TPTP Problem File: SLH0887^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Median_Method/0000_Median/prob_00150_005468__14678138_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1349 ( 506 unt; 76 typ; 0 def)
% Number of atoms : 3901 (1151 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 10743 ( 386 ~; 138 |; 195 &;8227 @)
% ( 0 <=>;1797 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 393 ( 393 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 13 con; 0-4 aty)
% Number of variables : 3500 ( 164 ^;3193 !; 143 ?;3500 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:43:44.906
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
formal3361831859752904756s_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Set__Oset_Itf__b_J,type,
set_b: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (67)
thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
archim7802044766580827645g_real: real > int ).
thf(sy_c_Formal__Power__Series_Ofps__tan_001t__Real__Oreal,type,
formal3683295897622742886n_real: real > formal3361831859752904756s_real ).
thf(sy_c_Formal__Power__Series_Oradical_001t__Real__Oreal,type,
formal8005797870169972230l_real: ( nat > real > real ) > nat > formal3361831859752904756s_real > nat > real ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
times_times_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Formal____Power____Series__Ofps_It__Real__Oreal_J,type,
zero_z7760665558314615101s_real: formal3361831859752904756s_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Harmonic__Numbers_Oharm_001t__Real__Oreal,type,
harmonic_harm_real: nat > real ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
ring_1_of_int_real: int > real ).
thf(sy_c_Median_Odown__ray_001t__Int__Oint,type,
down_ray_int: set_int > $o ).
thf(sy_c_Median_Odown__ray_001t__Nat__Onat,type,
down_ray_nat: set_nat > $o ).
thf(sy_c_Median_Odown__ray_001t__Real__Oreal,type,
down_ray_real: set_real > $o ).
thf(sy_c_Median_Odown__ray_001tf__b,type,
down_ray_b: set_b > $o ).
thf(sy_c_Median_Ointerval_001t__Int__Oint,type,
interval_int: set_int > $o ).
thf(sy_c_Median_Ointerval_001t__Nat__Onat,type,
interval_nat: set_nat > $o ).
thf(sy_c_Median_Ointerval_001t__Real__Oreal,type,
interval_real: set_real > $o ).
thf(sy_c_Median_Ointerval_001tf__b,type,
interval_b: set_b > $o ).
thf(sy_c_Median_Oup__ray_001t__Int__Oint,type,
up_ray_int: set_int > $o ).
thf(sy_c_Median_Oup__ray_001t__Nat__Onat,type,
up_ray_nat: set_nat > $o ).
thf(sy_c_Median_Oup__ray_001t__Real__Oreal,type,
up_ray_real: set_real > $o ).
thf(sy_c_Median_Oup__ray_001tf__b,type,
up_ray_b: set_b > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
semiri5074537144036343181t_real: nat > real ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001tf__b,type,
ord_less_b: b > b > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001tf__b,type,
ord_less_eq_b: b > b > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Int__Oint,type,
order_Greatest_int: ( int > $o ) > int ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Nat__Onat,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Real__Oreal,type,
order_Greatest_real: ( real > $o ) > real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001tf__b,type,
order_Greatest_b: ( b > $o ) > b ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
divide_divide_real: real > real > real ).
thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
collect_real: ( real > $o ) > set_real ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_c_member_001tf__b,type,
member_b: b > set_b > $o ).
thf(sy_v_g____,type,
g: nat > nat > b ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_ja____,type,
ja: nat ).
thf(sy_v_k____,type,
k: nat ).
thf(sy_v_na____,type,
na: nat ).
% Relevant facts (1265)
thf(fact_0_Suc_Oprems_I2_J,axiom,
ord_less_nat @ ia @ ja ).
% Suc.prems(2)
thf(fact_1_k__def,axiom,
k = na ).
% k_def
thf(fact_2_Suc_Oprems_I1_J,axiom,
ord_less_nat @ ja @ ( suc @ na ) ).
% Suc.prems(1)
thf(fact_3_order__refl,axiom,
! [X: b] : ( ord_less_eq_b @ X @ X ) ).
% order_refl
thf(fact_4_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_5_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_6_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_7_dual__order_Orefl,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% dual_order.refl
thf(fact_8_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_9_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_11_minf_I8_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ~ ( ord_less_eq_b @ T @ X2 ) ) ).
% minf(8)
thf(fact_12_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% minf(8)
thf(fact_13_minf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% minf(8)
thf(fact_14_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_15_minf_I6_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ord_less_eq_b @ X2 @ T ) ) ).
% minf(6)
thf(fact_16_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_eq_nat @ X2 @ T ) ) ).
% minf(6)
thf(fact_17_minf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_eq_real @ X2 @ T ) ) ).
% minf(6)
thf(fact_18_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_19_pinf_I8_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ord_less_eq_b @ T @ X2 ) ) ).
% pinf(8)
thf(fact_20_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_eq_nat @ T @ X2 ) ) ).
% pinf(8)
thf(fact_21_pinf_I8_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ T @ X2 ) ) ).
% pinf(8)
thf(fact_22_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_23_pinf_I6_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ~ ( ord_less_eq_b @ X2 @ T ) ) ).
% pinf(6)
thf(fact_24_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% pinf(6)
thf(fact_25_pinf_I6_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% pinf(6)
thf(fact_26_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_27_verit__comp__simplify1_I3_J,axiom,
! [B: b,A2: b] :
( ( ~ ( ord_less_eq_b @ B @ A2 ) )
= ( ord_less_b @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_28_verit__comp__simplify1_I3_J,axiom,
! [B: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B @ A2 ) )
= ( ord_less_nat @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_29_verit__comp__simplify1_I3_J,axiom,
! [B: real,A2: real] :
( ( ~ ( ord_less_eq_real @ B @ A2 ) )
= ( ord_less_real @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_30_verit__comp__simplify1_I3_J,axiom,
! [B: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B @ A2 ) )
= ( ord_less_int @ A2 @ B ) ) ).
% verit_comp_simplify1(3)
thf(fact_31_leD,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ~ ( ord_less_b @ X @ Y ) ) ).
% leD
thf(fact_32_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_33_leD,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_real @ X @ Y ) ) ).
% leD
thf(fact_34_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_35_leI,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ord_less_eq_b @ Y @ X ) ) ).
% leI
thf(fact_36_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_37_leI,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% leI
thf(fact_38_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_39_nless__le,axiom,
! [A: b,B2: b] :
( ( ~ ( ord_less_b @ A @ B2 ) )
= ( ~ ( ord_less_eq_b @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_40_nless__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_41_nless__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_real @ A @ B2 ) )
= ( ~ ( ord_less_eq_real @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_42_nless__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_int @ A @ B2 ) )
= ( ~ ( ord_less_eq_int @ A @ B2 )
| ( A = B2 ) ) ) ).
% nless_le
thf(fact_43_antisym__conv1,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ( ord_less_eq_b @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_44_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_45_antisym__conv1,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_46_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_47_order__antisym__conv,axiom,
! [Y: b,X: b] :
( ( ord_less_eq_b @ Y @ X )
=> ( ( ord_less_eq_b @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_48_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_49_order__antisym__conv,axiom,
! [Y: real,X: real] :
( ( ord_less_eq_real @ Y @ X )
=> ( ( ord_less_eq_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_50_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_51_linorder__le__cases,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_eq_b @ X @ Y )
=> ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_52_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_53_linorder__le__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_54_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_55_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_56_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_62_ord__le__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_63_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_64_ord__le__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_65_ord__eq__le__subst,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_66_ord__eq__le__subst,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_ord__eq__le__subst,axiom,
! [A: int,F: b > int,B2: b,C: b] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_69_ord__eq__le__subst,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_70_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_71_ord__eq__le__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_72_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_73_ord__eq__le__subst,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_74_ord__eq__le__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_75_linorder__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
| ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_linear
thf(fact_76_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_77_linorder__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_78_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_79_verit__la__disequality,axiom,
! [A: b,B2: b] :
( ( A = B2 )
| ~ ( ord_less_eq_b @ A @ B2 )
| ~ ( ord_less_eq_b @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_80_verit__la__disequality,axiom,
! [A: nat,B2: nat] :
( ( A = B2 )
| ~ ( ord_less_eq_nat @ A @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_81_verit__la__disequality,axiom,
! [A: real,B2: real] :
( ( A = B2 )
| ~ ( ord_less_eq_real @ A @ B2 )
| ~ ( ord_less_eq_real @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_82_verit__la__disequality,axiom,
! [A: int,B2: int] :
( ( A = B2 )
| ~ ( ord_less_eq_int @ A @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A ) ) ).
% verit_la_disequality
thf(fact_83_order__eq__refl,axiom,
! [X: b,Y: b] :
( ( X = Y )
=> ( ord_less_eq_b @ X @ Y ) ) ).
