TPTP Problem File: SLH0439^1.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 : Dedekind_Real/0000_Dedekind_Real/prob_01280_041742__5749446_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1349 ( 593 unt; 70 typ; 0 def)
% Number of atoms : 3419 (1413 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10171 ( 371 ~; 86 |; 136 &;8083 @)
% ( 0 <=>;1495 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 642 ( 642 >; 0 *; 0 +; 0 <<)
% Number of symbols : 68 ( 65 usr; 11 con; 0-3 aty)
% Number of variables : 3545 ( 252 ^;3211 !; 82 ?;3545 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:21.492
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Opreal_J,type,
set_Dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (65)
thf(sy_c_Dedekind__Real_Opsup,type,
dedekind_psup: set_Dedekind_preal > dedekind_preal ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Dedekind____Real__Opreal,type,
invers3090987106763476162_preal: dedekind_preal > dedekind_preal ).
thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Dedekind____Real__Oreal,type,
invers3762989301784728874d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ogroup_001t__Dedekind____Real__Oreal,type,
group_Dedekind_real: ( dedekind_real > dedekind_real > dedekind_real ) > dedekind_real > ( dedekind_real > dedekind_real ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Dedekind____Real__Opreal,type,
minus_7336623429200594941_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Dedekind____Real__Oreal,type,
minus_5539002012860128047d_real: dedekind_real > dedekind_real > dedekind_real ).
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__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Oreal,type,
one_on6069100329679821595d_real: dedekind_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Opreal,type,
plus_p3173629198307831117_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Dedekind____Real__Oreal,type,
plus_p4060926892116697567d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Osgn__class_Osgn_001t__Dedekind____Real__Oreal,type,
sgn_sg210920135175580611d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Oreal,type,
times_2157731159493324635d_real: dedekind_real > dedekind_real > dedekind_real ).
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__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Dedekind____Real__Oreal,type,
uminus7714077491378687647d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Dedekind____Real__Oreal,type,
zero_z580800474297136991d_real: dedekind_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Dedekind____Real__Oreal,type,
if_Dedekind_real: $o > dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Dedekind____Real__Oreal,type,
neg_nu7286324685880884523d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Dedekind____Real__Oreal,type,
neg_nu2849763295382964967d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Dedekind____Real__Oreal,type,
neg_nu5286250102780226903d_real: num > num > dedekind_real ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Dedekind____Real__Oreal,type,
numera6440378960330123505d_real: num > dedekind_real ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero_001t__Dedekind____Real__Oreal,type,
ring_1453304368769599722d_real: dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Opreal,type,
ord_le5708704896291381698_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Dedekind____Real__Oreal,type,
ord_le2991122432403439658d_real: dedekind_real > dedekind_real > $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__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Dedekind____Real__Opreal_J_J,type,
ord_le6171167672038760972_preal: ( $o > $o > dedekind_preal ) > ( $o > $o > dedekind_preal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Dedekind____Real__Oreal_J_J,type,
ord_le8943158644485376652d_real: ( $o > $o > dedekind_real ) > ( $o > $o > dedekind_real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Nat__Onat_J_J,type,
ord_less_eq_o_o_nat: ( $o > $o > nat ) > ( $o > $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_062_I_Eo_Mt__Num__Onum_J_J,type,
ord_less_eq_o_o_num: ( $o > $o > num ) > ( $o > $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Dedekind____Real__Opreal_J,type,
ord_le6023059474077774659_preal: ( $o > dedekind_preal ) > ( $o > dedekind_preal ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Dedekind____Real__Oreal_J,type,
ord_le267469853237711487d_real: ( $o > dedekind_real ) > ( $o > dedekind_real ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Nat__Onat_J,type,
ord_less_eq_o_nat: ( $o > nat ) > ( $o > nat ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_Mt__Num__Onum_J,type,
ord_less_eq_o_num: ( $o > num ) > ( $o > num ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Opreal,type,
ord_le5604041210740703414_preal: dedekind_preal > dedekind_preal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Dedekind____Real__Oreal,type,
ord_le2716243287969276982d_real: dedekind_real > dedekind_real > $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__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Dedekind____Real__Opreal_J,type,
order_6170817298126604234_preal: ( ( $o > dedekind_preal ) > $o ) > $o > dedekind_preal ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Dedekind____Real__Oreal_J,type,
order_1363326640887983160d_real: ( ( $o > dedekind_real ) > $o ) > $o > dedekind_real ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Nat__Onat_J,type,
order_Greatest_o_nat: ( ( $o > nat ) > $o ) > $o > nat ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001_062_I_Eo_Mt__Num__Onum_J,type,
order_Greatest_o_num: ( ( $o > num ) > $o ) > $o > num ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Dedekind____Real__Opreal,type,
order_958373252487505263_preal: ( dedekind_preal > $o ) > dedekind_preal ).
thf(sy_c_Orderings_Oorder__class_OGreatest_001t__Dedekind____Real__Oreal,type,
order_1105151008113787197d_real: ( dedekind_real > $o ) > dedekind_real ).
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__Num__Onum,type,
order_Greatest_num: ( num > $o ) > num ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Opreal,type,
divide4190755330972744004_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Oreal,type,
divide9119111104558704680d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Opreal,type,
collec1132657498972982273_preal: ( dedekind_preal > $o ) > set_Dedekind_preal ).
thf(sy_c_member_001t__Dedekind____Real__Opreal,type,
member6871284927547481791_preal: dedekind_preal > set_Dedekind_preal > $o ).
thf(sy_v_x,type,
x: dedekind_real ).
thf(sy_v_y,type,
y: dedekind_real ).
thf(sy_v_z,type,
z: dedekind_real ).
% Relevant facts (1273)
thf(fact_0_le,axiom,
ord_le2716243287969276982d_real @ x @ y ).
% le
thf(fact_1__092_060open_062z_A_L_Ax_A_N_A_Iz_A_L_Ay_J_A_061_Az_A_L_A_N_Az_A_L_A_Ix_A_N_Ay_J_092_060close_062,axiom,
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ z @ x ) @ ( plus_p4060926892116697567d_real @ z @ y ) )
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ z @ ( uminus7714077491378687647d_real @ z ) ) @ ( minus_5539002012860128047d_real @ x @ y ) ) ) ).
% \<open>z + x - (z + y) = z + - z + (x - y)\<close>
thf(fact_2_real__le__refl,axiom,
! [W: dedekind_real] : ( ord_le2716243287969276982d_real @ W @ W ) ).
% real_le_refl
thf(fact_3_add__eq__exists,axiom,
! [A: dedekind_real,B: dedekind_real] :
? [X: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ X )
= B ) ).
% add_eq_exists
thf(fact_4_real__le__trans,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real] :
( ( ord_le2716243287969276982d_real @ I @ J )
=> ( ( ord_le2716243287969276982d_real @ J @ K )
=> ( ord_le2716243287969276982d_real @ I @ K ) ) ) ).
% real_le_trans
thf(fact_5_real__le__antisym,axiom,
! [Z: dedekind_real,W: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z @ W )
=> ( ( ord_le2716243287969276982d_real @ W @ Z )
=> ( Z = W ) ) ) ).
% real_le_antisym
thf(fact_6_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_7_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_8_add__left__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_9_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_10_add__right__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_11_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_12_order__refl,axiom,
! [X2: $o > nat] : ( ord_less_eq_o_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_13_order__refl,axiom,
! [X2: $o > num] : ( ord_less_eq_o_num @ X2 @ X2 ) ).
% order_refl
thf(fact_14_order__refl,axiom,
! [X2: $o > dedekind_preal] : ( ord_le6023059474077774659_preal @ X2 @ X2 ) ).
% order_refl
thf(fact_15_order__refl,axiom,
! [X2: $o > dedekind_real] : ( ord_le267469853237711487d_real @ X2 @ X2 ) ).
% order_refl
thf(fact_16_order__refl,axiom,
! [X2: dedekind_real] : ( ord_le2716243287969276982d_real @ X2 @ X2 ) ).
% order_refl
thf(fact_17_order__refl,axiom,
! [X2: dedekind_preal] : ( ord_le5604041210740703414_preal @ X2 @ X2 ) ).
% order_refl
thf(fact_18_order__refl,axiom,
! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% order_refl
thf(fact_19_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_20_dual__order_Orefl,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% dual_order.refl
thf(fact_21_dual__order_Orefl,axiom,
! [A: $o > num] : ( ord_less_eq_o_num @ A @ A ) ).
% dual_order.refl
thf(fact_22_dual__order_Orefl,axiom,
! [A: $o > dedekind_preal] : ( ord_le6023059474077774659_preal @ A @ A ) ).
% dual_order.refl
thf(fact_23_dual__order_Orefl,axiom,
! [A: $o > dedekind_real] : ( ord_le267469853237711487d_real @ A @ A ) ).
% dual_order.refl
thf(fact_24_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_25_dual__order_Orefl,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% dual_order.refl
thf(fact_26_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_27_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_28_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( K = L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_29_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_30_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_31_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_32_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_33_add__mono__thms__linordered__semiring_I1_J,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 @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_34_add__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ C @ D )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_35_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_36_add__left__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ C @ A ) @ ( plus_p3173629198307831117_preal @ C @ B ) ) ) ).
% add_left_mono
thf(fact_37_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_38_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_39_add_Oinverse__inverse,axiom,
! [A: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_40_neg__equal__iff__equal,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= ( uminus7714077491378687647d_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_41_add__diff__cancel__right_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_42_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_43_add__diff__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_44_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_45_add__diff__cancel__left_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_46_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_47_add__diff__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( minus_5539002012860128047d_real @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_48_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_49_diff__add__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_50_add__diff__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_51_minus__add__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_52_minus__add__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ ( plus_p4060926892116697567d_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_53_add__minus__cancel,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_54_minus__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( minus_5539002012860128047d_real @ A @ B ) )
= ( minus_5539002012860128047d_real @ B @ A ) ) ).
% minus_diff_eq
thf(fact_55_uminus__add__conv__diff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( minus_5539002012860128047d_real @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_56_diff__minus__eq__add,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( plus_p4060926892116697567d_real @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_57_diff__eq__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ( minus_5539002012860128047d_real @ A @ B )
= ( minus_5539002012860128047d_real @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_58_equation__minus__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ B ) )
= ( B
= ( uminus7714077491378687647d_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_59_minus__equation__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= B )
= ( ( uminus7714077491378687647d_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_60_minus__diff__commute,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ B ) @ A )
= ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_61_diff__right__commute,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ C ) @ B )
= ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_62_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_63_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_5539002012860128047d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( plus_p4060926892116697567d_real @ A2 @ ( uminus7714077491378687647d_real @ B2 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_64_diff__conv__add__uminus,axiom,
( minus_5539002012860128047d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( plus_p4060926892116697567d_real @ A2 @ ( uminus7714077491378687647d_real @ B2 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_65_group__cancel_Osub2,axiom,
! [B3: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B3
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( minus_5539002012860128047d_real @ A @ B3 )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ K ) @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_66_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: dedekind_real,V22: dedekind_real,U22: dedekind_real,Y2: dedekind_real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_p4060926892116697567d_real @ X22 @ V22 )
= ( plus_p4060926892116697567d_real @ U22 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_67_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: dedekind_preal,V22: dedekind_preal,U22: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V22 )
= ( plus_p3173629198307831117_preal @ U22 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_68_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: num,V22: num,U22: num,Y2: num] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_plus_num @ X22 @ V22 )
= ( plus_plus_num @ U22 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_69_real__trans__lemma,axiom,
! [X2: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: nat,V22: nat,U22: nat,Y2: nat] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_plus_nat @ X22 @ V22 )
= ( plus_plus_nat @ U22 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_70_mem__Collect__eq,axiom,
! [A: dedekind_preal,P: dedekind_preal > $o] :
( ( member6871284927547481791_preal @ A @ ( collec1132657498972982273_preal @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_71_Collect__mem__eq,axiom,
! [A3: set_Dedekind_preal] :
( ( collec1132657498972982273_preal
@ ^ [X3: dedekind_preal] : ( member6871284927547481791_preal @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_72_Collect__cong,axiom,
! [P: dedekind_preal > $o,Q: dedekind_preal > $o] :
( ! [X: dedekind_preal] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collec1132657498972982273_preal @ P )
= ( collec1132657498972982273_preal @ Q ) ) ) ).
