! [V0x_2E0,V1y1_2E0,V2y2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y2_2E0))))
     => s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1y1_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y2_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__LT__WELLDEF,axiom,
    ! [V0x1_2E0,V1x2_2E0,V2y1_2E0,V3y2_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0)))) )
     => s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2y1_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__lt_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V3y2_2E0))) ) ).

fof(thm_2Erealax_2ETREAL__INV__WELLDEF,axiom,
    ! [V0x1_2E0,V1x2_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V0x1_2E0))),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),c_2Erealax_2Etreal__inv_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V1x2_2E0)))))) ) ).

fof(thm_2Erealax_2Ereal__ABS__REP__CLASS,axiom,
    ( ! [V0a_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,V0a_2E0))))) = s(tyop_2Erealax_2Ereal,V0a_2E0)
    & ! [V1c_2E0] :
        ( ? [V2r_2E0] :
            ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Etreal__eq_2E2(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0),s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))))
            & s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1c_2E0) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Etreal__eq_2E1(s(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),V2r_2E0))) )
      <=> s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),c_2Erealax_2Ereal__REP__CLASS_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__ABS__CLASS_2E1(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1c_2E0))))) = s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool),V1c_2E0) ) ) ).

fof(thm_2Erealax_2Ereal__QUOTIENT,axiom,
    p(s(tyop_2Emin_2Ebool,c_2Equotient_2EQUOTIENT_2E3(s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Emin_2Ebool)),c_2Erealax_2Etreal__eq_2E0),s(tyop_2Emin_2Efun(tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal),tyop_2Erealax_2Ereal),c_2Erealax_2Ereal__ABS_2E0),s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Epair_2Eprod(tyop_2Ehreal_2Ehreal,tyop_2Ehreal_2Ehreal)),c_2Erealax_2Ereal__REP_2E0)))) ).

fof(thm_2Erealax_2EREAL__ISO__EQ,axiom,
    ! [V0h_2E0,V1i_2E0] : s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V0h_2E0),s(tyop_2Ehreal_2Ehreal,V1i_2E0))) = s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0h_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V1i_2E0))))) ).

fof(thm_2Erealax_2EREAL__POS,axiom,
    ! [V0X_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V0X_2E0)))))) ).

fof(thm_2Erealax_2ESUP__ALLPOS__LEMMA1,axiom,
    ! [V0P_2E0,V1y_2E0] :
      ( ! [V2x_2E0] :
          ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V2x_2E0))))
         => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V2x_2E0)))) )
     => ( ? [V3x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V3x_2E0))))
            & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V3x_2E0)))) )
      <=> ? [V4X_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V4X_2E0))))))
            & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V4X_2E0)))))) ) ) ) ).

fof(thm_2Erealax_2ESUP__ALLPOS__LEMMA2,axiom,
    ! [V0P_2E0,V1X_2E0] : s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V1X_2E0))))) = s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V1X_2E0))))) ).

fof(thm_2Erealax_2ESUP__ALLPOS__LEMMA3,axiom,
    ! [V0P_2E0] :
      ( ( ! [V1x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0))))
           => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0)))) )
        & ? [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V2x_2E0))))
        & ? [V3z_2E0] :
          ! [V4x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V4x_2E0))))
           => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V4x_2E0),s(tyop_2Erealax_2Ereal,V3z_2E0)))) ) )
     => ( ? [V5X_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V5X_2E0))))))
        & ? [V6Y_2E0] :
          ! [V7X_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V7X_2E0))))))
           => p(s(tyop_2Emin_2Ebool,c_2Ehreal_2Ehreal__lt_2E2(s(tyop_2Ehreal_2Ehreal,V7X_2E0),s(tyop_2Ehreal_2Ehreal,V6Y_2E0)))) ) ) ) ).

fof(thm_2Erealax_2ESUP__ALLPOS__LEMMA4,axiom,
    ! [V0y_2E0] :
      ( ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V0y_2E0))))
     => ! [V1x_2E0] : p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0y_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__of__hreal_2E1(s(tyop_2Ehreal_2Ehreal,V1x_2E0)))))) ) ).