% order_eq_refl
thf(fact_84_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_85_order__eq__refl,axiom,
! [X: real,Y: real] :
( ( X = Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_86_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_87_order__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_88_order__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_89_order__subst2,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_90_order__subst2,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_91_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_92_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_93_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_94_order__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_95_order__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_96_order__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_97_order__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_98_order__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_99_order__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_100_order__subst1,axiom,
! [A: b,F: int > b,B2: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_b @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_101_order__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_102_order__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_103_order__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_104_order__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_105_order__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_106_order__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_107_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_108_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_109_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_110_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_111_antisym,axiom,
! [A: b,B2: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_112_antisym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_113_antisym,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_114_antisym,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A )
=> ( A = B2 ) ) ) ).
% antisym
thf(fact_115_dual__order_Otrans,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_eq_b @ C @ B2 )
=> ( ord_less_eq_b @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_116_dual__order_Otrans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_117_dual__order_Otrans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_eq_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_118_dual__order_Otrans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_119_dual__order_Oantisym,axiom,
! [B2: b,A: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_eq_b @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_120_dual__order_Oantisym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_121_dual__order_Oantisym,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_122_dual__order_Oantisym,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( A = B2 ) ) ) ).
% dual_order.antisym
thf(fact_123_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_124_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_125_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_126_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_127_linorder__wlog,axiom,
! [P: b > b > $o,A: b,B2: b] :
( ! [A4: b,B4: b] :
( ( ord_less_eq_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_128_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_129_linorder__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_eq_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_130_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ).
% linorder_wlog
thf(fact_131_order__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z3 )
=> ( ord_less_eq_b @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_132_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_133_order__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_eq_real @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_134_order__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_eq_int @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_135_order_Otrans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% order.trans
thf(fact_136_order_Otrans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_137_order_Otrans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% order.trans
thf(fact_138_order_Otrans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_139_order__antisym,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_140_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_141_order__antisym,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_142_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_143_ord__le__eq__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_144_ord__le__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_145_ord__le__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_146_ord__le__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_147_ord__eq__le__trans,axiom,
! [A: b,B2: b,C: b] :
( ( A = B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_eq_b @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_148_ord__eq__le__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_149_ord__eq__le__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_150_ord__eq__le__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_151_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: b,Z2: b] : ( Y3 = Z2 ) )
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ( ord_less_eq_b @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_152_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_153_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: real,Z2: real] : ( Y3 = Z2 ) )
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_154_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z2: int] : ( Y3 = Z2 ) )
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_155_le__cases3,axiom,
! [X: b,Y: b,Z3: b] :
( ( ( ord_less_eq_b @ X @ Y )
=> ~ ( ord_less_eq_b @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_b @ Y @ X )
=> ~ ( ord_less_eq_b @ X @ Z3 ) )
=> ( ( ( ord_less_eq_b @ X @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_b @ Z3 @ Y )
=> ~ ( ord_less_eq_b @ Y @ X ) )
=> ( ( ( ord_less_eq_b @ Y @ Z3 )
=> ~ ( ord_less_eq_b @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_b @ Z3 @ X )
=> ~ ( ord_less_eq_b @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_156_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_157_le__cases3,axiom,
! [X: real,Y: real,Z3: real] :
( ( ( ord_less_eq_real @ X @ Y )
=> ~ ( ord_less_eq_real @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_real @ Y @ X )
=> ~ ( ord_less_eq_real @ X @ Z3 ) )
=> ( ( ( ord_less_eq_real @ X @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_real @ Z3 @ Y )
=> ~ ( ord_less_eq_real @ Y @ X ) )
=> ( ( ( ord_less_eq_real @ Y @ Z3 )
=> ~ ( ord_less_eq_real @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_real @ Z3 @ X )
=> ~ ( ord_less_eq_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_158_le__cases3,axiom,
! [X: int,Y: int,Z3: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z3 ) )
=> ( ( ( ord_less_eq_int @ X @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z3 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z3 )
=> ~ ( ord_less_eq_int @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z3 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_159_nle__le,axiom,
! [A: b,B2: b] :
( ( ~ ( ord_less_eq_b @ A @ B2 ) )
= ( ( ord_less_eq_b @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_160_nle__le,axiom,
! [A: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_161_nle__le,axiom,
! [A: real,B2: real] :
( ( ~ ( ord_less_eq_real @ A @ B2 ) )
= ( ( ord_less_eq_real @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_162_nle__le,axiom,
! [A: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( B2 != A ) ) ) ).
% nle_le
thf(fact_163_verit__comp__simplify1_I2_J,axiom,
! [A: b] : ( ord_less_eq_b @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_164_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_165_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_166_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_167_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_168_order__less__imp__not__less,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_169_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_170_order__less__imp__not__less,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_171_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_172_order__less__imp__not__eq2,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_173_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_174_order__less__imp__not__eq2,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_175_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_176_order__less__imp__not__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_177_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_178_order__less__imp__not__eq,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_179_mem__Collect__eq,axiom,
! [A: real,P: real > $o] :
( ( member_real @ A @ ( collect_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_180_Collect__mem__eq,axiom,
! [A5: set_real] :
( ( collect_real
@ ^ [X4: real] : ( member_real @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_181_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_182_linorder__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
| ( X = Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_183_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_184_linorder__less__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
| ( X = Y )
| ( ord_less_b @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_185_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_186_order__less__imp__triv,axiom,
! [X: real,Y: real,P: $o] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_187_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_188_order__less__imp__triv,axiom,
! [X: b,Y: b,P: $o] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_b @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_189_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_190_order__less__not__sym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_191_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_192_order__less__not__sym,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_193_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_194_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_195_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_196_order__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_197_order__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_198_order__less__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_199_order__less__subst2,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_200_order__less__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_201_order__less__subst2,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_202_order__less__subst2,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_203_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_204_order__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_205_order__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_206_order__less__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_207_order__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_208_order__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_209_order__less__subst1,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_210_order__less__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_211_order__less__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_212_order__less__subst1,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_213_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_214_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_215_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_216_order__less__irrefl,axiom,
! [X: b] :
~ ( ord_less_b @ X @ X ) ).
% order_less_irrefl
thf(fact_217_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_218_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_219_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_220_ord__less__eq__subst,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_221_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_222_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_223_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > int,C: int] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_224_ord__less__eq__subst,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_225_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_226_ord__less__eq__subst,axiom,
! [A: int,B2: int,F: int > real,C: real] :
( ( ord_less_int @ A @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_227_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_228_ord__eq__less__subst,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_229_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_230_ord__eq__less__subst,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_231_ord__eq__less__subst,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_232_ord__eq__less__subst,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_233_ord__eq__less__subst,axiom,
! [A: int,F: real > int,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_234_ord__eq__less__subst,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_235_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_236_ord__eq__less__subst,axiom,
! [A: real,F: int > real,B2: int,C: int] :
( ( A
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_237_order__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_238_order__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_239_order__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_240_order__less__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_less_trans
thf(fact_241_order__less__asym_H,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_242_order__less__asym_H,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_243_order__less__asym_H,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_244_order__less__asym_H,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ~ ( ord_less_b @ B2 @ A ) ) ).
% order_less_asym'
thf(fact_245_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_246_linorder__neq__iff,axiom,
! [X: real,Y: real] :
( ( X != Y )
= ( ( ord_less_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_247_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_248_linorder__neq__iff,axiom,
! [X: b,Y: b] :
( ( X != Y )
= ( ( ord_less_b @ X @ Y )
| ( ord_less_b @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_249_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_250_order__less__asym,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ~ ( ord_less_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_251_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_252_order__less__asym,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ~ ( ord_less_b @ Y @ X ) ) ).
% order_less_asym
thf(fact_253_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_254_linorder__neqE,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_255_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_256_linorder__neqE,axiom,
! [X: b,Y: b] :
( ( X != Y )
=> ( ~ ( ord_less_b @ X @ Y )
=> ( ord_less_b @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_257_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_258_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_259_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_260_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ( A != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_261_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_262_order_Ostrict__implies__not__eq,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_263_order_Ostrict__implies__not__eq,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_264_order_Ostrict__implies__not__eq,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ( A != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_265_dual__order_Ostrict__trans,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_266_dual__order_Ostrict__trans,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_267_dual__order_Ostrict__trans,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_268_dual__order_Ostrict__trans,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_b @ B2 @ A )
=> ( ( ord_less_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_269_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_270_not__less__iff__gr__or__eq,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ( ord_less_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_271_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_272_not__less__iff__gr__or__eq,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_b @ X @ Y ) )
= ( ( ord_less_b @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_273_order_Ostrict__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_274_order_Ostrict__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_275_order_Ostrict__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_276_order_Ostrict__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_277_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_278_linorder__less__wlog,axiom,
! [P: real > real > $o,A: real,B2: real] :
( ! [A4: real,B4: real] :
( ( ord_less_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B4: real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_279_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_280_linorder__less__wlog,axiom,
! [P: b > b > $o,A: b,B2: b] :
( ! [A4: b,B4: b] :
( ( ord_less_b @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: b] : ( P @ A4 @ A4 )
=> ( ! [A4: b,B4: b] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_281_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_282_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_283_dual__order_Oirrefl,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_284_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_285_dual__order_Oirrefl,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% dual_order.irrefl
thf(fact_286_dual__order_Oasym,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ~ ( ord_less_nat @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_287_dual__order_Oasym,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ~ ( ord_less_real @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_288_dual__order_Oasym,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ~ ( ord_less_int @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_289_dual__order_Oasym,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ~ ( ord_less_b @ A @ B2 ) ) ).