% Collect_cong
thf(fact_73_diff__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_74_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_75_add__implies__diff,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ C @ B )
= A )
=> ( C
= ( minus_5539002012860128047d_real @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_76_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_77_diff__add__eq__diff__diff__swap,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( minus_5539002012860128047d_real @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_78_diff__add__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_79_diff__diff__eq2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_80_add__diff__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_81_eq__diff__eq,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( A
= ( minus_5539002012860128047d_real @ C @ B ) )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_82_diff__eq__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( minus_5539002012860128047d_real @ A @ B )
= C )
= ( A
= ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_83_group__cancel_Osub1,axiom,
! [A3: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A3
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( minus_5539002012860128047d_real @ A3 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_84_add_Oinverse__distrib__swap,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_85_group__cancel_Oneg1,axiom,
! [A3: dedekind_real,K: dedekind_real,A: dedekind_real] :
( ( A3
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( uminus7714077491378687647d_real @ A3 )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ K ) @ ( uminus7714077491378687647d_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_86_order__antisym__conv,axiom,
! [Y: $o > nat,X2: $o > nat] :
( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( ( ord_less_eq_o_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_87_order__antisym__conv,axiom,
! [Y: $o > num,X2: $o > num] :
( ( ord_less_eq_o_num @ Y @ X2 )
=> ( ( ord_less_eq_o_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_88_order__antisym__conv,axiom,
! [Y: $o > dedekind_preal,X2: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ Y @ X2 )
=> ( ( ord_le6023059474077774659_preal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_89_order__antisym__conv,axiom,
! [Y: $o > dedekind_real,X2: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ Y @ X2 )
=> ( ( ord_le267469853237711487d_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_90_order__antisym__conv,axiom,
! [Y: dedekind_real,X2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X2 )
=> ( ( ord_le2716243287969276982d_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_91_order__antisym__conv,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( ( ord_le5604041210740703414_preal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_92_order__antisym__conv,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_93_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_94_linorder__le__cases,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_95_linorder__le__cases,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_96_linorder__le__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_eq_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_97_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_98_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_100_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_101_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_102_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_103_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_104_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_105_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_106_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_107_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_108_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_109_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_110_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_111_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_113_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_114_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_115_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_116_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_117_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_linorder__linear,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
| ( ord_le2716243287969276982d_real @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_119_linorder__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
| ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_120_linorder__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_121_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_122_order__eq__refl,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( X2 = Y )
=> ( ord_less_eq_o_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_123_order__eq__refl,axiom,
! [X2: $o > num,Y: $o > num] :
( ( X2 = Y )
=> ( ord_less_eq_o_num @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_124_order__eq__refl,axiom,
! [X2: $o > dedekind_preal,Y: $o > dedekind_preal] :
( ( X2 = Y )
=> ( ord_le6023059474077774659_preal @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_125_order__eq__refl,axiom,
! [X2: $o > dedekind_real,Y: $o > dedekind_real] :
( ( X2 = Y )
=> ( ord_le267469853237711487d_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_126_order__eq__refl,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( X2 = Y )
=> ( ord_le2716243287969276982d_real @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_127_order__eq__refl,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 = Y )
=> ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_128_order__eq__refl,axiom,
! [X2: num,Y: num] :
( ( X2 = Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_129_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_130_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_131_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_132_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_133_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_134_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_135_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_136_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_137_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_138_order__subst2,axiom,
! [A: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_139_order__subst2,axiom,
! [A: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_140_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_141_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_142_order__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_143_order__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_144_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_145_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_146_order__subst1,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_147_order__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_eq_nat @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_148_order__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_149_order__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_150_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ A2 @ B2 )
& ( ord_less_eq_o_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_151_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > num,Z2: $o > num] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > num,B2: $o > num] :
( ( ord_less_eq_o_num @ A2 @ B2 )
& ( ord_less_eq_o_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_152_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > dedekind_preal,Z2: $o > dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > dedekind_preal,B2: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ A2 @ B2 )
& ( ord_le6023059474077774659_preal @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_153_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > dedekind_real,Z2: $o > dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > dedekind_real,B2: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ A2 @ B2 )
& ( ord_le267469853237711487d_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_154_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z2: dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ( ord_le2716243287969276982d_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_155_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z2: dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_156_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_157_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_158_le__fun__def,axiom,
( ord_less_eq_o_nat
= ( ^ [F2: $o > nat,G: $o > nat] :
! [X3: $o] : ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_159_le__fun__def,axiom,
( ord_less_eq_o_num
= ( ^ [F2: $o > num,G: $o > num] :
! [X3: $o] : ( ord_less_eq_num @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_160_le__fun__def,axiom,
( ord_le6023059474077774659_preal
= ( ^ [F2: $o > dedekind_preal,G: $o > dedekind_preal] :
! [X3: $o] : ( ord_le5604041210740703414_preal @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_161_le__fun__def,axiom,
( ord_le267469853237711487d_real
= ( ^ [F2: $o > dedekind_real,G: $o > dedekind_real] :
! [X3: $o] : ( ord_le2716243287969276982d_real @ ( F2 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_fun_def
thf(fact_162_le__funI,axiom,
! [F: $o > dedekind_real,G2: $o > dedekind_real] :
( ! [X: $o] : ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le267469853237711487d_real @ F @ G2 ) ) ).
% le_funI
thf(fact_163_le__funI,axiom,
! [F: $o > dedekind_preal,G2: $o > dedekind_preal] :
( ! [X: $o] : ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_le6023059474077774659_preal @ F @ G2 ) ) ).
% le_funI
thf(fact_164_le__funI,axiom,
! [F: $o > num,G2: $o > num] :
( ! [X: $o] : ( ord_less_eq_num @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_num @ F @ G2 ) ) ).
% le_funI
thf(fact_165_le__funI,axiom,
! [F: $o > nat,G2: $o > nat] :
( ! [X: $o] : ( ord_less_eq_nat @ ( F @ X ) @ ( G2 @ X ) )
=> ( ord_less_eq_o_nat @ F @ G2 ) ) ).
% le_funI
thf(fact_166_le__funE,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_167_le__funE,axiom,
! [F: $o > num,G2: $o > num,X2: $o] :
( ( ord_less_eq_o_num @ F @ G2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_168_le__funE,axiom,
! [F: $o > dedekind_preal,G2: $o > dedekind_preal,X2: $o] :
( ( ord_le6023059474077774659_preal @ F @ G2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_169_le__funE,axiom,
! [F: $o > dedekind_real,G2: $o > dedekind_real,X2: $o] :
( ( ord_le267469853237711487d_real @ F @ G2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funE
thf(fact_170_le__funD,axiom,
! [F: $o > nat,G2: $o > nat,X2: $o] :
( ( ord_less_eq_o_nat @ F @ G2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_171_le__funD,axiom,
! [F: $o > num,G2: $o > num,X2: $o] :
( ( ord_less_eq_o_num @ F @ G2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_172_le__funD,axiom,
! [F: $o > dedekind_preal,G2: $o > dedekind_preal,X2: $o] :
( ( ord_le6023059474077774659_preal @ F @ G2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_173_le__funD,axiom,
! [F: $o > dedekind_real,G2: $o > dedekind_real,X2: $o] :
( ( ord_le267469853237711487d_real @ F @ G2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ).
% le_funD
thf(fact_174_antisym,axiom,
! [A: $o > nat,B: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_175_antisym,axiom,
! [A: $o > num,B: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( ord_less_eq_o_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_176_antisym,axiom,
! [A: $o > dedekind_preal,B: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ A @ B )
=> ( ( ord_le6023059474077774659_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_177_antisym,axiom,
! [A: $o > dedekind_real,B: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ A @ B )
=> ( ( ord_le267469853237711487d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_178_antisym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_179_antisym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_180_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_181_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_182_dual__order_Otrans,axiom,
! [B: $o > nat,A: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ C @ B )
=> ( ord_less_eq_o_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_183_dual__order_Otrans,axiom,
! [B: $o > num,A: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ B @ A )
=> ( ( ord_less_eq_o_num @ C @ B )
=> ( ord_less_eq_o_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_184_dual__order_Otrans,axiom,
! [B: $o > dedekind_preal,A: $o > dedekind_preal,C: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ B @ A )
=> ( ( ord_le6023059474077774659_preal @ C @ B )
=> ( ord_le6023059474077774659_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_185_dual__order_Otrans,axiom,
! [B: $o > dedekind_real,A: $o > dedekind_real,C: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ B @ A )
=> ( ( ord_le267469853237711487d_real @ C @ B )
=> ( ord_le267469853237711487d_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_186_dual__order_Otrans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_187_dual__order_Otrans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_188_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_189_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_190_dual__order_Oantisym,axiom,
! [B: $o > nat,A: $o > nat] :
( ( ord_less_eq_o_nat @ B @ A )
=> ( ( ord_less_eq_o_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_191_dual__order_Oantisym,axiom,
! [B: $o > num,A: $o > num] :
( ( ord_less_eq_o_num @ B @ A )
=> ( ( ord_less_eq_o_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_192_dual__order_Oantisym,axiom,
! [B: $o > dedekind_preal,A: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ B @ A )
=> ( ( ord_le6023059474077774659_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_193_dual__order_Oantisym,axiom,
! [B: $o > dedekind_real,A: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ B @ A )
=> ( ( ord_le267469853237711487d_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_194_dual__order_Oantisym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_195_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_196_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_197_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_198_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > nat,B2: $o > nat] :
( ( ord_less_eq_o_nat @ B2 @ A2 )
& ( ord_less_eq_o_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_199_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > num,Z2: $o > num] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > num,B2: $o > num] :
( ( ord_less_eq_o_num @ B2 @ A2 )
& ( ord_less_eq_o_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_200_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > dedekind_preal,Z2: $o > dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > dedekind_preal,B2: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ B2 @ A2 )
& ( ord_le6023059474077774659_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_201_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: $o > dedekind_real,Z2: $o > dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [A2: $o > dedekind_real,B2: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ B2 @ A2 )
& ( ord_le267469853237711487d_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_202_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z2: dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ( ord_le2716243287969276982d_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_203_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z2: dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_204_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_205_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_206_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A4: dedekind_real,B4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_real,B4: dedekind_real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_207_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_208_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_209_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ).
% linorder_wlog
thf(fact_210_order__trans,axiom,
! [X2: $o > nat,Y: $o > nat,Z: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ Z )
=> ( ord_less_eq_o_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_211_order__trans,axiom,
! [X2: $o > num,Y: $o > num,Z: $o > num] :
( ( ord_less_eq_o_num @ X2 @ Y )
=> ( ( ord_less_eq_o_num @ Y @ Z )
=> ( ord_less_eq_o_num @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_212_order__trans,axiom,
! [X2: $o > dedekind_preal,Y: $o > dedekind_preal,Z: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ X2 @ Y )
=> ( ( ord_le6023059474077774659_preal @ Y @ Z )
=> ( ord_le6023059474077774659_preal @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_213_order__trans,axiom,
! [X2: $o > dedekind_real,Y: $o > dedekind_real,Z: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ X2 @ Y )
=> ( ( ord_le267469853237711487d_real @ Y @ Z )
=> ( ord_le267469853237711487d_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_214_order__trans,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z )
=> ( ord_le2716243287969276982d_real @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_215_order__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ( ord_le5604041210740703414_preal @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_216_order__trans,axiom,
! [X2: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_217_order__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% order_trans
thf(fact_218_order_Otrans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_219_order_Otrans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( ord_less_eq_o_num @ B @ C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% order.trans
thf(fact_220_order_Otrans,axiom,
! [A: $o > dedekind_preal,B: $o > dedekind_preal,C: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ A @ B )
=> ( ( ord_le6023059474077774659_preal @ B @ C )
=> ( ord_le6023059474077774659_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_221_order_Otrans,axiom,
! [A: $o > dedekind_real,B: $o > dedekind_real,C: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ A @ B )
=> ( ( ord_le267469853237711487d_real @ B @ C )
=> ( ord_le267469853237711487d_real @ A @ C ) ) ) ).
% order.trans
thf(fact_222_order_Otrans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% order.trans
thf(fact_223_order_Otrans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_224_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_225_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_226_order__antisym,axiom,
! [X2: $o > nat,Y: $o > nat] :
( ( ord_less_eq_o_nat @ X2 @ Y )
=> ( ( ord_less_eq_o_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_227_order__antisym,axiom,
! [X2: $o > num,Y: $o > num] :
( ( ord_less_eq_o_num @ X2 @ Y )
=> ( ( ord_less_eq_o_num @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_228_order__antisym,axiom,
! [X2: $o > dedekind_preal,Y: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ X2 @ Y )
=> ( ( ord_le6023059474077774659_preal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_229_order__antisym,axiom,
! [X2: $o > dedekind_real,Y: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ X2 @ Y )
=> ( ( ord_le267469853237711487d_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_230_order__antisym,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_231_order__antisym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_232_order__antisym,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_233_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_234_ord__le__eq__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( ord_less_eq_o_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_235_ord__le__eq__trans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( ord_less_eq_o_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_236_ord__le__eq__trans,axiom,
! [A: $o > dedekind_preal,B: $o > dedekind_preal,C: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le6023059474077774659_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_237_ord__le__eq__trans,axiom,
! [A: $o > dedekind_real,B: $o > dedekind_real,C: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le267469853237711487d_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_238_ord__le__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_239_ord__le__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_240_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_241_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_242_ord__eq__le__trans,axiom,
! [A: $o > nat,B: $o > nat,C: $o > nat] :
( ( A = B )
=> ( ( ord_less_eq_o_nat @ B @ C )
=> ( ord_less_eq_o_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_243_ord__eq__le__trans,axiom,
! [A: $o > num,B: $o > num,C: $o > num] :
( ( A = B )
=> ( ( ord_less_eq_o_num @ B @ C )
=> ( ord_less_eq_o_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_244_ord__eq__le__trans,axiom,
! [A: $o > dedekind_preal,B: $o > dedekind_preal,C: $o > dedekind_preal] :
( ( A = B )
=> ( ( ord_le6023059474077774659_preal @ B @ C )
=> ( ord_le6023059474077774659_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_245_ord__eq__le__trans,axiom,
! [A: $o > dedekind_real,B: $o > dedekind_real,C: $o > dedekind_real] :
( ( A = B )
=> ( ( ord_le267469853237711487d_real @ B @ C )
=> ( ord_le267469853237711487d_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_246_ord__eq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_247_ord__eq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_248_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_249_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_250_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > nat,Z2: $o > nat] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > nat,Y5: $o > nat] :
( ( ord_less_eq_o_nat @ X3 @ Y5 )
& ( ord_less_eq_o_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_251_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > num,Z2: $o > num] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > num,Y5: $o > num] :
( ( ord_less_eq_o_num @ X3 @ Y5 )
& ( ord_less_eq_o_num @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_252_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > dedekind_preal,Z2: $o > dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > dedekind_preal,Y5: $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ X3 @ Y5 )
& ( ord_le6023059474077774659_preal @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_253_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: $o > dedekind_real,Z2: $o > dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [X3: $o > dedekind_real,Y5: $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ X3 @ Y5 )
& ( ord_le267469853237711487d_real @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_254_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z2: dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [X3: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y5 )
& ( ord_le2716243287969276982d_real @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_255_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z2: dedekind_preal] : ( Y4 = Z2 ) )
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y5 )
& ( ord_le5604041210740703414_preal @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_256_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z2: num] : ( Y4 = Z2 ) )
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_257_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_258_le__cases3,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ Z ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ X2 )
=> ~ ( ord_le2716243287969276982d_real @ X2 @ Z ) )
=> ( ( ( ord_le2716243287969276982d_real @ X2 @ Z )
=> ~ ( ord_le2716243287969276982d_real @ Z @ Y ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ X2 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ Z )
=> ~ ( ord_le2716243287969276982d_real @ Z @ X2 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z @ X2 )
=> ~ ( ord_le2716243287969276982d_real @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_259_le__cases3,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ Z ) )
=> ( ( ( ord_le5604041210740703414_preal @ X2 @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ Y ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ X2 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ~ ( ord_le5604041210740703414_preal @ Z @ X2 ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z @ X2 )
=> ~ ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_260_le__cases3,axiom,
! [X2: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X2 @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Z ) )
=> ( ( ( ord_less_eq_num @ X2 @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X2 ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_num @ Z @ X2 )
=> ~ ( ord_less_eq_num @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_261_le__cases3,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_262_nle__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ A @ B ) )
= ( ( ord_le2716243287969276982d_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_263_nle__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_264_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_265_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_266_add__right__imp__eq,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_267_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_268_add__left__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_269_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_270_add_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_271_add_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ B @ ( plus_p3173629198307831117_preal @ A @ C ) )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.left_commute
thf(fact_272_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_273_add_Ocommute,axiom,
( plus_p4060926892116697567d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( plus_p4060926892116697567d_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_274_add_Ocommute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( plus_p3173629198307831117_preal @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_275_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_276_add_Oright__cancel,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= ( plus_p4060926892116697567d_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_277_add_Oleft__cancel,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= ( plus_p4060926892116697567d_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_278_add_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_279_add_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.assoc
thf(fact_280_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_281_group__cancel_Oadd2,axiom,
! [B3: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B3
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( plus_p4060926892116697567d_real @ A @ B3 )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_282_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_283_group__cancel_Oadd1,axiom,
! [A3: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A3
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( plus_p4060926892116697567d_real @ A3 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_284_group__cancel_Oadd1,axiom,
! [A3: nat,K: nat,A: nat,B: nat] :
( ( A3
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A3 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_285_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3173629198307831117_preal @ I @ K )
= ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_286_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_287_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_288_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_289_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_290_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_291_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_292_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_293_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_294_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_295_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_296_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_297_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_298_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_299_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_300_real__diff__def,axiom,
( minus_5539002012860128047d_real
= ( ^ [R: dedekind_real,S: dedekind_real] : ( plus_p4060926892116697567d_real @ R @ ( uminus7714077491378687647d_real @ S ) ) ) ) ).
