TPTP Problem File: SLH0080^1.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_01537_050236__5789800_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1356 ( 811 unt; 88 typ; 0 def)
% Number of atoms : 2863 (1088 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8027 ( 274 ~; 14 |; 115 &;6821 @)
% ( 0 <=>; 803 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 77 usr; 15 con; 0-2 aty)
% Number of variables : 2618 ( 130 ^;2443 !; 45 ?;2618 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:41.982
%------------------------------------------------------------------------------
% Could-be-implicit typings (11)
thf(ty_n_t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
set_Dedekind_real: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
set_num: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (77)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Dedekind____Real__Oreal,type,
condit7337920006933021298d_real: set_Dedekind_real > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Int__Oint,type,
condit2212336002400062152ve_int: set_int > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Num__Onum,type,
condit7995530295919666870ve_num: set_num > $o ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Real__Oreal,type,
condit7084745239686109512e_real: set_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__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
minus_minus_real: real > real > real ).
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__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
one_one_real: real ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__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__Int__Oint,type,
plus_plus_int: int > int > int ).
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_Oplus__class_Oplus_001t__Real__Oreal,type,
plus_plus_real: real > real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Dedekind____Real__Oreal,type,
uminus7714077491378687647d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Dedekind____Real__Oreal,type,
zero_z580800474297136991d_real: dedekind_real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Dedekind____Real__Oreal,type,
inf_in6372492281993387880d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Real__Oreal,type,
inf_inf_real: real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
inf_in841446065189194142d_real: set_Dedekind_real > set_Dedekind_real > set_Dedekind_real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
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__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
neg_nu6075765906172075777c_real: real > real ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
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__Int__Oint,type,
numeral_numeral_int: num > int ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
numeral_numeral_real: num > real ).
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__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal,type,
ord_less_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
ord_le1729950913864802912d_real: set_Dedekind_real > set_Dedekind_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Dedekind____Real__Oreal_M_Eo_J,type,
ord_le60669685330047769real_o: ( dedekind_real > $o ) > ( dedekind_real > $o ) > $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__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal,type,
ord_less_eq_real: real > real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Dedekind____Real__Oreal_J,type,
ord_le5071027967042688620d_real: set_Dedekind_real > set_Dedekind_real > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
power_power_real: real > nat > real ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
dvd_dvd_int: int > int > $o ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Dedekind____Real__Oreal,type,
collec1873404778653292075d_real: ( dedekind_real > $o ) > set_Dedekind_real ).
thf(sy_c_member_001t__Dedekind____Real__Oreal,type,
member_Dedekind_real: dedekind_real > set_Dedekind_real > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Num__Onum,type,
member_num: num > set_num > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_S,type,
s: set_Dedekind_real ).
thf(sy_v_X____,type,
x: dedekind_real ).
thf(sy_v_Y____,type,
y: dedekind_real ).
thf(sy_v_t____,type,
t: dedekind_real ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_y____,type,
y2: dedekind_real ).
% Relevant facts (1266)
thf(fact_0_notempty__S,axiom,
? [X: dedekind_real] : ( member_Dedekind_real @ X @ s ) ).
% notempty_S
thf(fact_1_y__in__S,axiom,
member_Dedekind_real @ y2 @ s ).
% y_in_S
thf(fact_2_X__in__S,axiom,
member_Dedekind_real @ x @ s ).
% X_in_S
thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062X_O_AX_A_092_060in_062_AS_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X: dedekind_real] :
~ ( member_Dedekind_real @ X @ s ) ).
% \<open>\<And>thesis. (\<And>X. X \<in> S \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_4_t__is__Lub,axiom,
! [Y: dedekind_real] :
( ( ? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Aa: dedekind_real] :
( ( member_Dedekind_real @ Aa @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Aa @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
& ( ord_le2991122432403439658d_real @ Y @ X2 ) ) )
= ( ord_le2991122432403439658d_real @ Y @ t ) ) ).
% t_is_Lub
thf(fact_5_exists__Ub,axiom,
condit7337920006933021298d_real @ s ).