% dual_order.asym
thf(fact_290_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_291_linorder__cases,axiom,
! [X: real,Y: real] :
( ~ ( ord_less_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_292_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_293_linorder__cases,axiom,
! [X: b,Y: b] :
( ~ ( ord_less_b @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_b @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_294_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_295_antisym__conv3,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_real @ Y @ X )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_296_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_297_antisym__conv3,axiom,
! [Y: b,X: b] :
( ~ ( ord_less_b @ Y @ X )
=> ( ( ~ ( ord_less_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_298_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_299_ord__less__eq__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_300_ord__less__eq__trans,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_301_ord__less__eq__trans,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_302_ord__less__eq__trans,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( B2 = C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_303_ord__eq__less__trans,axiom,
! [A: nat,B2: nat,C: nat] :
( ( A = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_304_ord__eq__less__trans,axiom,
! [A: real,B2: real,C: real] :
( ( A = B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_305_ord__eq__less__trans,axiom,
! [A: int,B2: int,C: int] :
( ( A = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_306_ord__eq__less__trans,axiom,
! [A: b,B2: b,C: b] :
( ( A = B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_307_order_Oasym,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ( ord_less_nat @ B2 @ A ) ) ).
% order.asym
thf(fact_308_order_Oasym,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ~ ( ord_less_real @ B2 @ A ) ) ).
% order.asym
thf(fact_309_order_Oasym,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ~ ( ord_less_int @ B2 @ A ) ) ).
% order.asym
thf(fact_310_order_Oasym,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ~ ( ord_less_b @ B2 @ A ) ) ).
% order.asym
thf(fact_311_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_312_less__imp__neq,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_313_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_314_less__imp__neq,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_315_dense,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ? [Z: real] :
( ( ord_less_real @ X @ Z )
& ( ord_less_real @ Z @ Y ) ) ) ).
% dense
thf(fact_316_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_317_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_318_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_319_lt__ex,axiom,
! [X: real] :
? [Y2: real] : ( ord_less_real @ Y2 @ X ) ).
% lt_ex
thf(fact_320_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_321_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_322_verit__comp__simplify1_I1_J,axiom,
! [A: real] :
~ ( ord_less_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_323_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_324_verit__comp__simplify1_I1_J,axiom,
! [A: b] :
~ ( ord_less_b @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_325_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_326_pinf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_327_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_328_pinf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_329_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_330_pinf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_331_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_332_pinf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_333_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_334_pinf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_335_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_336_pinf_I3_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_337_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_338_pinf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_339_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_340_pinf_I4_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_341_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ~ ( ord_less_nat @ X2 @ T ) ) ).
% pinf(5)
thf(fact_342_pinf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ~ ( ord_less_real @ X2 @ T ) ) ).
% pinf(5)
thf(fact_343_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_344_pinf_I5_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ~ ( ord_less_b @ X2 @ T ) ) ).
% pinf(5)
thf(fact_345_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z @ X2 )
=> ( ord_less_nat @ T @ X2 ) ) ).
% pinf(7)
thf(fact_346_pinf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_real @ T @ X2 ) ) ).
% pinf(7)
thf(fact_347_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_348_pinf_I7_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ Z @ X2 )
=> ( ord_less_b @ T @ X2 ) ) ).
% pinf(7)
thf(fact_349_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_350_minf_I1_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_351_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_352_minf_I1_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P4 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_353_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_354_minf_I2_J,axiom,
! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_355_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_356_minf_I2_J,axiom,
! [P: b > $o,P4: b > $o,Q: b > $o,Q2: b > $o] :
( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: b] :
! [X3: b] :
( ( ord_less_b @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P4 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_357_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_358_minf_I3_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_359_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_360_minf_I3_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_361_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_362_minf_I4_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_363_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_364_minf_I4_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_365_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ( ord_less_nat @ X2 @ T ) ) ).
% minf(5)
thf(fact_366_minf_I5_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ( ord_less_real @ X2 @ T ) ) ).
% minf(5)
thf(fact_367_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_368_minf_I5_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ( ord_less_b @ X2 @ T ) ) ).
% minf(5)
thf(fact_369_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z )
=> ~ ( ord_less_nat @ T @ X2 ) ) ).
% minf(7)
thf(fact_370_minf_I7_J,axiom,
! [T: real] :
? [Z: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z )
=> ~ ( ord_less_real @ T @ X2 ) ) ).
% minf(7)
thf(fact_371_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_372_minf_I7_J,axiom,
! [T: b] :
? [Z: b] :
! [X2: b] :
( ( ord_less_b @ X2 @ Z )
=> ~ ( ord_less_b @ T @ X2 ) ) ).
% minf(7)
thf(fact_373_order__le__imp__less__or__eq,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_b @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_374_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_375_order__le__imp__less__or__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_376_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_377_linorder__le__less__linear,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
| ( ord_less_b @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_378_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_379_linorder__le__less__linear,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
| ( ord_less_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_380_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_381_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_382_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_383_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > b,C: b] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_384_order__less__le__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_385_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_386_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_387_order__less__le__subst2,axiom,
! [A: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_388_order__less__le__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_389_order__less__le__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_390_order__less__le__subst2,axiom,
! [A: real,B2: real,F: real > real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_392_order__less__le__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_393_order__less__le__subst1,axiom,
! [A: real,F: b > real,B2: b,C: b] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_394_order__less__le__subst1,axiom,
! [A: int,F: b > int,B2: b,C: b] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_395_order__less__le__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_396_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_397_order__less__le__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_398_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_399_order__less__le__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_400_order__less__le__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_401_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_402_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > nat,C: nat] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_403_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > real,C: real] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_order__le__less__subst2,axiom,
! [A: b,B2: b,F: b > int,C: int] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_eq_b @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_405_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > b,C: b] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_406_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_407_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_real @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_408_order__le__less__subst2,axiom,
! [A: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_409_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > b,C: b] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_b @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_410_order__le__less__subst2,axiom,
! [A: real,B2: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_eq_real @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_411_order__le__less__subst1,axiom,
! [A: b,F: nat > b,B2: nat,C: nat] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_412_order__le__less__subst1,axiom,
! [A: b,F: real > b,B2: real,C: real] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_413_order__le__less__subst1,axiom,
! [A: b,F: int > b,B2: int,C: int] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_414_order__le__less__subst1,axiom,
! [A: b,F: b > b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_b @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_b @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_415_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_416_order__le__less__subst1,axiom,
! [A: nat,F: real > nat,B2: real,C: real] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_417_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_418_order__le__less__subst1,axiom,
! [A: nat,F: b > nat,B2: b,C: b] :
( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
=> ( ( ord_less_b @ B2 @ C )
=> ( ! [X3: b,Y2: b] :
( ( ord_less_b @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_419_order__le__less__subst1,axiom,
! [A: real,F: nat > real,B2: nat,C: nat] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_420_order__le__less__subst1,axiom,
! [A: real,F: real > real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ ( F @ B2 ) )
=> ( ( ord_less_real @ B2 @ C )
=> ( ! [X3: real,Y2: real] :
( ( ord_less_real @ X3 @ Y2 )
=> ( ord_less_real @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_421_order__less__le__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_b @ X @ Y )
=> ( ( ord_less_eq_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_422_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_423_order__less__le__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ( ord_less_eq_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_424_order__less__le__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_less_le_trans
thf(fact_425_order__le__less__trans,axiom,
! [X: b,Y: b,Z3: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ord_less_b @ Y @ Z3 )
=> ( ord_less_b @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_426_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_427_order__le__less__trans,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ord_less_real @ Y @ Z3 )
=> ( ord_less_real @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_428_order__le__less__trans,axiom,
! [X: int,Y: int,Z3: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X @ Z3 ) ) ) ).
% order_le_less_trans
thf(fact_429_order__neq__le__trans,axiom,
! [A: b,B2: b] :
( ( A != B2 )
=> ( ( ord_less_eq_b @ A @ B2 )
=> ( ord_less_b @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_430_order__neq__le__trans,axiom,
! [A: nat,B2: nat] :
( ( A != B2 )
=> ( ( ord_less_eq_nat @ A @ B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_431_order__neq__le__trans,axiom,
! [A: real,B2: real] :
( ( A != B2 )
=> ( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_432_order__neq__le__trans,axiom,
! [A: int,B2: int] :
( ( A != B2 )
=> ( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_433_order__le__neq__trans,axiom,
! [A: b,B2: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_b @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_434_order__le__neq__trans,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_435_order__le__neq__trans,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_436_order__le__neq__trans,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( A != B2 )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_437_order__less__imp__le,axiom,
! [X: b,Y: b] :
( ( ord_less_b @ X @ Y )
=> ( ord_less_eq_b @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_438_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_439_order__less__imp__le,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ X @ Y )
=> ( ord_less_eq_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_440_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_441_linorder__not__less,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_b @ X @ Y ) )
= ( ord_less_eq_b @ Y @ X ) ) ).