% real_diff_def
thf(fact_301_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_302_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_303_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_304_add__right__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add_right_mono
thf(fact_305_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_306_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_307_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_308_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_309_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_310_verit__minus__simplify_I4_J,axiom,
! [B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_311_minus__diff__minus,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( uminus7714077491378687647d_real @ ( minus_5539002012860128047d_real @ A @ B ) ) ) ).
% minus_diff_minus
thf(fact_312_is__num__normalize_I8_J,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_313_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
= ( ord_le5604041210740703414_preal @ R2 @ S2 ) ) ).
% preal_add_le_cancel_left
thf(fact_314_preal__add__le__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
= ( ord_le5604041210740703414_preal @ R2 @ S2 ) ) ).
% preal_add_le_cancel_right
thf(fact_315_real__le__lemma,axiom,
! [U1: dedekind_preal,V22: dedekind_preal,U22: dedekind_preal,V1: dedekind_preal,X1: dedekind_preal,Y1: dedekind_preal,X22: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U1 @ V22 ) @ ( plus_p3173629198307831117_preal @ U22 @ V1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X1 @ V1 )
= ( plus_p3173629198307831117_preal @ U1 @ Y1 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V22 )
= ( plus_p3173629198307831117_preal @ U22 @ Y2 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y2 ) @ ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_316_preal__add__assoc,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y ) @ Z )
= ( plus_p3173629198307831117_preal @ X2 @ ( plus_p3173629198307831117_preal @ Y @ Z ) ) ) ).
% preal_add_assoc
thf(fact_317_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y5 @ X3 ) ) ) ).
% preal_add_commute
thf(fact_318_preal__trans__lemma,axiom,
! [X2: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y: dedekind_preal,Y2: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X2 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y ) )
=> ( ( ( plus_p3173629198307831117_preal @ X2 @ Y2 )
= ( plus_p3173629198307831117_preal @ X22 @ Y ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y2 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_319_preal__eq__le__imp__le,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ A @ B )
= ( plus_p3173629198307831117_preal @ C @ D ) )
=> ( ( ord_le5604041210740703414_preal @ C @ A )
=> ( ord_le5604041210740703414_preal @ B @ D ) ) ) ).
% preal_eq_le_imp_le
thf(fact_320_preal__add__left__cancel,axiom,
! [C: dedekind_preal,A: dedekind_preal,B: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ C @ A )
= ( plus_p3173629198307831117_preal @ C @ B ) )
=> ( A = B ) ) ).
% preal_add_left_cancel
thf(fact_321_preal__add__right__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R2 @ T )
= ( plus_p3173629198307831117_preal @ S2 @ T ) )
=> ( R2 = S2 ) ) ).
% preal_add_right_cancel
thf(fact_322_verit__comp__simplify1_I2_J,axiom,
! [A: $o > nat] : ( ord_less_eq_o_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_323_verit__comp__simplify1_I2_J,axiom,
! [A: $o > num] : ( ord_less_eq_o_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_324_verit__comp__simplify1_I2_J,axiom,
! [A: $o > dedekind_preal] : ( ord_le6023059474077774659_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_325_verit__comp__simplify1_I2_J,axiom,
! [A: $o > dedekind_real] : ( ord_le267469853237711487d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_326_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_327_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_328_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_329_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_330_verit__la__disequality,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A = B )
| ~ ( ord_le2716243287969276982d_real @ A @ B )
| ~ ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_331_verit__la__disequality,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A = B )
| ~ ( ord_le5604041210740703414_preal @ A @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_332_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_333_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_334_is__num__normalize_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_335_add__diff__add,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real,D: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) )
= ( plus_p4060926892116697567d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ ( minus_5539002012860128047d_real @ C @ D ) ) ) ).
% add_diff_add
thf(fact_336_preal__psup__le,axiom,
! [P: set_Dedekind_preal,Y6: dedekind_preal,X2: dedekind_preal] :
( ! [X4: dedekind_preal] :
( ( member6871284927547481791_preal @ X4 @ P )
=> ( ord_le5604041210740703414_preal @ X4 @ Y6 ) )
=> ( ( member6871284927547481791_preal @ X2 @ P )
=> ( ord_le5604041210740703414_preal @ X2 @ ( dedekind_psup @ P ) ) ) ) ).
% preal_psup_le
thf(fact_337_GreatestI2__order,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat,Q: ( $o > nat ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: $o > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_o_nat @ Y3 @ X2 ) )
=> ( ! [X: $o > nat] :
( ( P @ X )
=> ( ! [Y7: $o > nat] :
( ( P @ Y7 )
=> ( ord_less_eq_o_nat @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_338_GreatestI2__order,axiom,
! [P: ( $o > num ) > $o,X2: $o > num,Q: ( $o > num ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: $o > num] :
( ( P @ Y3 )
=> ( ord_less_eq_o_num @ Y3 @ X2 ) )
=> ( ! [X: $o > num] :
( ( P @ X )
=> ( ! [Y7: $o > num] :
( ( P @ Y7 )
=> ( ord_less_eq_o_num @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_o_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_339_GreatestI2__order,axiom,
! [P: ( $o > dedekind_preal ) > $o,X2: $o > dedekind_preal,Q: ( $o > dedekind_preal ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: $o > dedekind_preal] :
( ( P @ Y3 )
=> ( ord_le6023059474077774659_preal @ Y3 @ X2 ) )
=> ( ! [X: $o > dedekind_preal] :
( ( P @ X )
=> ( ! [Y7: $o > dedekind_preal] :
( ( P @ Y7 )
=> ( ord_le6023059474077774659_preal @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_6170817298126604234_preal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_340_GreatestI2__order,axiom,
! [P: ( $o > dedekind_real ) > $o,X2: $o > dedekind_real,Q: ( $o > dedekind_real ) > $o] :
( ( P @ X2 )
=> ( ! [Y3: $o > dedekind_real] :
( ( P @ Y3 )
=> ( ord_le267469853237711487d_real @ Y3 @ X2 ) )
=> ( ! [X: $o > dedekind_real] :
( ( P @ X )
=> ( ! [Y7: $o > dedekind_real] :
( ( P @ Y7 )
=> ( ord_le267469853237711487d_real @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1363326640887983160d_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_341_GreatestI2__order,axiom,
! [P: dedekind_real > $o,X2: dedekind_real,Q: dedekind_real > $o] :
( ( P @ X2 )
=> ( ! [Y3: dedekind_real] :
( ( P @ Y3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ X2 ) )
=> ( ! [X: dedekind_real] :
( ( P @ X )
=> ( ! [Y7: dedekind_real] :
( ( P @ Y7 )
=> ( ord_le2716243287969276982d_real @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_1105151008113787197d_real @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_342_GreatestI2__order,axiom,
! [P: dedekind_preal > $o,X2: dedekind_preal,Q: dedekind_preal > $o] :
( ( P @ X2 )
=> ( ! [Y3: dedekind_preal] :
( ( P @ Y3 )
=> ( ord_le5604041210740703414_preal @ Y3 @ X2 ) )
=> ( ! [X: dedekind_preal] :
( ( P @ X )
=> ( ! [Y7: dedekind_preal] :
( ( P @ Y7 )
=> ( ord_le5604041210740703414_preal @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_958373252487505263_preal @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_343_GreatestI2__order,axiom,
! [P: num > $o,X2: num,Q: num > $o] :
( ( P @ X2 )
=> ( ! [Y3: num] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X2 ) )
=> ( ! [X: num] :
( ( P @ X )
=> ( ! [Y7: num] :
( ( P @ Y7 )
=> ( ord_less_eq_num @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_num @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_344_GreatestI2__order,axiom,
! [P: nat > $o,X2: nat,Q: nat > $o] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest_nat @ P ) ) ) ) ) ).
% GreatestI2_order
thf(fact_345_Greatest__equality,axiom,
! [P: ( $o > nat ) > $o,X2: $o > nat] :
( ( P @ X2 )
=> ( ! [Y3: $o > nat] :
( ( P @ Y3 )
=> ( ord_less_eq_o_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_o_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_346_Greatest__equality,axiom,
! [P: ( $o > num ) > $o,X2: $o > num] :
( ( P @ X2 )
=> ( ! [Y3: $o > num] :
( ( P @ Y3 )
=> ( ord_less_eq_o_num @ Y3 @ X2 ) )
=> ( ( order_Greatest_o_num @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_347_Greatest__equality,axiom,
! [P: ( $o > dedekind_preal ) > $o,X2: $o > dedekind_preal] :
( ( P @ X2 )
=> ( ! [Y3: $o > dedekind_preal] :
( ( P @ Y3 )
=> ( ord_le6023059474077774659_preal @ Y3 @ X2 ) )
=> ( ( order_6170817298126604234_preal @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_348_Greatest__equality,axiom,
! [P: ( $o > dedekind_real ) > $o,X2: $o > dedekind_real] :
( ( P @ X2 )
=> ( ! [Y3: $o > dedekind_real] :
( ( P @ Y3 )
=> ( ord_le267469853237711487d_real @ Y3 @ X2 ) )
=> ( ( order_1363326640887983160d_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_349_Greatest__equality,axiom,
! [P: dedekind_real > $o,X2: dedekind_real] :
( ( P @ X2 )
=> ( ! [Y3: dedekind_real] :
( ( P @ Y3 )
=> ( ord_le2716243287969276982d_real @ Y3 @ X2 ) )
=> ( ( order_1105151008113787197d_real @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_350_Greatest__equality,axiom,
! [P: dedekind_preal > $o,X2: dedekind_preal] :
( ( P @ X2 )
=> ( ! [Y3: dedekind_preal] :
( ( P @ Y3 )
=> ( ord_le5604041210740703414_preal @ Y3 @ X2 ) )
=> ( ( order_958373252487505263_preal @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_351_Greatest__equality,axiom,
! [P: num > $o,X2: num] :
( ( P @ X2 )
=> ( ! [Y3: num] :
( ( P @ Y3 )
=> ( ord_less_eq_num @ Y3 @ X2 ) )
=> ( ( order_Greatest_num @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_352_Greatest__equality,axiom,
! [P: nat > $o,X2: nat] :
( ( P @ X2 )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) )
=> ( ( order_Greatest_nat @ P )
= X2 ) ) ) ).
% Greatest_equality
thf(fact_353_real__le__eq__diff,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X3: dedekind_real,Y5: dedekind_real] : ( ord_le2716243287969276982d_real @ ( minus_5539002012860128047d_real @ X3 @ Y5 ) @ zero_z580800474297136991d_real ) ) ) ).
% real_le_eq_diff
thf(fact_354_diff__left__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ( minus_5539002012860128047d_real @ A @ B )
= ( minus_5539002012860128047d_real @ A @ C ) )
=> ( B = C ) ) ).