fof(thm_2Erealax_2EREAL__SUP__ALLPOS,axiom,
    ! [V0P_2E0] :
      ( ( ! [V1x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0))))
           => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V1x_2E0)))) )
        & ? [V2x_2E0] : p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V2x_2E0))))
        & ? [V3z_2E0] :
          ! [V4x_2E0] :
            ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V4x_2E0))))
           => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V4x_2E0),s(tyop_2Erealax_2Ereal,V3z_2E0)))) ) )
     => ? [V5s_2E0] :
        ! [V6y_2E0] :
          ( ? [V7x_2E0] :
              ( p(s(tyop_2Emin_2Ebool,app_2E2(s(tyop_2Emin_2Efun(tyop_2Erealax_2Ereal,tyop_2Emin_2Ebool),V0P_2E0),s(tyop_2Erealax_2Ereal,V7x_2E0))))
              & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V6y_2E0),s(tyop_2Erealax_2Ereal,V7x_2E0)))) )
        <=> p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V6y_2E0),s(tyop_2Erealax_2Ereal,V5s_2E0)))) ) ) ).

fof(thm_2Erealax_2EREAL__10,axiom,
    s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__1_2E0) != s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0) ).

fof(thm_2Erealax_2EREAL__ADD__SYM,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))) ).

fof(thm_2Erealax_2EREAL__MUL__SYM,axiom,
    ! [V0x_2E0,V1y_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))) ).

fof(thm_2Erealax_2EREAL__ADD__ASSOC,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))) ).

fof(thm_2Erealax_2EREAL__MUL__ASSOC,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,V2z_2E0))) ).

fof(thm_2Erealax_2EREAL__LDISTRIB,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))) ).

fof(thm_2Erealax_2EREAL__ADD__LID,axiom,
    ! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ).

fof(thm_2Erealax_2EREAL__MUL__LID,axiom,
    ! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__1_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,V0x_2E0) ).

fof(thm_2Erealax_2EREAL__ADD__LINV,axiom,
    ! [V0x_2E0] : s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__neg_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0) ).

fof(thm_2Erealax_2EREAL__MUL__LINV,axiom,
    ! [V0x_2E0] :
      ( s(tyop_2Erealax_2Ereal,V0x_2E0) != s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0)
     => s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E1(s(tyop_2Erealax_2Ereal,V0x_2E0))),s(tyop_2Erealax_2Ereal,V0x_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__1_2E0) ) ).

fof(thm_2Erealax_2EREAL__LT__TOTAL,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( s(tyop_2Erealax_2Ereal,V0x_2E0) = s(tyop_2Erealax_2Ereal,V1y_2E0)
      | p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
      | p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0)))) ) ).

fof(thm_2Erealax_2EREAL__LT__REFL,axiom,
    ! [V0x_2E0] : ~ p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0)))) ).

fof(thm_2Erealax_2EREAL__LT__TRANS,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))) ) ).

fof(thm_2Erealax_2EREAL__LT__IADD,axiom,
    ! [V0x_2E0,V1y_2E0,V2z_2E0] :
      ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,V1y_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0))))
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0))),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__add_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V2z_2E0)))))) ) ).

fof(thm_2Erealax_2EREAL__LT__MUL,axiom,
    ! [V0x_2E0,V1y_2E0] :
      ( ( p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V0x_2E0))))
        & p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0)))) )
     => p(s(tyop_2Emin_2Ebool,c_2Erealax_2Ereal__lt_2E2(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0),s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__mul_2E2(s(tyop_2Erealax_2Ereal,V0x_2E0),s(tyop_2Erealax_2Ereal,V1y_2E0)))))) ) ).

fof(thm_2Erealax_2EREAL__INV__0,axiom,
    s(tyop_2Erealax_2Ereal,c_2Erealax_2Einv_2E1(s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0))) = s(tyop_2Erealax_2Ereal,c_2Erealax_2Ereal__0_2E0) ).

fof(reserved_2Eho_2Eeq__ext,axiom,
    ! [A_27a,A_27b,V0f_2E0,V1g_2E0] :
      ( ! [V2x_2E0] : s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0),s(A_27a,V2x_2E0))) = s(A_27b,app_2E2(s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0),s(A_27a,V2x_2E0)))
     => s(tyop_2Emin_2Efun(A_27a,A_27b),V0f_2E0) = s(tyop_2Emin_2Efun(A_27a,A_27b),V1g_2E0) ) ).

%------------------------------------------------------------------------------