% linorder_not_less
thf(fact_442_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_443_linorder__not__less,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_real @ X @ Y ) )
= ( ord_less_eq_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_444_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_445_linorder__not__le,axiom,
! [X: b,Y: b] :
( ( ~ ( ord_less_eq_b @ X @ Y ) )
= ( ord_less_b @ Y @ X ) ) ).
% linorder_not_le
thf(fact_446_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_447_linorder__not__le,axiom,
! [X: real,Y: real] :
( ( ~ ( ord_less_eq_real @ X @ Y ) )
= ( ord_less_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_448_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_449_order__less__le,axiom,
( ord_less_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_450_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_451_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_452_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_453_order__le__less,axiom,
( ord_less_eq_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_b @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_454_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_455_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_456_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_457_dual__order_Ostrict__implies__order,axiom,
! [B2: b,A: b] :
( ( ord_less_b @ B2 @ A )
=> ( ord_less_eq_b @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_458_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ord_less_eq_nat @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_459_dual__order_Ostrict__implies__order,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ B2 @ A )
=> ( ord_less_eq_real @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_460_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ A )
=> ( ord_less_eq_int @ B2 @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_461_order_Ostrict__implies__order,axiom,
! [A: b,B2: b] :
( ( ord_less_b @ A @ B2 )
=> ( ord_less_eq_b @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_462_order_Ostrict__implies__order,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_eq_nat @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_463_order_Ostrict__implies__order,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_eq_real @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_464_order_Ostrict__implies__order,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_eq_int @ A @ B2 ) ) ).
% order.strict_implies_order
thf(fact_465_dual__order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ~ ( ord_less_eq_b @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_466_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_467_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_468_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_469_dual__order_Ostrict__trans2,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_b @ B2 @ A )
=> ( ( ord_less_eq_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_470_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_nat @ B2 @ A )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_471_dual__order_Ostrict__trans2,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_472_dual__order_Ostrict__trans2,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_473_dual__order_Ostrict__trans1,axiom,
! [B2: b,A: b,C: b] :
( ( ord_less_eq_b @ B2 @ A )
=> ( ( ord_less_b @ C @ B2 )
=> ( ord_less_b @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_474_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_475_dual__order_Ostrict__trans1,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_real @ C @ B2 )
=> ( ord_less_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_476_dual__order_Ostrict__trans1,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_477_dual__order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_eq_b @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_478_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_479_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_480_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_481_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [B3: b,A3: b] :
( ( ord_less_b @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_482_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_483_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_484_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_485_dense__le__bounded,axiom,
! [X: real,Y: real,Z3: real] :
( ( ord_less_real @ X @ Y )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y )
=> ( ord_less_eq_real @ W @ Z3 ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_le_bounded
thf(fact_486_dense__ge__bounded,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ Z3 @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z3 @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y @ W ) ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ) ).
% dense_ge_bounded
thf(fact_487_order_Ostrict__iff__not,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ~ ( ord_less_eq_b @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_488_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_489_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_490_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_491_order_Ostrict__trans2,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_b @ A @ B2 )
=> ( ( ord_less_eq_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_492_order_Ostrict__trans2,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_493_order_Ostrict__trans2,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_494_order_Ostrict__trans2,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_495_order_Ostrict__trans1,axiom,
! [A: b,B2: b,C: b] :
( ( ord_less_eq_b @ A @ B2 )
=> ( ( ord_less_b @ B2 @ C )
=> ( ord_less_b @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_496_order_Ostrict__trans1,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_497_order_Ostrict__trans1,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ B2 @ C )
=> ( ord_less_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_498_order_Ostrict__trans1,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_499_order_Ostrict__iff__order,axiom,
( ord_less_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_eq_b @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_500_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_501_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_502_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_503_order_Oorder__iff__strict,axiom,
( ord_less_eq_b
= ( ^ [A3: b,B3: b] :
( ( ord_less_b @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_504_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_505_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_506_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_507_not__le__imp__less,axiom,
! [Y: b,X: b] :
( ~ ( ord_less_eq_b @ Y @ X )
=> ( ord_less_b @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_508_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_509_not__le__imp__less,axiom,
! [Y: real,X: real] :
( ~ ( ord_less_eq_real @ Y @ X )
=> ( ord_less_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_510_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_511_less__le__not__le,axiom,
( ord_less_b
= ( ^ [X4: b,Y4: b] :
( ( ord_less_eq_b @ X4 @ Y4 )
& ~ ( ord_less_eq_b @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_512_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_513_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_514_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_515_dense__le,axiom,
! [Y: real,Z3: real] :
( ! [X3: real] :
( ( ord_less_real @ X3 @ Y )
=> ( ord_less_eq_real @ X3 @ Z3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_le
thf(fact_516_dense__ge,axiom,
! [Z3: real,Y: real] :
( ! [X3: real] :
( ( ord_less_real @ Z3 @ X3 )
=> ( ord_less_eq_real @ Y @ X3 ) )
=> ( ord_less_eq_real @ Y @ Z3 ) ) ).
% dense_ge
thf(fact_517_antisym__conv2,axiom,
! [X: b,Y: b] :
( ( ord_less_eq_b @ X @ Y )
=> ( ( ~ ( ord_less_b @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_518_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_519_antisym__conv2,axiom,
! [X: real,Y: real] :
( ( ord_less_eq_real @ X @ Y )
=> ( ( ~ ( ord_less_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_520_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_521_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_522_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_523_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_524_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_525_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_526_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_527_lift__Suc__mono__less__iff,axiom,
! [F: nat > b,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_b @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_528_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_529_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_530_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_531_lift__Suc__mono__less,axiom,
! [F: nat > b,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_b @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_532_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_533_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_534_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_535_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_536_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_537_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_538_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_539_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_540_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_541_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_542_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_543_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_544_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z: nat] :
( ( R @ X3 @ Y2 )
=> ( ( R @ Y2 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_545_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_546_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_547_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_548_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_549_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_550_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_551_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_552_Suc__le__D,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_553_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_554_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_555_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_556_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_557_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_558_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_559_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_560_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_561_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_562_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_563_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_564_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_565_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_566_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_567_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_568_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_569_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_570_lift__Suc__mono__le,axiom,
! [F: nat > b,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_b @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_b @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_571_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_572_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_573_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_574_lift__Suc__antimono__le,axiom,
! [F: nat > b,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_b @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_b @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_575_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_576_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_577_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_578_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_579_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_580_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_581_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_582_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_583_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_584_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_585_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_586_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_587_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_588_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less_nat @ N2 @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_589_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_590_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_591_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_592_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_593_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_594_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_595_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_596_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_597_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_598_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_599_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_600_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_601_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_602_eucl__less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_eq_real @ X4 @ Y4 )
& ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).
% eucl_less_le_not_le
thf(fact_603_complete__interval,axiom,
! [A: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B2 )
& ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_604_complete__interval,axiom,
! [A: real,B2: real,P: real > $o] :
( ( ord_less_real @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: real] :
( ( ord_less_eq_real @ A @ C2 )
& ( ord_less_eq_real @ C2 @ B2 )
& ! [X2: real] :
( ( ( ord_less_eq_real @ A @ X2 )
& ( ord_less_real @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: real] :
( ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_real @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_605_complete__interval,axiom,
! [A: int,B2: int,P: int > $o] :
( ( ord_less_int @ A @ B2 )
=> ( ( P @ A )
=> ( ~ ( P @ B2 )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B2 )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ C2 ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_606_up__ray__def,axiom,
( up_ray_b
= ( ^ [I5: set_b] :
! [X4: b,Y4: b] :
( ( member_b @ X4 @ I5 )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( member_b @ Y4 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_607_up__ray__def,axiom,
( up_ray_nat
= ( ^ [I5: set_nat] :
! [X4: nat,Y4: nat] :
( ( member_nat @ X4 @ I5 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( member_nat @ Y4 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_608_up__ray__def,axiom,
( up_ray_real
= ( ^ [I5: set_real] :
! [X4: real,Y4: real] :
( ( member_real @ X4 @ I5 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( member_real @ Y4 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_609_up__ray__def,axiom,
( up_ray_int
= ( ^ [I5: set_int] :
! [X4: int,Y4: int] :
( ( member_int @ X4 @ I5 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( member_int @ Y4 @ I5 ) ) ) ) ) ).
% up_ray_def
thf(fact_610_down__ray__def,axiom,
( down_ray_b
= ( ^ [I5: set_b] :
! [X4: b,Y4: b] :
( ( member_b @ Y4 @ I5 )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( member_b @ X4 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_611_down__ray__def,axiom,
( down_ray_nat
= ( ^ [I5: set_nat] :
! [X4: nat,Y4: nat] :
( ( member_nat @ Y4 @ I5 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( member_nat @ X4 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_612_down__ray__def,axiom,
( down_ray_real
= ( ^ [I5: set_real] :
! [X4: real,Y4: real] :
( ( member_real @ Y4 @ I5 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( member_real @ X4 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_613_down__ray__def,axiom,
( down_ray_int
= ( ^ [I5: set_int] :
! [X4: int,Y4: int] :
( ( member_int @ Y4 @ I5 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( member_int @ X4 @ I5 ) ) ) ) ) ).