% diff_left_imp_eq
thf(fact_355_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_nat
= ( ^ [X5: $o > $o > nat,Y8: $o > $o > nat] :
( ( ord_less_eq_o_nat @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq_o_nat @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_356_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_o_num
= ( ^ [X5: $o > $o > num,Y8: $o > $o > num] :
( ( ord_less_eq_o_num @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq_o_num @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_357_le__rel__bool__arg__iff,axiom,
( ord_le6171167672038760972_preal
= ( ^ [X5: $o > $o > dedekind_preal,Y8: $o > $o > dedekind_preal] :
( ( ord_le6023059474077774659_preal @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le6023059474077774659_preal @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_358_le__rel__bool__arg__iff,axiom,
( ord_le8943158644485376652d_real
= ( ^ [X5: $o > $o > dedekind_real,Y8: $o > $o > dedekind_real] :
( ( ord_le267469853237711487d_real @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le267469853237711487d_real @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_359_le__rel__bool__arg__iff,axiom,
( ord_le267469853237711487d_real
= ( ^ [X5: $o > dedekind_real,Y8: $o > dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le2716243287969276982d_real @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_360_le__rel__bool__arg__iff,axiom,
( ord_le6023059474077774659_preal
= ( ^ [X5: $o > dedekind_preal,Y8: $o > dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_le5604041210740703414_preal @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_361_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_num
= ( ^ [X5: $o > num,Y8: $o > num] :
( ( ord_less_eq_num @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq_num @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_362_le__rel__bool__arg__iff,axiom,
( ord_less_eq_o_nat
= ( ^ [X5: $o > nat,Y8: $o > nat] :
( ( ord_less_eq_nat @ ( X5 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq_nat @ ( X5 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_363_verit__minus__simplify_I3_J,axiom,
! [B: dedekind_real] :
( ( minus_5539002012860128047d_real @ zero_z580800474297136991d_real @ B )
= ( uminus7714077491378687647d_real @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_364_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_365_mult__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_366_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_367_mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_368_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_369_mult__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_370_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_371_mult__zero__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_372_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_373_mult__zero__left,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_374_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_375_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_376_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_377_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_378_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_379_add__cancel__right__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ A @ B ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_right
thf(fact_380_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_381_add__cancel__right__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_right_left
thf(fact_382_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_383_add__cancel__left__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_right
thf(fact_384_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_385_add__cancel__left__left,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ B @ A )
= A )
= ( B = zero_z580800474297136991d_real ) ) ).
% add_cancel_left_left
thf(fact_386_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_387_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_388_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_389_diff__self,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ A )
= zero_z580800474297136991d_real ) ).
% diff_self
thf(fact_390_diff__0__right,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% diff_0_right
thf(fact_391_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_392_diff__zero,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% diff_zero
thf(fact_393_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_394_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ A @ A )
= zero_z580800474297136991d_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_395_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_396_add_Oinverse__neutral,axiom,
( ( uminus7714077491378687647d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% add.inverse_neutral
thf(fact_397_neg__0__equal__iff__equal,axiom,
! [A: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( uminus7714077491378687647d_real @ A ) )
= ( zero_z580800474297136991d_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_398_neg__equal__0__iff__equal,axiom,
! [A: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_399_mult__minus__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( uminus7714077491378687647d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ).
% mult_minus_left
thf(fact_400_minus__mult__minus,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( times_2157731159493324635d_real @ A @ B ) ) ).
% minus_mult_minus
thf(fact_401_mult__minus__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( uminus7714077491378687647d_real @ ( times_2157731159493324635d_real @ A @ B ) ) ) ).
% mult_minus_right
thf(fact_402_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_403_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_404_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_405_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_406_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_407_ab__left__minus,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= zero_z580800474297136991d_real ) ).
% ab_left_minus
thf(fact_408_add_Oright__inverse,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( uminus7714077491378687647d_real @ A ) )
= zero_z580800474297136991d_real ) ).
% add.right_inverse
thf(fact_409_diff__0,axiom,
! [A: dedekind_real] :
( ( minus_5539002012860128047d_real @ zero_z580800474297136991d_real @ A )
= ( uminus7714077491378687647d_real @ A ) ) ).
% diff_0
thf(fact_410_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_411_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( 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 @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_412_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_413_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_414_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_415_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_416_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_417_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_418_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_419_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_420_mult__right__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_421_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_422_mult__left__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_423_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_424_no__zero__divisors,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real ) ) ) ).
% no_zero_divisors
thf(fact_425_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_426_divisors__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divisors_zero
thf(fact_427_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_428_mult__not__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real )
=> ( ( A != zero_z580800474297136991d_real )
& ( B != zero_z580800474297136991d_real ) ) ) ).
% mult_not_zero
thf(fact_429_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_430_zero__reorient,axiom,
! [X2: dedekind_real] :
( ( zero_z580800474297136991d_real = X2 )
= ( X2 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_431_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_432_mult_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ B @ ( times_3000655703912201937_preal @ A @ C ) )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_433_mult_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( times_2157731159493324635d_real @ A @ C ) )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_434_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_435_mult_Ocommute,axiom,
( times_3000655703912201937_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( times_3000655703912201937_preal @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_436_mult_Ocommute,axiom,
( times_2157731159493324635d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_437_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_438_mult_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_439_mult_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_440_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_441_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_442_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_443_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_444_combine__common__factor,axiom,
! [A: dedekind_preal,E: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ E ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ B @ E ) @ C ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_445_combine__common__factor,axiom,
! [A: dedekind_real,E: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ E ) @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ B @ E ) @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_446_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_447_distrib__right,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ C ) @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% distrib_right
thf(fact_448_distrib__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_449_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_450_distrib__left,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ B ) @ ( times_3000655703912201937_preal @ A @ C ) ) ) ).
% distrib_left
thf(fact_451_distrib__left,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_452_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_453_comm__semiring__class_Odistrib,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ C ) @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_454_comm__semiring__class_Odistrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_455_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_456_ring__class_Oring__distribs_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_457_ring__class_Oring__distribs_I2_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_458_left__diff__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_459_right__diff__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_460_left__diff__distrib_H,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ B @ C ) @ A )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ B @ A ) @ ( times_2157731159493324635d_real @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_461_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_462_right__diff__distrib_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_463_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_464_square__eq__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ A )
= ( times_2157731159493324635d_real @ B @ B ) )
= ( ( A = B )
| ( A
= ( uminus7714077491378687647d_real @ B ) ) ) ) ).
% square_eq_iff
thf(fact_465_minus__mult__commute,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( times_2157731159493324635d_real @ A @ ( uminus7714077491378687647d_real @ B ) ) ) ).
% minus_mult_commute
thf(fact_466_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_467_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_468_add_Ogroup__left__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_469_add_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.comm_neutral
thf(fact_470_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_471_comm__monoid__add__class_Oadd__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_472_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_473_verit__sum__simplify,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% verit_sum_simplify
thf(fact_474_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_475_eq__iff__diff__eq__0,axiom,
( ( ^ [Y4: dedekind_real,Z2: dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( minus_5539002012860128047d_real @ A2 @ B2 )
= zero_z580800474297136991d_real ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_476_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y2: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X2: dedekind_preal,Y: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y2 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ X1 ) @ ( times_3000655703912201937_preal @ Y @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ Y2 ) @ ( times_3000655703912201937_preal @ Y @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ X22 ) @ ( times_3000655703912201937_preal @ Y @ Y2 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ Y1 ) @ ( times_3000655703912201937_preal @ Y @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_477_preal__add__mult__distrib2,axiom,
! [W: dedekind_preal,X2: dedekind_preal,Y: dedekind_preal] :
( ( times_3000655703912201937_preal @ W @ ( plus_p3173629198307831117_preal @ X2 @ Y ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W @ X2 ) @ ( times_3000655703912201937_preal @ W @ Y ) ) ) ).
% preal_add_mult_distrib2
thf(fact_478_preal__add__mult__distrib,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,W: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X2 @ Y ) @ W )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X2 @ W ) @ ( times_3000655703912201937_preal @ Y @ W ) ) ) ).
% preal_add_mult_distrib
thf(fact_479_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,W: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z22 ) @ W )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W ) @ ( times_2157731159493324635d_real @ Z22 @ W ) ) ) ).
% real_add_mult_distrib
thf(fact_480_mult__diff__mult,axiom,
! [X2: dedekind_real,Y: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ X2 @ Y ) @ ( times_2157731159493324635d_real @ A @ B ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ X2 @ ( minus_5539002012860128047d_real @ Y @ B ) ) @ ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ X2 @ A ) @ B ) ) ) ).
% mult_diff_mult
thf(fact_481_square__diff__square__factored,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ X2 @ X2 ) @ ( times_2157731159493324635d_real @ Y @ Y ) )
= ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ X2 @ Y ) @ ( minus_5539002012860128047d_real @ X2 @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_482_eq__add__iff2,axiom,
! [A: dedekind_real,E: dedekind_real,C: dedekind_real,B: dedekind_real,D: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ E ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ B @ E ) @ D ) )
= ( C
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_483_eq__add__iff1,axiom,
! [A: dedekind_real,E: dedekind_real,C: dedekind_real,B: dedekind_real,D: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ E ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ B @ E ) @ D ) )
= ( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_484_add__nonpos__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_485_add__nonneg__eq__0__iff,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_486_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_487_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_488_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_489_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_490_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_491_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_492_neg__eq__iff__add__eq__0,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= B )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_493_eq__neg__iff__add__eq__0,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ B ) )
= ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_494_add_Oinverse__unique,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_495_ab__group__add__class_Oab__left__minus,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= zero_z580800474297136991d_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_496_add__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( B
= ( uminus7714077491378687647d_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_497_add__scale__eq__noteq,axiom,
! [R2: dedekind_real,A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( R2 != zero_z580800474297136991d_real )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_p4060926892116697567d_real @ A @ ( times_2157731159493324635d_real @ R2 @ C ) )
!= ( plus_p4060926892116697567d_real @ B @ ( times_2157731159493324635d_real @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_498_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_499_inf__period_I1_J,axiom,
! [P: dedekind_real > $o,D2: dedekind_real,Q: dedekind_real > $o] :
( ! [X: dedekind_real,K2: dedekind_real] :
( ( P @ X )
= ( P @ ( minus_5539002012860128047d_real @ X @ ( times_2157731159493324635d_real @ K2 @ D2 ) ) ) )
=> ( ! [X: dedekind_real,K2: dedekind_real] :
( ( Q @ X )
= ( Q @ ( minus_5539002012860128047d_real @ X @ ( times_2157731159493324635d_real @ K2 @ D2 ) ) ) )
=> ! [X6: dedekind_real,K3: dedekind_real] :
( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P @ ( minus_5539002012860128047d_real @ X6 @ ( times_2157731159493324635d_real @ K3 @ D2 ) ) )
& ( Q @ ( minus_5539002012860128047d_real @ X6 @ ( times_2157731159493324635d_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_500_inf__period_I2_J,axiom,
! [P: dedekind_real > $o,D2: dedekind_real,Q: dedekind_real > $o] :
( ! [X: dedekind_real,K2: dedekind_real] :
( ( P @ X )
= ( P @ ( minus_5539002012860128047d_real @ X @ ( times_2157731159493324635d_real @ K2 @ D2 ) ) ) )
=> ( ! [X: dedekind_real,K2: dedekind_real] :
( ( Q @ X )
= ( Q @ ( minus_5539002012860128047d_real @ X @ ( times_2157731159493324635d_real @ K2 @ D2 ) ) ) )
=> ! [X6: dedekind_real,K3: dedekind_real] :
( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P @ ( minus_5539002012860128047d_real @ X6 @ ( times_2157731159493324635d_real @ K3 @ D2 ) ) )
| ( Q @ ( minus_5539002012860128047d_real @ X6 @ ( times_2157731159493324635d_real @ K3 @ D2 ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_501_crossproduct__eq,axiom,
! [W: dedekind_real,Y: dedekind_real,X2: dedekind_real,Z: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ W @ Y ) @ ( times_2157731159493324635d_real @ X2 @ Z ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ W @ Z ) @ ( times_2157731159493324635d_real @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_502_crossproduct__eq,axiom,
! [W: nat,Y: nat,X2: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X2 @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X2 @ Y ) ) )
= ( ( W = X2 )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_503_crossproduct__noteq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ D ) )
!= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ D ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_504_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_505_add__0__iff,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B
= ( plus_p4060926892116697567d_real @ B @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% add_0_iff
thf(fact_506_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_507_real__mult__assoc,axiom,
! [Z1: dedekind_real,Z22: dedekind_real,Z3: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ Z1 @ Z22 ) @ Z3 )
= ( times_2157731159493324635d_real @ Z1 @ ( times_2157731159493324635d_real @ Z22 @ Z3 ) ) ) ).
% real_mult_assoc
thf(fact_508_preal__mult__assoc,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ X2 @ Y ) @ Z )
= ( times_3000655703912201937_preal @ X2 @ ( times_3000655703912201937_preal @ Y @ Z ) ) ) ).
% preal_mult_assoc
thf(fact_509_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z4: dedekind_real,W2: dedekind_real] : ( times_2157731159493324635d_real @ W2 @ Z4 ) ) ) ).
% real_mult_commute
thf(fact_510_preal__mult__commute,axiom,
( times_3000655703912201937_preal
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] : ( times_3000655703912201937_preal @ Y5 @ X3 ) ) ) ).
% preal_mult_commute
thf(fact_511_add_Ogroup__axioms,axiom,
group_Dedekind_real @ plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ uminus7714077491378687647d_real ).
% add.group_axioms
thf(fact_512_diff__numeral__special_I12_J,axiom,
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= zero_z580800474297136991d_real ) ).
% diff_numeral_special(12)
thf(fact_513_add__neg__numeral__special_I7_J,axiom,
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= zero_z580800474297136991d_real ) ).
% add_neg_numeral_special(7)
thf(fact_514_add__neg__numeral__special_I8_J,axiom,
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ one_on6069100329679821595d_real )
= zero_z580800474297136991d_real ) ).