% down_ray_def
thf(fact_614_interval__def,axiom,
( interval_b
= ( ^ [I5: set_b] :
! [X4: b,Y4: b,Z5: b] :
( ( member_b @ X4 @ I5 )
=> ( ( member_b @ Z5 @ I5 )
=> ( ( ord_less_eq_b @ X4 @ Y4 )
=> ( ( ord_less_eq_b @ Y4 @ Z5 )
=> ( member_b @ Y4 @ I5 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_615_interval__def,axiom,
( interval_nat
= ( ^ [I5: set_nat] :
! [X4: nat,Y4: nat,Z5: nat] :
( ( member_nat @ X4 @ I5 )
=> ( ( member_nat @ Z5 @ I5 )
=> ( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ Z5 )
=> ( member_nat @ Y4 @ I5 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_616_interval__def,axiom,
( interval_real
= ( ^ [I5: set_real] :
! [X4: real,Y4: real,Z5: real] :
( ( member_real @ X4 @ I5 )
=> ( ( member_real @ Z5 @ I5 )
=> ( ( ord_less_eq_real @ X4 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ Z5 )
=> ( member_real @ Y4 @ I5 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_617_interval__def,axiom,
( interval_int
= ( ^ [I5: set_int] :
! [X4: int,Y4: int,Z5: int] :
( ( member_int @ X4 @ I5 )
=> ( ( member_int @ Z5 @ I5 )
=> ( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ Z5 )
=> ( member_int @ Y4 @ I5 ) ) ) ) ) ) ) ).
% interval_def
thf(fact_618_GreatestI2__order,axiom,
! [P: b > $o,X: b,Q: b > $o] :
( ( P @ X )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X ) )
=> ( ! [X3: b] :
( ( P @ X3 )
=> ( ! [Y5: b] :
( ( P @ Y5 )
=> ( ord_less_eq_b @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_b @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_619_GreatestI2__order,axiom,
! [P: real > $o,X: real,Q: real > $o] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ! [X3: real] :
( ( P @ X3 )
=> ( ! [Y5: real] :
( ( P @ Y5 )
=> ( ord_less_eq_real @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_620_GreatestI2__order,axiom,
! [P: int > $o,X: int,Q: int > $o] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( ! [Y5: int] :
( ( P @ Y5 )
=> ( ord_less_eq_int @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_int @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_621_GreatestI2__order,axiom,
! [P: nat > $o,X: nat,Q: nat > $o] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_622_Greatest__equality,axiom,
! [P: b > $o,X: b] :
( ( P @ X )
=> ( ! [Y2: b] :
( ( P @ Y2 )
=> ( ord_less_eq_b @ Y2 @ X ) )
=> ( ( order_Greatest_b @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_623_Greatest__equality,axiom,
! [P: real > $o,X: real] :
( ( P @ X )
=> ( ! [Y2: real] :
( ( P @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( order_Greatest_real @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_624_Greatest__equality,axiom,
! [P: int > $o,X: int] :
( ( P @ X )
=> ( ! [Y2: int] :
( ( P @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X ) )
=> ( ( order_Greatest_int @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_625_Greatest__equality,axiom,
! [P: nat > $o,X: nat] :
( ( P @ X )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( order_Greatest_nat @ P )
= X ) ) ) ).
% Greatest_equality
thf(fact_626_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_12: nat] : ( P @ X_12 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_627_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_628_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_629_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_630_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_631_seq__mono__lemma,axiom,
! [M2: nat,D2: nat > real,E: nat > real] :
( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_real @ ( D2 @ N3 ) @ ( E @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ord_less_eq_real @ ( E @ N3 ) @ ( E @ M2 ) ) )
=> ! [N5: nat] :
( ( ord_less_eq_nat @ M2 @ N5 )
=> ( ord_less_real @ ( D2 @ N5 ) @ ( E @ M2 ) ) ) ) ) ).
% seq_mono_lemma
thf(fact_632_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_633_linorder__neqE__linordered__idom,axiom,
! [X: real,Y: real] :
( ( X != Y )
=> ( ~ ( ord_less_real @ X @ Y )
=> ( ord_less_real @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_634_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_635_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_636_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_637_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_638_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_639_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_640_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_641_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_642_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_643_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_644_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_645_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_646_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_647_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_648_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_649_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_650_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_651_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_652_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_653_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_654_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_655_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_656_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_657_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_658_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_659_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_660_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_661_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_662_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_663_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_664_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_665_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_666_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_667_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_668_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_669_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_670_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_671_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_672_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_673_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_674_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_675_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y4: real] :
( ( ord_less_real @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% less_eq_real_def
thf(fact_676_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_677_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_678_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_679_complete__real,axiom,
! [S2: set_real] :
( ? [X2: real] : ( member_real @ X2 @ S2 )
=> ( ? [Z4: real] :
! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ? [Y2: real] :
( ! [X2: real] :
( ( member_real @ X2 @ S2 )
=> ( ord_less_eq_real @ X2 @ Y2 ) )
& ! [Z4: real] :
( ! [X3: real] :
( ( member_real @ X3 @ S2 )
=> ( ord_less_eq_real @ X3 @ Z4 ) )
=> ( ord_less_eq_real @ Y2 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_680_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_681_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E2: real] :
( ( ord_less_real @ zero_zero_real @ E2 )
& ( ord_less_real @ E2 @ D1 )
& ( ord_less_real @ E2 @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_682_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_683_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_684_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_685_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_686_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_687_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_688_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_689_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_690_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_691_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_692_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_693_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_694_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_695_harm__pos__iff,axiom,
! [N2: nat] :
( ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% harm_pos_iff
thf(fact_696_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_697_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_698_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_699_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= ( semiri5074537144036343181t_real @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_700_mult__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_701_mult__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_702_mult__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_right
thf(fact_703_mult__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( ( C = zero_zero_nat )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_704_mult__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_705_mult__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( A = B2 ) ) ) ).
% mult_cancel_left
thf(fact_706_mult__eq__0__iff,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_707_mult__eq__0__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_708_mult__eq__0__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_709_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_710_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_711_mult__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_712_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_713_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_714_mult__zero__left,axiom,
! [A: real] :
( ( times_times_real @ zero_zero_real @ A )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_715_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_716_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_717_mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_718_mult__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M2 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_719_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_720_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_721_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_722_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_723_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_724_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_725_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_726_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_727_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri5074537144036343181t_real @ M2 )
= zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_728_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_729_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_730_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_731_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_732_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_733_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_734_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_mult
thf(fact_735_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_mult
thf(fact_736_of__nat__mult,axiom,
! [M2: nat,N2: nat] :
( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% of_nat_mult
thf(fact_737_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_738_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_739_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_740_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_741_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_742_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_743_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_744_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_745_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_746_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_747_mult_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B2 ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_748_mult_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B2 ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_749_mult_Oassoc,axiom,
! [A: real,B2: real,C: real] :
( ( times_times_real @ ( times_times_real @ A @ B2 ) @ C )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.assoc
thf(fact_750_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_751_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_752_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_753_mult_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( times_times_nat @ B2 @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_754_mult_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( times_times_int @ B2 @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_755_mult_Oleft__commute,axiom,
! [B2: real,A: real,C: real] :
( ( times_times_real @ B2 @ ( times_times_real @ A @ C ) )
= ( times_times_real @ A @ ( times_times_real @ B2 @ C ) ) ) ).
% mult.left_commute
thf(fact_756_mult__of__nat__commute,axiom,
! [X: nat,Y: nat] :
( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
= ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_757_mult__of__nat__commute,axiom,
! [X: nat,Y: int] :
( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
= ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_758_mult__of__nat__commute,axiom,
! [X: nat,Y: real] :
( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
= ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% mult_of_nat_commute
thf(fact_759_ex__less__of__nat__mult,axiom,
! [X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% ex_less_of_nat_mult
thf(fact_760_real__arch__simple,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% real_arch_simple
thf(fact_761_reals__Archimedean2,axiom,
! [X: real] :
? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% reals_Archimedean2
thf(fact_762_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_763_mult__right__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_764_mult__right__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_765_mult__right__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A @ C )
= ( times_times_real @ B2 @ C ) )
= ( A = B2 ) ) ) ).
% mult_right_cancel
thf(fact_766_mult__left__cancel,axiom,
! [C: nat,A: nat,B2: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_767_mult__left__cancel,axiom,
! [C: int,A: int,B2: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_768_mult__left__cancel,axiom,
! [C: real,A: real,B2: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A )
= ( times_times_real @ C @ B2 ) )
= ( A = B2 ) ) ) ).
% mult_left_cancel
thf(fact_769_no__zero__divisors,axiom,
! [A: nat,B2: nat] :
( ( A != zero_zero_nat )
=> ( ( B2 != zero_zero_nat )
=> ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_770_no__zero__divisors,axiom,
! [A: int,B2: int] :
( ( A != zero_zero_int )
=> ( ( B2 != zero_zero_int )
=> ( ( times_times_int @ A @ B2 )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_771_no__zero__divisors,axiom,
! [A: real,B2: real] :
( ( A != zero_zero_real )
=> ( ( B2 != zero_zero_real )
=> ( ( times_times_real @ A @ B2 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_772_divisors__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B2 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_773_divisors__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B2 = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_774_divisors__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
= zero_zero_real )
=> ( ( A = zero_zero_real )
| ( B2 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_775_mult__not__zero,axiom,
! [A: nat,B2: nat] :
( ( ( times_times_nat @ A @ B2 )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B2 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_776_mult__not__zero,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ B2 )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B2 != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_777_mult__not__zero,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ B2 )
!= zero_zero_real )
=> ( ( A != zero_zero_real )
& ( B2 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_778_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_779_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_780_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_781_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M2 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_782_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_783_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_784_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_785_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_786_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_787_harm__nonneg,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( harmonic_harm_real @ N2 ) ) ).
% harm_nonneg
thf(fact_788_harm__expand_I1_J,axiom,
( ( harmonic_harm_real @ zero_zero_nat )
= zero_zero_real ) ).
% harm_expand(1)
thf(fact_789_harm__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_real @ ( harmonic_harm_real @ M2 ) @ ( harmonic_harm_real @ N2 ) ) ) ).
% harm_mono
thf(fact_790_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_791_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_792_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_793_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_794_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_795_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).
% of_nat_less_0_iff
thf(fact_796_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_797_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_798_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
!= zero_zero_real ) ).
% of_nat_neq_0
thf(fact_799_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_800_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% of_nat_mono
thf(fact_801_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_802_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_803_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_804_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_805_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_806_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_807_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_808_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_809_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_810_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_811_zero__le__mult__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_812_zero__le__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_813_mult__nonneg__nonpos2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_814_mult__nonneg__nonpos2,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_815_mult__nonneg__nonpos2,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_816_mult__nonpos__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_817_mult__nonpos__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonpos_nonneg
thf(fact_818_mult__nonpos__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_819_mult__nonneg__nonpos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B2 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_820_mult__nonneg__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos
thf(fact_821_mult__nonneg__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_822_mult__nonneg__nonneg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_823_mult__nonneg__nonneg,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_824_mult__nonneg__nonneg,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_825_split__mult__neg__le,axiom,
! [A: nat,B2: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B2 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_826_split__mult__neg__le,axiom,
! [A: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_827_split__mult__neg__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_828_mult__le__0__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_829_mult__le__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_le_0_iff
thf(fact_830_mult__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_831_mult__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_832_mult__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_833_mult__right__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_834_mult__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_835_mult__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_836_mult__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_837_mult__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono
thf(fact_838_mult__nonpos__nonpos,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B2 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_839_mult__nonpos__nonpos,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_840_mult__left__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_eq_real @ B2 @ A )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_841_mult__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_eq_int @ B2 @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_842_split__mult__pos__le,axiom,
! [A: real,B2: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
& ( ord_less_eq_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_eq_real @ A @ zero_zero_real )
& ( ord_less_eq_real @ B2 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_843_split__mult__pos__le,axiom,
! [A: int,B2: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B2 @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ).
% split_mult_pos_le
thf(fact_844_zero__le__square,axiom,
! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% zero_le_square
thf(fact_845_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_846_mult__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_847_mult__mono_H,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_848_mult__mono_H,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_849_mult__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_850_mult__mono,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_851_mult__mono,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_mono
thf(fact_852_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_853_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_854_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_855_mult__less__cancel__right__disj,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_856_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_857_mult__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_858_mult__strict__right__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_859_mult__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_860_mult__strict__right__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_861_mult__strict__right__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_862_mult__less__cancel__left__disj,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A @ B2 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_863_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B2 ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_864_mult__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_865_mult__strict__left__mono,axiom,
! [A: real,B2: real,C: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_866_mult__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono
thf(fact_867_mult__strict__left__mono__neg,axiom,
! [B2: real,A: real,C: real] :
( ( ord_less_real @ B2 @ A )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_868_mult__strict__left__mono__neg,axiom,
! [B2: int,A: int,C: int] :
( ( ord_less_int @ B2 @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_869_mult__less__cancel__left__pos,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ A @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_870_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_871_mult__less__cancel__left__neg,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_real @ B2 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_872_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_int @ B2 @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_873_zero__less__mult__pos2,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B2 @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_874_zero__less__mult__pos2,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B2 @ A ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_875_zero__less__mult__pos2,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B2 @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos2
thf(fact_876_zero__less__mult__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_877_zero__less__mult__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ord_less_real @ zero_zero_real @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_878_zero__less__mult__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B2 ) ) ) ).
% zero_less_mult_pos
thf(fact_879_zero__less__mult__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ zero_zero_real @ B2 ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ B2 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_880_zero__less__mult__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B2 ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B2 @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_881_mult__pos__neg2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B2 @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_882_mult__pos__neg2,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B2 @ A ) @ zero_zero_real ) ) ) ).
% mult_pos_neg2
thf(fact_883_mult__pos__neg2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B2 @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_884_mult__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_885_mult__pos__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_886_mult__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_pos_pos
thf(fact_887_mult__pos__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_888_mult__pos__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_pos_neg
thf(fact_889_mult__pos__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_890_mult__neg__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ ( times_times_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_891_mult__neg__pos,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real ) ) ) ).
% mult_neg_pos
thf(fact_892_mult__neg__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_893_mult__less__0__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ B2 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A )
& ( ord_less_real @ B2 @ zero_zero_real ) )
| ( ( ord_less_real @ A @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_894_mult__less__0__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ B2 ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B2 @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% mult_less_0_iff
thf(fact_895_not__square__less__zero,axiom,
! [A: real] :
~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_896_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_897_mult__neg__neg,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B2 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_898_mult__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B2 ) ) ) ) ).
% mult_neg_neg
thf(fact_899_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_900_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_901_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_902_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M2 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_903_mult__less__le__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_904_mult__less__le__imp__less,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_eq_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_905_mult__less__le__imp__less,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_906_mult__le__less__imp__less,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_907_mult__le__less__imp__less,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_908_mult__le__less__imp__less,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_909_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_910_mult__right__le__imp__le,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_911_mult__right__le__imp__le,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_right_le_imp_le
thf(fact_912_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_913_mult__left__le__imp__le,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_914_mult__left__le__imp__le,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_left_le_imp_le
thf(fact_915_mult__le__cancel__left__pos,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ A @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_916_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ A @ B2 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_917_mult__le__cancel__left__neg,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ord_less_eq_real @ B2 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_918_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ord_less_eq_int @ B2 @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_919_mult__less__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_920_mult__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_921_mult__strict__mono_H,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_922_mult__strict__mono_H,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_923_mult__strict__mono_H,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_924_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_925_mult__right__less__imp__less,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_926_mult__right__less__imp__less,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_right_less_imp_less
thf(fact_927_mult__less__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_928_mult__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B2 @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_929_mult__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D2 )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_930_mult__strict__mono,axiom,
! [A: real,B2: real,C: real,D2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ( ord_less_real @ C @ D2 )
=> ( ( ord_less_real @ zero_zero_real @ B2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_931_mult__strict__mono,axiom,
! [A: int,B2: int,C: int,D2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ D2 ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_932_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B2 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_933_mult__left__less__imp__less,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_934_mult__left__less__imp__less,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B2 ) ) ) ).
% mult_left_less_imp_less
thf(fact_935_mult__le__cancel__right,axiom,
! [A: real,C: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B2 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_936_mult__le__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B2 @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_937_mult__le__cancel__left,axiom,
! [C: real,A: real,B2: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B2 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A @ B2 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B2 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_938_mult__le__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B2 ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B2 ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B2 @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_939_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_940_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_941_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_942_harm__pos,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_real @ zero_zero_real @ ( harmonic_harm_real @ N2 ) ) ) ).