% add_neg_numeral_special(8)
thf(fact_515_inverse__diff__inverse,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( uminus7714077491378687647d_real @ ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ ( minus_5539002012860128047d_real @ A @ B ) ) @ ( invers3762989301784728874d_real @ B ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_516_inverse__eq__iff__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
= ( A = B ) ) ).
% inverse_eq_iff_eq
thf(fact_517_inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ).
% inverse_inverse_eq
thf(fact_518_division__ring__divide__zero,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% division_ring_divide_zero
thf(fact_519_divide__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ C )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_520_divide__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ C @ A )
= ( divide9119111104558704680d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_521_divide__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divide_eq_0_iff
thf(fact_522_div__by__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% div_by_0
thf(fact_523_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_524_div__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% div_0
thf(fact_525_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_526_mult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% mult_1
thf(fact_527_mult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% mult_1
thf(fact_528_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_529_mult_Oright__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.right_neutral
thf(fact_530_mult_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.right_neutral
thf(fact_531_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_532_times__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_533_divide__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_534_divide__divide__eq__left,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ A @ B ) @ C )
= ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_535_times__divide__eq__left,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( times_2157731159493324635d_real @ ( divide9119111104558704680d_real @ B @ C ) @ A )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_536_div__by__1,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% div_by_1
thf(fact_537_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_538_inverse__nonzero__iff__nonzero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_539_inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% inverse_zero
thf(fact_540_inverse__mult__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( invers3762989301784728874d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) ) ) ).
% inverse_mult_distrib
thf(fact_541_inverse__eq__1__iff,axiom,
! [X2: dedekind_real] :
( ( ( invers3762989301784728874d_real @ X2 )
= one_on6069100329679821595d_real )
= ( X2 = one_on6069100329679821595d_real ) ) ).
% inverse_eq_1_iff
thf(fact_542_inverse__1,axiom,
( ( invers3762989301784728874d_real @ one_on6069100329679821595d_real )
= one_on6069100329679821595d_real ) ).
% inverse_1
thf(fact_543_inverse__minus__eq,axiom,
! [A: dedekind_real] :
( ( invers3762989301784728874d_real @ ( uminus7714077491378687647d_real @ A ) )
= ( uminus7714077491378687647d_real @ ( invers3762989301784728874d_real @ A ) ) ) ).
% inverse_minus_eq
thf(fact_544_inverse__divide,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( invers3762989301784728874d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ B @ A ) ) ).
% inverse_divide
thf(fact_545_mult__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right2
thf(fact_546_mult__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right1
thf(fact_547_mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left2
thf(fact_548_mult__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left1
thf(fact_549_diff__numeral__special_I9_J,axiom,
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real )
= zero_z580800474297136991d_real ) ).
% diff_numeral_special(9)
thf(fact_550_mult__divide__mult__cancel__left__if,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( C = zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= zero_z580800474297136991d_real ) )
& ( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_551_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_552_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_553_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_554_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_555_nonzero__mult__div__cancel__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_556_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_557_nonzero__mult__div__cancel__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_558_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_559_divide__self__if,axiom,
! [A: dedekind_real] :
( ( ( A = zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= zero_z580800474297136991d_real ) )
& ( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ) ).
% divide_self_if
thf(fact_560_divide__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% divide_self
thf(fact_561_one__eq__divide__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( one_on6069100329679821595d_real
= ( divide9119111104558704680d_real @ A @ B ) )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_562_divide__eq__1__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_563_div__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% div_self
thf(fact_564_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_565_mult__minus1,axiom,
! [Z: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ Z )
= ( uminus7714077491378687647d_real @ Z ) ) ).
% mult_minus1
thf(fact_566_mult__minus1__right,axiom,
! [Z: dedekind_real] :
( ( times_2157731159493324635d_real @ Z @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ Z ) ) ).
% mult_minus1_right
thf(fact_567_divide__minus1,axiom,
! [X2: dedekind_real] :
( ( divide9119111104558704680d_real @ X2 @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ X2 ) ) ).
% divide_minus1
thf(fact_568_nonzero__divide__mult__cancel__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ B @ ( times_2157731159493324635d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_569_nonzero__divide__mult__cancel__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_570_right__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ ( invers3762989301784728874d_real @ A ) )
= one_on6069100329679821595d_real ) ) ).
% right_inverse
thf(fact_571_left__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ A )
= one_on6069100329679821595d_real ) ) ).
% left_inverse
thf(fact_572_field__class_Ofield__inverse,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ A )
= one_on6069100329679821595d_real ) ) ).
% field_class.field_inverse
thf(fact_573_field__class_Ofield__divide__inverse,axiom,
( divide9119111104558704680d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ A2 @ ( invers3762989301784728874d_real @ B2 ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_574_times__divide__times__eq,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real,W: dedekind_real] :
( ( times_2157731159493324635d_real @ ( divide9119111104558704680d_real @ X2 @ Y ) @ ( divide9119111104558704680d_real @ Z @ W ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ X2 @ Z ) @ ( times_2157731159493324635d_real @ Y @ W ) ) ) ).
% times_divide_times_eq
thf(fact_575_divide__inverse,axiom,
( divide9119111104558704680d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ A2 @ ( invers3762989301784728874d_real @ B2 ) ) ) ) ).
% divide_inverse
thf(fact_576_inverse__unique,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= one_on6069100329679821595d_real )
=> ( ( invers3762989301784728874d_real @ A )
= B ) ) ).
% inverse_unique
thf(fact_577_divide__divide__times__eq,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real,W: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ X2 @ Y ) @ ( divide9119111104558704680d_real @ Z @ W ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ X2 @ W ) @ ( times_2157731159493324635d_real @ Y @ Z ) ) ) ).
% divide_divide_times_eq
thf(fact_578_divide__inverse__commute,axiom,
( divide9119111104558704680d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ B2 ) @ A2 ) ) ) ).
% divide_inverse_commute
thf(fact_579_divide__divide__eq__left_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ A @ B ) @ C )
= ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_580_mult__commute__imp__mult__inverse__commute,axiom,
! [Y: dedekind_real,X2: dedekind_real] :
( ( ( times_2157731159493324635d_real @ Y @ X2 )
= ( times_2157731159493324635d_real @ X2 @ Y ) )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ Y ) @ X2 )
= ( times_2157731159493324635d_real @ X2 @ ( invers3762989301784728874d_real @ Y ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_581_real__divide__def,axiom,
( divide9119111104558704680d_real
= ( ^ [R: dedekind_real,S: dedekind_real] : ( times_2157731159493324635d_real @ R @ ( invers3762989301784728874d_real @ S ) ) ) ) ).
% real_divide_def
thf(fact_582_divide__preal__def,axiom,
( divide4190755330972744004_preal
= ( ^ [R: dedekind_preal,S: dedekind_preal] : ( times_3000655703912201937_preal @ R @ ( invers3090987106763476162_preal @ S ) ) ) ) ).
% divide_preal_def
thf(fact_583_preal__mult__inverse,axiom,
! [R2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( invers3090987106763476162_preal @ R2 ) @ R2 )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse
thf(fact_584_preal__mult__inverse__right,axiom,
! [R2: dedekind_preal] :
( ( times_3000655703912201937_preal @ R2 @ ( invers3090987106763476162_preal @ R2 ) )
= one_on9143529541772854033_preal ) ).
% preal_mult_inverse_right
thf(fact_585_divide__eq__minus__1__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( ( B != zero_z580800474297136991d_real )
& ( A
= ( uminus7714077491378687647d_real @ B ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_586_add__divide__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ C ) @ ( divide9119111104558704680d_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_587_diff__divide__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ C )
= ( minus_5539002012860128047d_real @ ( divide9119111104558704680d_real @ A @ C ) @ ( divide9119111104558704680d_real @ B @ C ) ) ) ).
% diff_divide_distrib
thf(fact_588_minus__divide__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ B ) ) ).
% minus_divide_left
thf(fact_589_minus__divide__divide,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ).
% minus_divide_divide
thf(fact_590_minus__divide__right,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ A @ ( uminus7714077491378687647d_real @ B ) ) ) ).
% minus_divide_right
thf(fact_591_inverse__eq__divide,axiom,
( invers3762989301784728874d_real
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real ) ) ).
% inverse_eq_divide
thf(fact_592_inverse__eq__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
=> ( A = B ) ) ).
% inverse_eq_imp_eq
thf(fact_593_one__reorient,axiom,
! [X2: dedekind_preal] :
( ( one_on9143529541772854033_preal = X2 )
= ( X2 = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_594_one__reorient,axiom,
! [X2: dedekind_real] :
( ( one_on6069100329679821595d_real = X2 )
= ( X2 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_595_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_596_group_Oinverse__distrib__swap,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( Inverse @ ( F @ A @ B ) )
= ( F @ ( Inverse @ B ) @ ( Inverse @ A ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_597_group_Ogroup__left__neutral,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( F @ Z @ A )
= A ) ) ).
% group.group_left_neutral
thf(fact_598_group_Oinverse__neutral,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( Inverse @ Z )
= Z ) ) ).
% group.inverse_neutral
thf(fact_599_group_Oinverse__inverse,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( Inverse @ ( Inverse @ A ) )
= A ) ) ).
% group.inverse_inverse
thf(fact_600_group_Oinverse__unique,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( ( F @ A @ B )
= Z )
=> ( ( Inverse @ A )
= B ) ) ) ).
% group.inverse_unique
thf(fact_601_group_Oright__inverse,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( F @ A @ ( Inverse @ A ) )
= Z ) ) ).
% group.right_inverse
thf(fact_602_group_Oright__cancel,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( ( F @ B @ A )
= ( F @ C @ A ) )
= ( B = C ) ) ) ).
% group.right_cancel
thf(fact_603_group_Oleft__inverse,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( F @ ( Inverse @ A ) @ A )
= Z ) ) ).
% group.left_inverse
thf(fact_604_group_Oleft__cancel,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,Z: dedekind_real,Inverse: dedekind_real > dedekind_real,A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( group_Dedekind_real @ F @ Z @ Inverse )
=> ( ( ( F @ A @ B )
= ( F @ A @ C ) )
= ( B = C ) ) ) ).
% group.left_cancel
thf(fact_605_nonzero__imp__inverse__nonzero,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ A )
!= zero_z580800474297136991d_real ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_606_nonzero__inverse__inverse__eq,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( invers3762989301784728874d_real @ A ) )
= A ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_607_nonzero__inverse__eq__imp__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= ( invers3762989301784728874d_real @ B ) )
=> ( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( A = B ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_608_nonzero__inverse__eq__divide,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ A )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ A ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_609_inverse__zero__imp__zero,axiom,
! [A: dedekind_real] :
( ( ( invers3762989301784728874d_real @ A )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ).
% inverse_zero_imp_zero
thf(fact_610_right__inverse__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_611_field__class_Ofield__inverse__zero,axiom,
( ( invers3762989301784728874d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% field_class.field_inverse_zero
thf(fact_612_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_613_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_614_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_615_nonzero__inverse__mult__distrib,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( times_2157731159493324635d_real @ A @ B ) )
= ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ B ) @ ( invers3762989301784728874d_real @ A ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_616_mult_Ocomm__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.comm_neutral
thf(fact_617_mult_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.comm_neutral
thf(fact_618_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_619_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_620_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_621_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_622_nonzero__inverse__minus__eq,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( invers3762989301784728874d_real @ ( uminus7714077491378687647d_real @ A ) )
= ( uminus7714077491378687647d_real @ ( invers3762989301784728874d_real @ A ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_623_frac__eq__eq,axiom,
! [Y: dedekind_real,Z: dedekind_real,X2: dedekind_real,W: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( Z != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ X2 @ Y )
= ( divide9119111104558704680d_real @ W @ Z ) )
= ( ( times_2157731159493324635d_real @ X2 @ Z )
= ( times_2157731159493324635d_real @ W @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_624_divide__eq__eq,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( ( divide9119111104558704680d_real @ B @ C )
= A )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ A @ C ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq
thf(fact_625_eq__divide__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ C )
= B ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq
thf(fact_626_divide__eq__imp,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( B
= ( times_2157731159493324635d_real @ A @ C ) )
=> ( ( divide9119111104558704680d_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_627_eq__divide__imp,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= B )
=> ( A
= ( divide9119111104558704680d_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_628_nonzero__divide__eq__eq,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ B @ C )
= A )
= ( B
= ( times_2157731159493324635d_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_629_nonzero__eq__divide__eq,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( A
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( times_2157731159493324635d_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_630_nonzero__minus__divide__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ A @ ( uminus7714077491378687647d_real @ B ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_631_nonzero__minus__divide__divide,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( uminus7714077491378687647d_real @ A ) @ ( uminus7714077491378687647d_real @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_632_real__mult__inverse__left,axiom,
! [X2: dedekind_real] :
( ( X2 != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ X2 ) @ X2 )
= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left
thf(fact_633_preal__mult__1,axiom,
! [Z: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z )
= Z ) ).
% preal_mult_1
thf(fact_634_real__mult__1,axiom,
! [Z: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z )
= Z ) ).
% real_mult_1
thf(fact_635_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_636_inverse__add,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ ( invers3762989301784728874d_real @ A ) ) @ ( invers3762989301784728874d_real @ B ) ) ) ) ) ).
% inverse_add
thf(fact_637_division__ring__inverse__add,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ ( plus_p4060926892116697567d_real @ A @ B ) ) @ ( invers3762989301784728874d_real @ B ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_638_division__ring__inverse__diff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( invers3762989301784728874d_real @ A ) @ ( invers3762989301784728874d_real @ B ) )
= ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ ( invers3762989301784728874d_real @ A ) @ ( minus_5539002012860128047d_real @ B @ A ) ) @ ( invers3762989301784728874d_real @ B ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_639_add__divide__eq__if__simps_I2_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ Z ) @ B )
= B ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ Z ) @ B )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ A @ ( times_2157731159493324635d_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_640_add__divide__eq__if__simps_I1_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ A @ ( divide9119111104558704680d_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ A @ ( divide9119111104558704680d_real @ B @ Z ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_641_add__frac__eq,axiom,
! [Y: dedekind_real,Z: dedekind_real,X2: dedekind_real,W: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X2 @ Y ) @ ( divide9119111104558704680d_real @ W @ Z ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ X2 @ Z ) @ ( times_2157731159493324635d_real @ W @ Y ) ) @ ( times_2157731159493324635d_real @ Y @ Z ) ) ) ) ) ).
% add_frac_eq
thf(fact_642_add__frac__num,axiom,
! [Y: dedekind_real,X2: dedekind_real,Z: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X2 @ Y ) @ Z )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X2 @ ( times_2157731159493324635d_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_643_add__num__frac,axiom,
! [Y: dedekind_real,Z: dedekind_real,X2: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ Z @ ( divide9119111104558704680d_real @ X2 @ Y ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X2 @ ( times_2157731159493324635d_real @ Z @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_644_add__divide__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ X2 @ ( divide9119111104558704680d_real @ Y @ Z ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% add_divide_eq_iff
thf(fact_645_divide__add__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X2 @ Z ) @ Y )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X2 @ ( times_2157731159493324635d_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_add_eq_iff
thf(fact_646_add__divide__eq__if__simps_I4_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ A @ ( divide9119111104558704680d_real @ B @ Z ) )
= A ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ A @ ( divide9119111104558704680d_real @ B @ Z ) )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_647_diff__frac__eq,axiom,
! [Y: dedekind_real,Z: dedekind_real,X2: dedekind_real,W: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( divide9119111104558704680d_real @ X2 @ Y ) @ ( divide9119111104558704680d_real @ W @ Z ) )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ X2 @ Z ) @ ( times_2157731159493324635d_real @ W @ Y ) ) @ ( times_2157731159493324635d_real @ Y @ Z ) ) ) ) ) ).
% diff_frac_eq
thf(fact_648_diff__divide__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ X2 @ ( divide9119111104558704680d_real @ Y @ Z ) )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ X2 @ Z ) @ Y ) @ Z ) ) ) ).
% diff_divide_eq_iff
thf(fact_649_divide__diff__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( divide9119111104558704680d_real @ X2 @ Z ) @ Y )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ X2 @ ( times_2157731159493324635d_real @ Y @ Z ) ) @ Z ) ) ) ).
% divide_diff_eq_iff
thf(fact_650_eq__minus__divide__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ B @ C ) ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ C )
= ( uminus7714077491378687647d_real @ B ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_651_minus__divide__eq__eq,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ B @ C ) )
= A )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ B )
= ( times_2157731159493324635d_real @ A @ C ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_652_nonzero__neg__divide__eq__eq,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) )
= C )
= ( ( uminus7714077491378687647d_real @ A )
= ( times_2157731159493324635d_real @ C @ B ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_653_nonzero__neg__divide__eq__eq2,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( C
= ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ B ) ) )
= ( ( times_2157731159493324635d_real @ C @ B )
= ( uminus7714077491378687647d_real @ A ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_654_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_655_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_656_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_657_square__eq__1__iff,axiom,
! [X2: dedekind_real] :
( ( ( times_2157731159493324635d_real @ X2 @ X2 )
= one_on6069100329679821595d_real )
= ( ( X2 = one_on6069100329679821595d_real )
| ( X2
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ) ) ).
% square_eq_1_iff
thf(fact_658_minus__divide__add__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ X2 @ Z ) ) @ Y )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ X2 ) @ ( times_2157731159493324635d_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_659_add__divide__eq__if__simps_I3_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ Z ) ) @ B )
= B ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ Z ) ) @ B )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ ( times_2157731159493324635d_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_660_minus__divide__diff__eq__iff,axiom,
! [Z: dedekind_real,X2: dedekind_real,Y: dedekind_real] :
( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ X2 @ Z ) ) @ Y )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ X2 ) @ ( times_2157731159493324635d_real @ Y @ Z ) ) @ Z ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_661_add__divide__eq__if__simps_I5_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( divide9119111104558704680d_real @ A @ Z ) @ B )
= ( uminus7714077491378687647d_real @ B ) ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( divide9119111104558704680d_real @ A @ Z ) @ B )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ A @ ( times_2157731159493324635d_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_662_add__divide__eq__if__simps_I6_J,axiom,
! [Z: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z = zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ Z ) ) @ B )
= ( uminus7714077491378687647d_real @ B ) ) )
& ( ( Z != zero_z580800474297136991d_real )
=> ( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( divide9119111104558704680d_real @ A @ Z ) ) @ B )
= ( divide9119111104558704680d_real @ ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ A ) @ ( times_2157731159493324635d_real @ B @ Z ) ) @ Z ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_663_real__mult__inverse__left__ex,axiom,
! [X2: dedekind_real] :
( ( X2 != zero_z580800474297136991d_real )
=> ~ ! [Y3: dedekind_real] :
( ( times_2157731159493324635d_real @ Y3 @ X2 )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_664_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_665_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_666_square__diff__one__factored,axiom,
! [X2: dedekind_real] :
( ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ X2 @ X2 ) @ one_on6069100329679821595d_real )
= ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ X2 @ one_on6069100329679821595d_real ) @ ( minus_5539002012860128047d_real @ X2 @ one_on6069100329679821595d_real ) ) ) ).
% square_diff_one_factored
thf(fact_667_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_668_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_669_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_670_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_671_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_672_div__mult__mult1__if,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( C = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= zero_zero_nat ) )
& ( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_mult1_if
thf(fact_673_div__mult__mult2,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_674_div__add__self1,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self1
thf(fact_675_div__add__self2,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% div_add_self2
thf(fact_676_bits__div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% bits_div_by_1
thf(fact_677_bits__div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% bits_div_0
thf(fact_678_bits__div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% bits_div_by_0
thf(fact_679_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7286324685880884523d_real @ zero_z580800474297136991d_real )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% dbl_dec_simps(2)
thf(fact_680_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7286324685880884523d_real @ one_on6069100329679821595d_real )
= one_on6069100329679821595d_real ) ).
% dbl_dec_simps(3)
thf(fact_681_dbl__dec__def,axiom,
( neg_nu7286324685880884523d_real
= ( ^ [X3: dedekind_real] : ( minus_5539002012860128047d_real @ ( plus_p4060926892116697567d_real @ X3 @ X3 ) @ one_on6069100329679821595d_real ) ) ) ).
% dbl_dec_def
thf(fact_682_dbl__inc__simps_I4_J,axiom,
( ( neg_nu2849763295382964967d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% dbl_inc_simps(4)
thf(fact_683_dbl__inc__simps_I2_J,axiom,
( ( neg_nu2849763295382964967d_real @ zero_z580800474297136991d_real )
= one_on6069100329679821595d_real ) ).
% dbl_inc_simps(2)
thf(fact_684_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
= ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_less_cancel_left
thf(fact_685_preal__add__less__cancel__right,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
= ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_less_cancel_right
thf(fact_686_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_687_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_688_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_689_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_690_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_691_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_692_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_693_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_694_less__imp__neq,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_695_less__imp__neq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_696_less__imp__neq,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_697_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_698_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_699_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_700_order_Oasym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order.asym
thf(fact_701_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_702_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_703_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_704_ord__eq__less__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_705_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_706_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_707_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_708_ord__less__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_709_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_710_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X: nat] :
( ! [Y7: nat] :
( ( ord_less_nat @ Y7 @ X )
=> ( P @ Y7 ) )
=> ( P @ X ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_711_antisym__conv3,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_num @ Y @ X2 )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_712_antisym__conv3,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y @ X2 )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_713_antisym__conv3,axiom,
! [Y: dedekind_real,X2: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y @ X2 )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_714_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_715_linorder__cases,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_num @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_716_linorder__cases,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_717_linorder__cases,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_718_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_719_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_720_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_721_dual__order_Oasym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ~ ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_722_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_723_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_724_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_725_dual__order_Oirrefl,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_726_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_727_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X7: nat] : ( P2 @ X7 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_728_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_729_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_preal] : ( P @ A4 @ A4 )
=> ( ! [A4: dedekind_preal,B4: dedekind_preal] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_730_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A4: dedekind_real,B4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: dedekind_real] : ( P @ A4 @ A4 )
=> ( ! [A4: dedekind_real,B4: dedekind_real] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_731_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_732_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_733_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_734_order_Ostrict__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_735_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_736_not__less__iff__gr__or__eq,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ( ord_less_num @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_737_not__less__iff__gr__or__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( ( ord_le5708704896291381698_preal @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_738_not__less__iff__gr__or__eq,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X2 @ Y ) )
= ( ( ord_le2991122432403439658d_real @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_739_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_740_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_741_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_742_dual__order_Ostrict__trans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_743_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_744_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_745_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_746_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_747_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_748_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_749_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_750_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_751_linorder__neqE,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 != Y )
=> ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_752_linorder__neqE,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( X2 != Y )
=> ( ~ ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_753_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_754_order__less__asym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_755_order__less__asym,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_756_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_757_linorder__neq__iff,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( X2 != Y )
= ( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_758_linorder__neq__iff,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( X2 != Y )
= ( ( ord_le2991122432403439658d_real @ X2 @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_759_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_760_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_761_order__less__asym_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_762_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_763_order__less__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_764_order__less__trans,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z )
=> ( ord_le2991122432403439658d_real @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_765_order__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_trans
thf(fact_766_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_767_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_768_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_769_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_770_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_771_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_772_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_773_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_774_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_775_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_776_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_777_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_778_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_779_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_780_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_781_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_782_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_783_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_784_order__less__irrefl,axiom,
! [X2: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_785_order__less__irrefl,axiom,
! [X2: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_786_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_787_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_788_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_789_order__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_790_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_791_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_792_order__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_793_order__less__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_794_order__less__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_795_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_796_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_797_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_798_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_799_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_800_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_801_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_802_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_803_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_804_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_805_order__less__not__sym,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_806_order__less__not__sym,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_807_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_808_order__less__imp__triv,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_809_order__less__imp__triv,axiom,
! [X2: dedekind_real,Y: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_810_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_811_linorder__less__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( X2 = Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_812_linorder__less__linear,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
| ( X2 = Y )
| ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_813_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_814_order__less__imp__not__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_815_order__less__imp__not__eq,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_816_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_817_order__less__imp__not__eq2,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_818_order__less__imp__not__eq2,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_819_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_820_order__less__imp__not__less,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_821_order__less__imp__not__less,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_822_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_823_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_824_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_825_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_826_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ~ ( ord_le5708704896291381698_preal @ T @ X6 ) ) ).
% minf(7)
thf(fact_827_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ~ ( ord_le2991122432403439658d_real @ T @ X6 ) ) ).
% minf(7)
thf(fact_828_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_829_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( ord_le5708704896291381698_preal @ X6 @ T ) ) ).
% minf(5)
thf(fact_830_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( ord_le2991122432403439658d_real @ X6 @ T ) ) ).
% minf(5)
thf(fact_831_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_832_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_833_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_834_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_835_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_836_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_837_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_838_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_839_minf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_840_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_841_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_842_minf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_843_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z6 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_844_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( ord_le5708704896291381698_preal @ T @ X6 ) ) ).
% pinf(7)
thf(fact_845_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( ord_le2991122432403439658d_real @ T @ X6 ) ) ).
% pinf(7)
thf(fact_846_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_847_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ~ ( ord_le5708704896291381698_preal @ X6 @ T ) ) ).
% pinf(5)
thf(fact_848_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ~ ( ord_le2991122432403439658d_real @ X6 @ T ) ) ).
% pinf(5)
thf(fact_849_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_850_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_851_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_852_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_853_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_854_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_855_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_856_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_857_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_858_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_859_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_860_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: dedekind_real] :
! [X: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_861_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z6: nat] :
! [X: nat] :
( ( ord_less_nat @ Z6 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_862_preal__add__right__less__cancel,axiom,
! [R2: dedekind_preal,T: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_right_less_cancel
thf(fact_863_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) )
=> ( ord_le5708704896291381698_preal @ R2 @ S2 ) ) ).
% preal_add_left_less_cancel
thf(fact_864_preal__self__less__add__left,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] : ( ord_le5708704896291381698_preal @ R2 @ ( plus_p3173629198307831117_preal @ R2 @ S2 ) ) ).
% preal_self_less_add_left
thf(fact_865_preal__add__less2__mono2,axiom,
! [R2: dedekind_preal,S2: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R2 ) @ ( plus_p3173629198307831117_preal @ T @ S2 ) ) ) ).
% preal_add_less2_mono2
thf(fact_866_preal__add__less2__mono1,axiom,
! [R2: dedekind_preal,S2: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R2 @ T ) @ ( plus_p3173629198307831117_preal @ S2 @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_867_less__add__left,axiom,
! [R2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R2 @ S2 )
=> ( ( plus_p3173629198307831117_preal @ R2 @ ( minus_7336623429200594941_preal @ S2 @ R2 ) )
= S2 ) ) ).