% harm_pos
thf(fact_943_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_944_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_945_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: real,X: real,B2: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ B2 @ X ) )
= ( ( A = B2 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_946_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: real,X: real,Y: real] :
( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
= ( ( X = Y )
| ( A = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_947_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: real] :
( ( times_times_real @ A @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_948_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_949_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: real,X: real] :
( ( ( times_times_real @ A @ X )
= zero_zero_real )
= ( ( A = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_950_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_951_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_952_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z2: nat] : ( Y3 = Z2 ) )
= ( ^ [A3: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A3 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_953_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_954_int__ops_I7_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B2 ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_955_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y5: real] :
? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_956_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M5: nat] :
( ( ord_less_nat @ zero_zero_nat @ M5 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_957_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: real,X: real,Y: real] :
( ( A != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ A @ Y ) )
=> ( X = Y ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_958_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A: real,B2: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A @ X )
= ( times_times_real @ B2 @ X ) )
=> ( A = B2 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_959_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( ( K = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_960_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M2 )
= ( times_times_nat @ K @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_961_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_962_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_963_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_964_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_965_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_966_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_967_mult__le__cancel__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_968_mult__le__cancel__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_969_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_970_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_971_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_972_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_973_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_int @ I @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_974_mult__less__iff1,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_975_mult__less__iff1,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_976_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_977_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_978_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_979_mult__le__cancel__iff2,axiom,
! [Z3: real,X: real,Y: real] :
( ( ord_less_real @ zero_zero_real @ Z3 )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_980_mult__le__cancel__iff2,axiom,
! [Z3: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_981_bgauge__existence__lemma,axiom,
! [S: set_real,Q3: real > real > $o] :
( ( ! [X4: real] :
( ( member_real @ X4 @ S )
=> ? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( Q3 @ D3 @ X4 ) ) ) )
= ( ! [X4: real] :
? [D3: real] :
( ( ord_less_real @ zero_zero_real @ D3 )
& ( ( member_real @ X4 @ S )
=> ( Q3 @ D3 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_982_mult__delta__right,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ X @ ( if_nat @ B2 @ Y @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_983_mult__delta__right,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ X @ ( if_int @ B2 @ Y @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_984_mult__delta__right,axiom,
! [B2: $o,X: real,Y: real] :
( ( B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y @ zero_zero_real ) )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_real @ X @ ( if_real @ B2 @ Y @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_985_mult__delta__left,axiom,
! [B2: $o,X: nat,Y: nat] :
( ( B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= ( times_times_nat @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_nat @ ( if_nat @ B2 @ X @ zero_zero_nat ) @ Y )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_986_mult__delta__left,axiom,
! [B2: $o,X: int,Y: int] :
( ( B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= ( times_times_int @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_int @ ( if_int @ B2 @ X @ zero_zero_int ) @ Y )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_987_mult__delta__left,axiom,
! [B2: $o,X: real,Y: real] :
( ( B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y )
= ( times_times_real @ X @ Y ) ) )
& ( ~ B2
=> ( ( times_times_real @ ( if_real @ B2 @ X @ zero_zero_real ) @ Y )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_988_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_989_verit__minus__simplify_I4_J,axiom,
! [B2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_990_verit__minus__simplify_I4_J,axiom,
! [B2: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B2 ) )
= B2 ) ).
% verit_minus_simplify(4)
thf(fact_991_neg__equal__iff__equal,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_992_neg__equal__iff__equal,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) )
= ( A = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_993_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_994_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_995_neg__le__iff__le,axiom,
! [B2: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_996_neg__le__iff__le,axiom,
! [B2: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B2 ) ) ).
% neg_le_iff_le
thf(fact_997_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_998_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_999_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1000_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_1001_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1002_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_1003_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1004_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_1005_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1006_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_1007_neg__less__iff__less,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1008_neg__less__iff__less,axiom,
! [B2: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1009_mult__minus__right,axiom,
! [A: int,B2: int] :
( ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% mult_minus_right
thf(fact_1010_mult__minus__right,axiom,
! [A: real,B2: real] :
( ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% mult_minus_right
thf(fact_1011_minus__mult__minus,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B2 ) )
= ( times_times_int @ A @ B2 ) ) ).
% minus_mult_minus
thf(fact_1012_minus__mult__minus,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B2 ) )
= ( times_times_real @ A @ B2 ) ) ).
% minus_mult_minus
thf(fact_1013_mult__minus__left,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( uminus_uminus_int @ ( times_times_int @ A @ B2 ) ) ) ).
% mult_minus_left
thf(fact_1014_mult__minus__left,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( uminus_uminus_real @ ( times_times_real @ A @ B2 ) ) ) ).
% mult_minus_left
thf(fact_1015_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_1016_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1017_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1018_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_1019_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_1020_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1021_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1022_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_1023_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1024_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_1025_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_1026_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_1027_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1028_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_1029_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1030_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_1031_negative__eq__positive,axiom,
! [N2: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_1032_negative__zless,axiom,
! [N2: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_1033_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1034_minus__mult__commute,axiom,
! [A: int,B2: int] :
( ( times_times_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( times_times_int @ A @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_mult_commute
thf(fact_1035_minus__mult__commute,axiom,
! [A: real,B2: real] :
( ( times_times_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( times_times_real @ A @ ( uminus_uminus_real @ B2 ) ) ) ).
% minus_mult_commute
thf(fact_1036_square__eq__iff,axiom,
! [A: int,B2: int] :
( ( ( times_times_int @ A @ A )
= ( times_times_int @ B2 @ B2 ) )
= ( ( A = B2 )
| ( A
= ( uminus_uminus_int @ B2 ) ) ) ) ).
% square_eq_iff
thf(fact_1037_square__eq__iff,axiom,
! [A: real,B2: real] :
( ( ( times_times_real @ A @ A )
= ( times_times_real @ B2 @ B2 ) )
= ( ( A = B2 )
| ( A
= ( uminus_uminus_real @ B2 ) ) ) ) ).
% square_eq_iff
thf(fact_1038_verit__negate__coefficient_I3_J,axiom,
! [A: int,B2: int] :
( ( A = B2 )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1039_verit__negate__coefficient_I3_J,axiom,
! [A: real,B2: real] :
( ( A = B2 )
=> ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B2 ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1040_minus__equation__iff,axiom,
! [A: int,B2: int] :
( ( ( uminus_uminus_int @ A )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_1041_minus__equation__iff,axiom,
! [A: real,B2: real] :
( ( ( uminus_uminus_real @ A )
= B2 )
= ( ( uminus_uminus_real @ B2 )
= A ) ) ).
% minus_equation_iff
thf(fact_1042_equation__minus__iff,axiom,
! [A: int,B2: int] :
( ( A
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_1043_equation__minus__iff,axiom,
! [A: real,B2: real] :
( ( A
= ( uminus_uminus_real @ B2 ) )
= ( B2
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_1044_verit__negate__coefficient_I2_J,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1045_verit__negate__coefficient_I2_J,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ B2 )
=> ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1046_minus__less__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_1047_minus__less__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_less_iff
thf(fact_1048_less__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_1049_less__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_1050_le__minus__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B2 ) )
= ( ord_less_eq_real @ B2 @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_1051_le__minus__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_1052_minus__le__iff,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B2 )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_1053_minus__le__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A ) ) ).
% minus_le_iff
thf(fact_1054_le__imp__neg__le,axiom,
! [A: real,B2: real] :
( ( ord_less_eq_real @ A @ B2 )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B2 ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_1055_le__imp__neg__le,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_1056_int__cases,axiom,
! [Z3: int] :
( ! [N3: nat] :
( Z3
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( Z3
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_1057_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
=> ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_1058_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1059_int__zle__neg,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N2 = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_1060_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_1061_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_1062_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1063_not__zle__0__negative,axiom,
! [N2: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% not_zle_0_negative
thf(fact_1064_negative__zless__0,axiom,
! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_1065_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N3: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_1066_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_1067_real__eq__0__iff__le__ge__0,axiom,
! [X: real] :
( ( X = zero_zero_real )
= ( ( ord_less_eq_real @ zero_zero_real @ X )
& ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ X ) ) ) ) ).
% real_eq_0_iff_le_ge_0
thf(fact_1068_radical__0,axiom,
! [N2: nat,R2: nat > real > real,A: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ N2 )
= zero_zero_real ) ) ).
% radical_0
thf(fact_1069_nat__mult__distrib__neg,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1070_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1071_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1072_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ Z3 @ zero_zero_int )
=> ( ( nat2 @ Z3 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1073_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less_int @ zero_zero_int @ Z3 )
& ( ord_less_int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_1074_nat__zminus__int,axiom,
! [N2: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_1075_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1076_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_1077_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1078_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1079_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1080_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_1081_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P3 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_1082_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less_int @ zero_zero_int @ Z3 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_1083_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1084_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z3: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z3 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1085_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1086_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1087_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1088_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( W2
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1089_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N ) )
=> ( P @ N ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1090_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less_int @ zero_zero_int @ W2 )
| ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1091_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1092_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ W2 )
=> ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1093_nat__mult__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1094_radical_Osimps_I2_J,axiom,
! [R2: nat > real > real,A: formal3361831859752904756s_real,N2: nat] :
( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A @ ( suc @ N2 ) )
= zero_zero_real ) ).
% radical.simps(2)
thf(fact_1095_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
= ( ord_less_int @ one_one_int @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_1096_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1097_of__int__of__nat,axiom,
( ring_1_of_int_real
= ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1098_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_int @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_1099_of__nat__nat,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z3 )
=> ( ( semiri5074537144036343181t_real @ ( nat2 @ Z3 ) )
= ( ring_1_of_int_real @ Z3 ) ) ) ).
% of_nat_nat
thf(fact_1100_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_1101_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1102_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1103_mult_Oright__neutral,axiom,
! [A: real] :
( ( times_times_real @ A @ one_one_real )
= A ) ).
% mult.right_neutral
thf(fact_1104_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1105_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1106_mult__1,axiom,
! [A: real] :
( ( times_times_real @ one_one_real @ A )
= A ) ).