% less_add_left
thf(fact_868_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_869_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_870_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_871_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_872_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_873_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_874_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_875_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_876_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_877_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_878_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_879_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_880_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_881_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_882_order__le__imp__less__or__eq,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ( ord_le2991122432403439658d_real @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_883_order__le__imp__less__or__eq,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_884_order__le__imp__less__or__eq,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_885_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_886_linorder__le__less__linear,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_887_linorder__le__less__linear,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_888_linorder__le__less__linear,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
| ( ord_less_num @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_889_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_890_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_891_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_892_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_893_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_894_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_895_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_896_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_897_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_898_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_899_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_900_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_901_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_902_order__less__le__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_903_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_904_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_905_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_906_order__less__le__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_907_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_908_order__less__le__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_909_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_910_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_911_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_912_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_913_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_914_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_915_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_916_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_917_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_918_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le2716243287969276982d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_919_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X: num,Y3: num] :
( ( ord_less_eq_num @ X @ Y3 )
=> ( ord_le5604041210740703414_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_920_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_921_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_922_order__le__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le2991122432403439658d_real @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_923_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_924_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_925_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_le5708704896291381698_preal @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_926_order__le__less__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_927_order__le__less__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X: dedekind_real,Y3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_928_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ord_less_num @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_929_order__le__less__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X: dedekind_preal,Y3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y3 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_930_order__less__le__trans,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z )
=> ( ord_le2991122432403439658d_real @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_931_order__less__le__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_932_order__less__le__trans,axiom,
! [X2: num,Y: num,Z: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_933_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_less_le_trans
thf(fact_934_order__le__less__trans,axiom,
! [X2: dedekind_real,Y: dedekind_real,Z: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z )
=> ( ord_le2991122432403439658d_real @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_935_order__le__less__trans,axiom,
! [X2: dedekind_preal,Y: dedekind_preal,Z: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z )
=> ( ord_le5708704896291381698_preal @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_936_order__le__less__trans,axiom,
! [X2: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_937_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X2 @ Z ) ) ) ).
% order_le_less_trans
thf(fact_938_order__neq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != B )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_939_order__neq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A != B )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_940_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_941_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_942_order__le__neq__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( A != B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_943_order__le__neq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( A != B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_944_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_945_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_946_order__less__imp__le,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ord_le2716243287969276982d_real @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_947_order__less__imp__le,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_948_order__less__imp__le,axiom,
! [X2: num,Y: num] :
( ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_949_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_950_linorder__not__less,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X2 @ Y ) )
= ( ord_le2716243287969276982d_real @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_951_linorder__not__less,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_952_linorder__not__less,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( ord_less_eq_num @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_953_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_954_linorder__not__le,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X2 @ Y ) )
= ( ord_le2991122432403439658d_real @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_955_linorder__not__le,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X2 @ Y ) )
= ( ord_le5708704896291381698_preal @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_956_linorder__not__le,axiom,
! [X2: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X2 @ Y ) )
= ( ord_less_num @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_957_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_958_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_959_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_960_order__less__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_961_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_962_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X3: dedekind_real,Y5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_963_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_964_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_num @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_965_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_966_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_967_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_968_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_969_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_970_order_Ostrict__implies__order,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_971_order_Ostrict__implies__order,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_972_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_973_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_974_dual__order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ~ ( ord_le2716243287969276982d_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_975_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ~ ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_976_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_977_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_978_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_979_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_980_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_981_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_982_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_983_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_984_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_985_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_986_dual__order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_987_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_988_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_989_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_990_dual__order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_991_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_992_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_993_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_994_order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ~ ( ord_le2716243287969276982d_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_995_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ~ ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_996_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_997_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_998_order_Ostrict__trans2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_999_order_Ostrict__trans2,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1000_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1001_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_1002_order_Ostrict__trans1,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1003_order_Ostrict__trans1,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1004_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1005_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_1006_order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1007_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1008_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1009_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_1010_order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1011_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1012_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1013_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_1014_not__le__imp__less,axiom,
! [Y: dedekind_real,X2: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y @ X2 )
=> ( ord_le2991122432403439658d_real @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1015_not__le__imp__less,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ( ord_le5708704896291381698_preal @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1016_not__le__imp__less,axiom,
! [Y: num,X2: num] :
( ~ ( ord_less_eq_num @ Y @ X2 )
=> ( ord_less_num @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1017_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_1018_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y5 )
& ~ ( ord_le2716243287969276982d_real @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1019_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X3: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X3 @ Y5 )
& ~ ( ord_le5604041210740703414_preal @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1020_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X3: num,Y5: num] :
( ( ord_less_eq_num @ X3 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1021_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_1022_antisym__conv2,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1023_antisym__conv2,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1024_antisym__conv2,axiom,
! [X2: num,Y: num] :
( ( ord_less_eq_num @ X2 @ Y )
=> ( ( ~ ( ord_less_num @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1025_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_1026_antisym__conv1,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ( ord_le2716243287969276982d_real @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1027_antisym__conv1,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ( ord_le5604041210740703414_preal @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1028_antisym__conv1,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ( ord_less_eq_num @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1029_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_1030_nless__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ A @ B ) )
= ( ~ ( ord_le2716243287969276982d_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1031_nless__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1032_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1033_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_1034_leI,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X2 @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X2 ) ) ).
% leI
thf(fact_1035_leI,axiom,
! [X2: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X2 @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X2 ) ) ).
% leI
thf(fact_1036_leI,axiom,
! [X2: num,Y: num] :
( ~ ( ord_less_num @ X2 @ Y )
=> ( ord_less_eq_num @ Y @ X2 ) ) ).
% leI
thf(fact_1037_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_1038_leD,axiom,
! [Y: dedekind_real,X2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X2 )
=> ~ ( ord_le2991122432403439658d_real @ X2 @ Y ) ) ).
% leD
thf(fact_1039_leD,axiom,
! [Y: dedekind_preal,X2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X2 )
=> ~ ( ord_le5708704896291381698_preal @ X2 @ Y ) ) ).
% leD
thf(fact_1040_leD,axiom,
! [Y: num,X2: num] :
( ( ord_less_eq_num @ Y @ X2 )
=> ~ ( ord_less_num @ X2 @ Y ) ) ).
% leD
thf(fact_1041_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_1042_verit__comp__simplify1_I3_J,axiom,
! [B5: dedekind_real,A5: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ B5 @ A5 ) )
= ( ord_le2991122432403439658d_real @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1043_verit__comp__simplify1_I3_J,axiom,
! [B5: dedekind_preal,A5: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B5 @ A5 ) )
= ( ord_le5708704896291381698_preal @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1044_verit__comp__simplify1_I3_J,axiom,
! [B5: num,A5: num] :
( ( ~ ( ord_less_eq_num @ B5 @ A5 ) )
= ( ord_less_num @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1045_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_1046_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ~ ( ord_le2716243287969276982d_real @ X6 @ T ) ) ).
% pinf(6)
thf(fact_1047_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ~ ( ord_le5604041210740703414_preal @ X6 @ T ) ) ).
% pinf(6)
thf(fact_1048_pinf_I6_J,axiom,
! [T: num] :
? [Z5: num] :
! [X6: num] :
( ( ord_less_num @ Z5 @ X6 )
=> ~ ( ord_less_eq_num @ X6 @ T ) ) ).
% pinf(6)
thf(fact_1049_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_1050_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X6 )
=> ( ord_le2716243287969276982d_real @ T @ X6 ) ) ).
% pinf(8)
thf(fact_1051_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X6 )
=> ( ord_le5604041210740703414_preal @ T @ X6 ) ) ).
% pinf(8)
thf(fact_1052_pinf_I8_J,axiom,
! [T: num] :
? [Z5: num] :
! [X6: num] :
( ( ord_less_num @ Z5 @ X6 )
=> ( ord_less_eq_num @ T @ X6 ) ) ).
% pinf(8)
thf(fact_1053_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z5 @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_1054_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ( ord_le2716243287969276982d_real @ X6 @ T ) ) ).
% minf(6)
thf(fact_1055_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ( ord_le5604041210740703414_preal @ X6 @ T ) ) ).
% minf(6)
thf(fact_1056_minf_I6_J,axiom,
! [T: num] :
? [Z5: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z5 )
=> ( ord_less_eq_num @ X6 @ T ) ) ).
% minf(6)
thf(fact_1057_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_1058_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X6: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X6 @ Z5 )
=> ~ ( ord_le2716243287969276982d_real @ T @ X6 ) ) ).
% minf(8)
thf(fact_1059_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X6: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X6 @ Z5 )
=> ~ ( ord_le5604041210740703414_preal @ T @ X6 ) ) ).
% minf(8)
thf(fact_1060_minf_I8_J,axiom,
! [T: num] :
? [Z5: num] :
! [X6: num] :
( ( ord_less_num @ X6 @ Z5 )
=> ~ ( ord_less_eq_num @ T @ X6 ) ) ).
% minf(8)
thf(fact_1061_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_1062_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X3: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% real_less_def
thf(fact_1063_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1064_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1065_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1066_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1067_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1068_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1069_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1070_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1071_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1072_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1073_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1074_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_1075_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_1076_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_1077_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_1078_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1079_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_1080_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_1081_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_1082_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_1083_less__1__mult,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).
% less_1_mult
thf(fact_1084_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_1085_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1086_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1087_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( 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 @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1088_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1089_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1090_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1091_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( 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 @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1092_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1093_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1094_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1095_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_1096_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_1097_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_1098_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_1099_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_1100_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_1101_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1102_dbl__inc__def,axiom,
( neg_nu2849763295382964967d_real
= ( ^ [X3: dedekind_real] : ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X3 @ X3 ) @ one_on6069100329679821595d_real ) ) ) ).
% dbl_inc_def
thf(fact_1103_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1104_real__sgn__def,axiom,
( sgn_sg210920135175580611d_real
= ( ^ [X3: dedekind_real] : ( if_Dedekind_real @ ( X3 = zero_z580800474297136991d_real ) @ zero_z580800474297136991d_real @ ( if_Dedekind_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X3 ) @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ) ) ) ).
% real_sgn_def
thf(fact_1105_numeral__eq__iff,axiom,
! [M2: num,N: num] :
( ( ( numeral_numeral_nat @ M2 )
= ( numeral_numeral_nat @ N ) )
= ( M2 = N ) ) ).
% numeral_eq_iff
thf(fact_1106_numeral__le__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M2 @ N ) ) ).
% numeral_le_iff
thf(fact_1107_numeral__less__iff,axiom,
! [M2: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M2 @ N ) ) ).
% numeral_less_iff
thf(fact_1108_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ Z ) )
= ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1109_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1110_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1111_numeral__times__numeral,axiom,
! [M2: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).
% numeral_times_numeral
thf(fact_1112_add__numeral__left,axiom,
! [V: num,W: num,Z: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ W ) @ Z ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1113_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_1114_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1115_numeral__plus__numeral,axiom,
! [M2: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_1116_distrib__right__numeral,axiom,
! [A: dedekind_real,B: dedekind_real,V: num] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ ( numera6440378960330123505d_real @ V ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ ( numera6440378960330123505d_real @ V ) ) @ ( times_2157731159493324635d_real @ B @ ( numera6440378960330123505d_real @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1117_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_1118_distrib__left__numeral,axiom,
! [V: num,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ B ) @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1119_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_1120_right__diff__distrib__numeral,axiom,
! [V: num,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ ( minus_5539002012860128047d_real @ B @ C ) )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ B ) @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ C ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1121_left__diff__distrib__numeral,axiom,
! [A: dedekind_real,B: dedekind_real,V: num] :
( ( times_2157731159493324635d_real @ ( minus_5539002012860128047d_real @ A @ B ) @ ( numera6440378960330123505d_real @ V ) )
= ( minus_5539002012860128047d_real @ ( times_2157731159493324635d_real @ A @ ( numera6440378960330123505d_real @ V ) ) @ ( times_2157731159493324635d_real @ B @ ( numera6440378960330123505d_real @ V ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1122_mult__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( numera6440378960330123505d_real @ ( times_times_num @ M2 @ N ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1123_mult__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( numera6440378960330123505d_real @ N ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1124_mult__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1125_add__neg__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1126_diff__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1127_diff__numeral__simps_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( numera6440378960330123505d_real @ N ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1128_divide__eq__eq__numeral1_I1_J,axiom,
! [B: dedekind_real,W: num,A: dedekind_real] :
( ( ( divide9119111104558704680d_real @ B @ ( numera6440378960330123505d_real @ W ) )
= A )
= ( ( ( ( numera6440378960330123505d_real @ W )
!= zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ A @ ( numera6440378960330123505d_real @ W ) ) ) )
& ( ( ( numera6440378960330123505d_real @ W )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1129_eq__divide__eq__numeral1_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,W: num] :
( ( A
= ( divide9119111104558704680d_real @ B @ ( numera6440378960330123505d_real @ W ) ) )
= ( ( ( ( numera6440378960330123505d_real @ W )
!= zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ ( numera6440378960330123505d_real @ W ) )
= B ) )
& ( ( ( numera6440378960330123505d_real @ W )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1130_inverse__eq__divide__numeral,axiom,
! [W: num] :
( ( invers3762989301784728874d_real @ ( numera6440378960330123505d_real @ W ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ W ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_1131_eq__divide__eq__numeral1_I2_J,axiom,
! [A: dedekind_real,B: dedekind_real,W: num] :
( ( A
= ( divide9119111104558704680d_real @ B @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) ) )
= ( ( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
!= zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= B ) )
& ( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1132_divide__eq__eq__numeral1_I2_J,axiom,
! [B: dedekind_real,W: num,A: dedekind_real] :
( ( ( divide9119111104558704680d_real @ B @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= A )
= ( ( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
!= zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ A @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) ) ) )
& ( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
= zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1133_dbl__inc__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu2849763295382964967d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ K ) ) )
= ( uminus7714077491378687647d_real @ ( neg_nu7286324685880884523d_real @ ( numera6440378960330123505d_real @ K ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1134_dbl__dec__simps_I1_J,axiom,
! [K: num] :
( ( neg_nu7286324685880884523d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ K ) ) )
= ( uminus7714077491378687647d_real @ ( neg_nu2849763295382964967d_real @ ( numera6440378960330123505d_real @ K ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1135_inverse__eq__divide__neg__numeral,axiom,
! [W: num] :
( ( invers3762989301784728874d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_1136_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_1137_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_1138_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_1139_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_1140_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_1141_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_1142_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ X2 ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ X2 ) @ one_on6069100329679821595d_real ) ) ).
% one_plus_numeral_commute
thf(fact_1143_one__plus__numeral__commute,axiom,
! [X2: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X2 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_1144_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_1145_divide__eq__eq__numeral_I1_J,axiom,
! [B: dedekind_real,C: dedekind_real,W: num] :
( ( ( divide9119111104558704680d_real @ B @ C )
= ( numera6440378960330123505d_real @ W ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ C ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( ( numera6440378960330123505d_real @ W )
= zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1146_eq__divide__eq__numeral_I1_J,axiom,
! [W: num,B: dedekind_real,C: dedekind_real] :
( ( ( numera6440378960330123505d_real @ W )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ C )
= B ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( ( numera6440378960330123505d_real @ W )
= zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1147_eq__divide__eq__numeral_I2_J,axiom,
! [W: num,B: dedekind_real,C: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ C )
= B ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
= zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1148_divide__eq__eq__numeral_I2_J,axiom,
! [B: dedekind_real,C: dedekind_real,W: num] :
( ( ( divide9119111104558704680d_real @ B @ C )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ C ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) )
= zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1149_semiring__norm_I167_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y ) )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(167)
thf(fact_1150_semiring__norm_I171_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y ) )
= ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).
% semiring_norm(171)
thf(fact_1151_semiring__norm_I169_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ Y ) )
= ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(169)
thf(fact_1152_semiring__norm_I170_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ V ) @ ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y ) )
= ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).
% semiring_norm(170)
thf(fact_1153_numeral__times__minus__swap,axiom,
! [W: num,X2: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ W ) @ ( uminus7714077491378687647d_real @ X2 ) )
= ( times_2157731159493324635d_real @ X2 @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1154_div__mult2__numeral__eq,axiom,
! [A: nat,K: num,L: num] :
( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
= ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1155_diff__numeral__special_I3_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ N ) ) ) ).
% diff_numeral_special(3)
thf(fact_1156_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_1157_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_1158_one__plus__numeral,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1159_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_1160_numeral__plus__one,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1161_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_1162_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_1163_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_1164_diff__numeral__special_I4_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_1165_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1166_numeral__One,axiom,
( ( numera6440378960330123505d_real @ one )
= one_on6069100329679821595d_real ) ).
% numeral_One
thf(fact_1167_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_1168_divide__numeral__1,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ ( numera6440378960330123505d_real @ one ) )
= A ) ).
% divide_numeral_1
thf(fact_1169_inverse__numeral__1,axiom,
( ( invers3762989301784728874d_real @ ( numera6440378960330123505d_real @ one ) )
= ( numera6440378960330123505d_real @ one ) ) ).
% inverse_numeral_1
thf(fact_1170_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq_num @ X2 @ one )
= ( X2 = one ) ) ).
% le_num_One_iff
thf(fact_1171_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1172_mult__numeral__1__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( numera6440378960330123505d_real @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_1173_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_1174_mult__numeral__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ ( numera6440378960330123505d_real @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_1175_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_1176_mult__1s__ring__1_I1_J,axiom,
! [B: dedekind_real] :
( ( times_2157731159493324635d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) ) @ B )
= ( uminus7714077491378687647d_real @ B ) ) ).
% mult_1s_ring_1(1)
thf(fact_1177_mult__1s__ring__1_I2_J,axiom,
! [B: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) ) )
= ( uminus7714077491378687647d_real @ B ) ) ).
% mult_1s_ring_1(2)
thf(fact_1178_uminus__numeral__One,axiom,
( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% uminus_numeral_One
thf(fact_1179_minus__sub__one__diff__one,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( neg_nu5286250102780226903d_real @ M2 @ one ) ) @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) ) ).
% minus_sub_one_diff_one
thf(fact_1180_diff__numeral__special_I7_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ N @ one ) ) ).
% diff_numeral_special(7)
thf(fact_1181_sub__num__simps_I1_J,axiom,
( ( neg_nu5286250102780226903d_real @ one @ one )
= zero_z580800474297136991d_real ) ).
% sub_num_simps(1)
thf(fact_1182_diff__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ M2 @ N ) ) ).
% diff_numeral_simps(1)
thf(fact_1183_add__neg__numeral__simps_I2_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ N @ M2 ) ) ).
% add_neg_numeral_simps(2)
thf(fact_1184_add__neg__numeral__simps_I1_J,axiom,
! [M2: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ M2 @ N ) ) ).
% add_neg_numeral_simps(1)
thf(fact_1185_semiring__norm_I166_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ V ) ) @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ W ) @ Y ) )
= ( plus_p4060926892116697567d_real @ ( neg_nu5286250102780226903d_real @ W @ V ) @ Y ) ) ).
% semiring_norm(166)
thf(fact_1186_semiring__norm_I165_J,axiom,
! [V: num,W: num,Y: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) @ Y ) )
= ( plus_p4060926892116697567d_real @ ( neg_nu5286250102780226903d_real @ V @ W ) @ Y ) ) ).
% semiring_norm(165)
thf(fact_1187_diff__numeral__simps_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( neg_nu5286250102780226903d_real @ N @ M2 ) ) ).
% diff_numeral_simps(4)
thf(fact_1188_diff__numeral__special_I1_J,axiom,
! [N: num] :
( ( minus_5539002012860128047d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ one @ N ) ) ).
% diff_numeral_special(1)
thf(fact_1189_diff__numeral__special_I2_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ M2 ) @ one_on6069100329679821595d_real )
= ( neg_nu5286250102780226903d_real @ M2 @ one ) ) ).
% diff_numeral_special(2)
thf(fact_1190_add__neg__numeral__special_I4_J,axiom,
! [N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( numera6440378960330123505d_real @ N ) )
= ( neg_nu5286250102780226903d_real @ N @ one ) ) ).
% add_neg_numeral_special(4)
thf(fact_1191_add__neg__numeral__special_I3_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M2 ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( neg_nu5286250102780226903d_real @ M2 @ one ) ) ).
% add_neg_numeral_special(3)
thf(fact_1192_add__neg__numeral__special_I2_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ one_on6069100329679821595d_real )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% add_neg_numeral_special(2)
thf(fact_1193_add__neg__numeral__special_I1_J,axiom,
! [M2: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% add_neg_numeral_special(1)
thf(fact_1194_diff__numeral__special_I8_J,axiom,
! [M2: num] :
( ( minus_5539002012860128047d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M2 ) ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( neg_nu5286250102780226903d_real @ one @ M2 ) ) ).
% diff_numeral_special(8)
thf(fact_1195_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ( divide_divide_nat @ M2 @ N )
= M2 )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1196_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_1197_neg__numeral__class_Osub__def,axiom,
( neg_nu5286250102780226903d_real
= ( ^ [K4: num,L2: num] : ( minus_5539002012860128047d_real @ ( numera6440378960330123505d_real @ K4 ) @ ( numera6440378960330123505d_real @ L2 ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_1198_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N ) )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1199_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times_nat @ M2 @ N )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1200_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1201_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% div_less
thf(fact_1202_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1203_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1204_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_1205_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ord_less_eq_nat @ N @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1206_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide_nat @ M2 @ N )
= zero_zero_nat )
= ( ( ord_less_nat @ M2 @ N )
| ( N = zero_zero_nat ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1207_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( times_times_nat @ I @ N ) )
=> ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1208_div__less__iff__less__mult,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q3 ) @ N )
= ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1209_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1210_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1211_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1212_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1213_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide_nat @ M2 @ N ) )
= ( ( ( N = zero_zero_nat )
=> ( P @ zero_zero_nat ) )
& ( ( N != zero_zero_nat )
=> ! [I2: nat,J2: nat] :
( ( ( ord_less_nat @ J2 @ N )
& ( M2
= ( plus_plus_nat @ ( times_times_nat @ N @ I2 ) @ J2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_div
thf(fact_1214_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1215_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_1216_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K = zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= zero_zero_nat ) )
& ( ( K != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ K @ M2 ) @ ( times_times_nat @ K @ N ) )
= ( divide_divide_nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1217_div__mult2__eq,axiom,
! [M2: nat,N: nat,Q3: nat] :
( ( divide_divide_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
= ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) ).
% div_mult2_eq
thf(fact_1218_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1219_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1220_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M: nat,N2: nat] : ( if_nat @ ( M = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1221_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times_nat @ M2 @ N ) )
=> ( ( N = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1222_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1223_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1224_eq__numeral__iff__iszero_I8_J,axiom,
! [Y: num] :
( ( one_on6069100329679821595d_real
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ one @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_1225_iszero__neg__numeral,axiom,
! [W: num] :
( ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ W ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ W ) ) ) ).
% iszero_neg_numeral
thf(fact_1226_eq__numeral__iff__iszero_I10_J,axiom,
! [Y: num] :
( ( zero_z580800474297136991d_real
= ( numera6440378960330123505d_real @ Y ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ Y ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_1227_eq__numeral__iff__iszero_I9_J,axiom,
! [X2: num] :
( ( ( numera6440378960330123505d_real @ X2 )
= zero_z580800474297136991d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ X2 ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_1228_iszero__0,axiom,
ring_1453304368769599722d_real @ zero_z580800474297136991d_real ).
% iszero_0
thf(fact_1229_iszero__def,axiom,
( ring_1453304368769599722d_real
= ( ^ [Z4: dedekind_real] : ( Z4 = zero_z580800474297136991d_real ) ) ) ).
% iszero_def
thf(fact_1230_eq__iff__iszero__diff,axiom,
( ( ^ [Y4: dedekind_real,Z2: dedekind_real] : ( Y4 = Z2 ) )
= ( ^ [X3: dedekind_real,Y5: dedekind_real] : ( ring_1453304368769599722d_real @ ( minus_5539002012860128047d_real @ X3 @ Y5 ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_1231_not__iszero__1,axiom,
~ ( ring_1453304368769599722d_real @ one_on6069100329679821595d_real ) ).
% not_iszero_1
thf(fact_1232_not__iszero__neg__1,axiom,
~ ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% not_iszero_neg_1
thf(fact_1233_eq__numeral__iff__iszero_I12_J,axiom,
! [Y: num] :
( ( zero_z580800474297136991d_real
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ Y ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_1234_eq__numeral__iff__iszero_I11_J,axiom,
! [X2: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X2 ) )
= zero_z580800474297136991d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ X2 ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_1235_not__iszero__neg__Numeral1,axiom,
~ ( ring_1453304368769599722d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_1236_eq__numeral__iff__iszero_I4_J,axiom,
! [X2: num,Y: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X2 ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y ) ) )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ Y @ X2 ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_1237_eq__numeral__iff__iszero_I2_J,axiom,
! [X2: num,Y: num] :
( ( ( numera6440378960330123505d_real @ X2 )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ Y ) ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_1238_eq__numeral__iff__iszero_I3_J,axiom,
! [X2: num,Y: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X2 ) )
= ( numera6440378960330123505d_real @ Y ) )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_1239_eq__numeral__iff__iszero_I6_J,axiom,
! [Y: num] :
( ( one_on6069100329679821595d_real
= ( numera6440378960330123505d_real @ Y ) )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ one @ Y ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_1240_eq__numeral__iff__iszero_I5_J,axiom,
! [X2: num] :
( ( ( numera6440378960330123505d_real @ X2 )
= one_on6069100329679821595d_real )
= ( ring_1453304368769599722d_real @ ( neg_nu5286250102780226903d_real @ X2 @ one ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_1241_eq__numeral__iff__iszero_I7_J,axiom,
! [X2: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ X2 ) )
= one_on6069100329679821595d_real )
= ( ring_1453304368769599722d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ X2 @ one ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_1242_dbl__dec__simps_I4_J,axiom,
( ( neg_nu7286324685880884523d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit1 @ one ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_1243_Suc__0__div__numeral_I1_J,axiom,
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ one ) )
= one_one_nat ) ).
% Suc_0_div_numeral(1)
thf(fact_1244_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1245_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1246_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_1247_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_1248_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1249_Suc__0__div__numeral_I3_J,axiom,
! [N: num] :
( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) )
= zero_zero_nat ) ).
% Suc_0_div_numeral(3)
thf(fact_1250_dbl__inc__simps_I3_J,axiom,
( ( neg_nu2849763295382964967d_real @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( bit1 @ one ) ) ) ).
% dbl_inc_simps(3)
thf(fact_1251_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_1252_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V: num] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V ) )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1253_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_1254_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1255_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1256_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1257_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1258_lift__Suc__mono__le,axiom,
! [F: nat > dedekind_real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le2716243287969276982d_real @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2716243287969276982d_real @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1259_lift__Suc__mono__le,axiom,
! [F: nat > dedekind_preal,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le5604041210740703414_preal @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le5604041210740703414_preal @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1260_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1261_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_1262_lift__Suc__antimono__le,axiom,
! [F: nat > dedekind_real,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le2716243287969276982d_real @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le2716243287969276982d_real @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1263_lift__Suc__antimono__le,axiom,
! [F: nat > dedekind_preal,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_le5604041210740703414_preal @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_le5604041210740703414_preal @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1264_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1265_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_1266_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1267_numeral__Bit1,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bit1 @ N ) )
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ ( numera6440378960330123505d_real @ N ) ) @ one_on6069100329679821595d_real ) ) ).
% numeral_Bit1
thf(fact_1268_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_1269_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_1270_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_1271_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1272_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__Dedekind____Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Dedekind____Real__Oreal_T,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( if_Dedekind_real @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Dedekind____Real__Oreal_T,axiom,
! [X2: dedekind_real,Y: dedekind_real] :
( ( if_Dedekind_real @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ z @ x ) @ ( plus_p4060926892116697567d_real @ z @ y ) ).
%------------------------------------------------------------------------------