% mult_1
thf(fact_1107_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1108_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1109_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_1110_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1111_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1112_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1113_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_1114_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1115_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1116_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_1117_mult__cancel__right2,axiom,
! [A: real,C: real] :
( ( ( times_times_real @ A @ C )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_1118_mult__cancel__right1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ B2 @ C ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_1119_mult__cancel__right1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ B2 @ C ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_1120_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_1121_mult__cancel__left2,axiom,
! [C: real,A: real] :
( ( ( times_times_real @ C @ A )
= C )
= ( ( C = zero_zero_real )
| ( A = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_1122_mult__cancel__left1,axiom,
! [C: int,B2: int] :
( ( C
= ( times_times_int @ C @ B2 ) )
= ( ( C = zero_zero_int )
| ( B2 = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_1123_mult__cancel__left1,axiom,
! [C: real,B2: real] :
( ( C
= ( times_times_real @ C @ B2 ) )
= ( ( C = zero_zero_real )
| ( B2 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_1124_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_1125_of__int__0,axiom,
( ( ring_1_of_int_real @ zero_zero_int )
= zero_zero_real ) ).
% of_int_0
thf(fact_1126_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1127_of__int__0__eq__iff,axiom,
! [Z3: int] :
( ( zero_zero_real
= ( ring_1_of_int_real @ Z3 ) )
= ( Z3 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_1128_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_int @ Z3 )
= zero_zero_int )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1129_of__int__eq__0__iff,axiom,
! [Z3: int] :
( ( ( ring_1_of_int_real @ Z3 )
= zero_zero_real )
= ( Z3 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_1130_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_1131_of__int__le__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ W2 @ Z3 ) ) ).
% of_int_le_iff
thf(fact_1132_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1133_of__int__less__iff,axiom,
! [W2: int,Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ W2 @ Z3 ) ) ).
% of_int_less_iff
thf(fact_1134_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1135_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1136_of__int__le__0__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z3 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_1137_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_1138_of__int__0__le__iff,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_le_iff
thf(fact_1139_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ zero_zero_real )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1140_of__int__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z3 ) @ zero_zero_int )
= ( ord_less_int @ Z3 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_1141_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1142_of__int__0__less__iff,axiom,
! [Z3: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z3 ) )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% of_int_0_less_iff
thf(fact_1143_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq_int @ one_one_int @ Z3 )
= ( ord_less_int @ zero_zero_int @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1144_pos__zmult__eq__1__iff,axiom,
! [M2: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M2 )
=> ( ( ( times_times_int @ M2 @ N2 )
= one_one_int )
= ( ( M2 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1145_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1146_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1147_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1148_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1149_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1150_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1151_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1152_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1153_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1154_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1155_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I2 ) )
& ( ord_less_eq_real @ ( X3 @ I2 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X2: nat > real,I4: nat] : ( ord_less_eq_nat @ ( L2 @ X2 @ I4 ) @ one_one_nat )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q @ I4 )
& ( ( X2 @ I4 )
= zero_zero_real ) )
=> ( ( L2 @ X2 @ I4 )
= zero_zero_nat ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q @ I4 )
& ( ( X2 @ I4 )
= one_one_real ) )
=> ( ( L2 @ X2 @ I4 )
= one_one_nat ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q @ I4 )
& ( ( L2 @ X2 @ I4 )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X2 @ I4 ) @ ( F @ X2 @ I4 ) ) )
& ! [X2: nat > real,I4: nat] :
( ( ( P @ X2 )
& ( Q @ I4 )
& ( ( L2 @ X2 @ I4 )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X2 @ I4 ) @ ( X2 @ I4 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_1156_real__of__nat__ge__one__iff,axiom,
! [N2: nat] :
( ( ord_less_eq_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ).
% real_of_nat_ge_one_iff
thf(fact_1157_nat__ceiling__le__eq,axiom,
! [X: real,A: nat] :
( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
= ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1158_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_1159_real__root__increasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1160_real__root__zero,axiom,
! [N2: nat] :
( ( root @ N2 @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_1161_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1162_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_1163_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_1164_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1165_real__root__le__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_1166_real__root__less__iff,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) )
= ( ord_less_real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_1167_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1168_real__root__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1169_real__root__ge__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_1170_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1171_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1172_real__root__gt__0__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ zero_zero_real @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_1173_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1174_real__root__ge__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_1175_real__root__gt__1__iff,axiom,
! [N2: nat,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
= ( ord_less_real @ one_one_real @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_1176_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1177_int__ops_I8_J,axiom,
! [A: nat,B2: nat] :
( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B2 ) )
= ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ).
% int_ops(8)
thf(fact_1178_real__of__nat__div4,axiom,
! [N2: nat,X: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_1179_real__of__int__div4,axiom,
! [N2: int,X: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X ) ) ) ).
% real_of_int_div4
thf(fact_1180_real__root__pos__pos__le,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_1181_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1182_real__root__le__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ X @ Y )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_1183_real__root__less__mono,axiom,
! [N2: nat,X: real,Y: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ Y )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_1184_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N2 ) )
= ( divide_divide_nat @ M2 @ N2 ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1185_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1186_real__root__strict__decreasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1187_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N2: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N2 ) @ ( divide_divide_int @ A5 @ N2 ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1188_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1189_real__root__strict__increasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N6 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1190_real__root__decreasing,axiom,
! [N2: nat,N6: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N6 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1191_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1192_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1193_div__mult__self1__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M2 ) @ N2 )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1194_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1195_div__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1196_div__mult__self__is__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N2 ) @ N2 )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1197_div__le__mono,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% div_le_mono
thf(fact_1198_div__le__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ).
% div_le_dividend
thf(fact_1199_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( divide_divide_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N2 )
| ( N2 = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1200_Suc__div__le__mono,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N2 ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_div_le_mono
thf(fact_1201_div__times__less__eq__dividend,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N2 ) @ N2 ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1202_times__div__less__eq__dividend,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M2 @ N2 ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1203_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N2 ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1204_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1205_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1206_div__neg__pos__less0,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_neg_pos_less0
thf(fact_1207_div__le__mono2,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1208_div__greater__zero__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ N2 @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% div_greater_zero_iff
thf(fact_1209_div__eq__dividend__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N2 )
= M2 )
= ( N2 = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1210_div__less__dividend,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1211_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N2 )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1212_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1213_zdiv__mono1,axiom,
! [A: int,A2: int,B2: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1214_zdiv__mono2,axiom,
! [A: int,B: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1215_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1216_zdiv__mono1__neg,axiom,
! [A: int,A2: int,B2: int] :
( ( ord_less_eq_int @ A @ A2 )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B2 ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1217_zdiv__mono2__neg,axiom,
! [A: int,B: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1218_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1219_div__nonneg__neg__le0,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1220_div__nonpos__pos__le0,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1221_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1222_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1223_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1224_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A )
& ( ord_less_int @ zero_zero_int @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1225_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B2 @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B2 ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1226_nat__div__distrib,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1227_nat__div__distrib_H,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( divide_divide_int @ X @ Y ) )
= ( divide_divide_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1228_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N2 @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N2 ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1229_div__nat__eqI,axiom,
! [N2: nat,Q3: nat,M2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M2 )
=> ( ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
=> ( ( divide_divide_nat @ M2 @ N2 )
= Q3 ) ) ) ).
% div_nat_eqI
thf(fact_1230_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B2 )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1231_split__div_H,axiom,
! [P: nat > $o,M2: nat,N2: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N2 ) )
= ( ( ( N2 = zero_zero_nat )
& ( P @ zero_zero_nat ) )
| ? [Q4: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M2 )
& ( ord_less_nat @ M2 @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1232_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1233_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1234_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1235_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1236_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1237_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1238_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1239_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1240_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_1241_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1242_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1243_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1244_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less_int @ zero_zero_int @ L )
=> ( ( ord_less_eq_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% div_pos_geq
thf(fact_1245_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1246_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1247_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1248_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1249_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1250_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1251_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1252_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1253_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1254_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1255_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_1256_le__diff__iff_H,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1257_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1258_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1259_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1260_int__minus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1261_nat__diff__distrib,axiom,
! [Z6: int,Z3: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ord_less_eq_int @ Z6 @ Z3 )
=> ( ( nat2 @ ( minus_minus_int @ Z3 @ Z6 ) )
= ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1262_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1263_int__ops_I6_J,axiom,
! [A: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1264_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z3 ) @ Z3 )
!= zero_zero_int ) ).
% odd_nonzero
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ! [I2: nat,J2: nat] :
( ~ ( ord_less_nat @ J2 @ na )
| ~ ( ord_less_nat @ I2 @ J2 )
| ( ord_less_eq_b @ ( g @ k @ I2 ) @ ( g @ k @ J2 ) ) )
& ! [L2: nat] :
( ~ ( ord_less_nat @ L2 @ k )
| ( ord_less_eq_b @ ( g @ k @ L2 ) @ ( g @ k @ na ) ) ) ) ).
%------------------------------------------------------------------------------