% exists_Ub
thf(fact_6__092_060open_062_092_060forall_062y_092_060in_062_123z_O_A_092_060exists_062x_092_060in_062S_O_Az_A_061_Ax_A_L_A_N_AX_A_L_A1_125_A_092_060inter_062_A_123x_O_A0_A_060_Ax_125_O_A0_A_060_Ay_092_060close_062,axiom,
! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Y2: dedekind_real] :
( ( member_Dedekind_real @ Y2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Y2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X3 ) ) ).
% \<open>\<forall>y\<in>{z. \<exists>x\<in>S. z = x + - X + 1} \<inter> {x. 0 < x}. 0 < y\<close>
thf(fact_7__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060forall_062y_O_A_I_092_060exists_062x_092_060in_062_123z_O_A_092_060exists_062x_092_060in_062S_O_Az_A_061_Ax_A_L_A_N_AX_A_L_A1_125_A_092_060inter_062_A_123x_O_A0_A_060_Ax_125_O_Ay_A_060_Ax_J_A_061_A_Iy_A_060_At_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [T: dedekind_real] :
~ ! [Y: dedekind_real] :
( ( ? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Aa: dedekind_real] :
( ( member_Dedekind_real @ Aa @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Aa @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
& ( ord_le2991122432403439658d_real @ Y @ X2 ) ) )
= ( ord_le2991122432403439658d_real @ Y @ T ) ) ).
% \<open>\<And>thesis. (\<And>t. \<forall>y. (\<exists>x\<in>{z. \<exists>x\<in>S. z = x + - X + 1} \<inter> {x. 0 < x}. y < x) = (y < t) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_8_shifted__not__empty,axiom,
? [U: dedekind_real] :
( member_Dedekind_real @ U
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) ) ).
% shifted_not_empty
thf(fact_9_add__eq__exists,axiom,
! [A: dedekind_real,B: dedekind_real] :
? [X4: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_10_add__eq__exists,axiom,
! [A: real,B: real] :
? [X4: real] :
( ( plus_plus_real @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_11_add__eq__exists,axiom,
! [A: int,B: int] :
? [X4: int] :
( ( plus_plus_int @ A @ X4 )
= B ) ).
% add_eq_exists
thf(fact_12_posreal__complete,axiom,
! [P: set_Dedekind_real] :
( ! [X4: dedekind_real] :
( ( member_Dedekind_real @ X4 @ P )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X4 ) )
=> ( ? [X3: dedekind_real] : ( member_Dedekind_real @ X3 @ P )
=> ( ? [Y: dedekind_real] :
! [X4: dedekind_real] :
( ( member_Dedekind_real @ X4 @ P )
=> ( ord_le2991122432403439658d_real @ X4 @ Y ) )
=> ? [S: dedekind_real] :
! [Y: dedekind_real] :
( ( ? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ P )
& ( ord_le2991122432403439658d_real @ Y @ X2 ) ) )
= ( ord_le2991122432403439658d_real @ Y @ S ) ) ) ) ) ).
% posreal_complete
thf(fact_13_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_14_real__less__sum__gt__zero,axiom,
! [W: dedekind_real,S2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W @ S2 )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S2 @ ( uminus7714077491378687647d_real @ W ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_15_real__sum__gt__zero__less,axiom,
! [S2: dedekind_real,W: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S2 @ ( uminus7714077491378687647d_real @ W ) ) )
=> ( ord_le2991122432403439658d_real @ W @ S2 ) ) ).
% real_sum_gt_zero_less
thf(fact_16_S__Ub__is__SHIFT__Ub,axiom,
! [X5: dedekind_real] :
( ! [X4: dedekind_real] :
( ( member_Dedekind_real @ X4 @ s )
=> ( ord_le2716243287969276982d_real @ X4 @ X5 ) )
=> ! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Y2: dedekind_real] :
( ( member_Dedekind_real @ Y2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Y2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
=> ( ord_le2716243287969276982d_real @ X3 @ ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X5 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) ) ).
% S_Ub_is_SHIFT_Ub
thf(fact_17_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_18_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
= zero_zero_real ) ).
% add_neg_numeral_special(8)
thf(fact_19_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_20_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_21_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
= zero_zero_real ) ).
% add_neg_numeral_special(7)
thf(fact_22_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_23_add_Oright__inverse,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ ( uminus7714077491378687647d_real @ A ) )
= zero_z580800474297136991d_real ) ).
% add.right_inverse
thf(fact_24_add_Oright__inverse,axiom,
! [A: real] :
( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
= zero_zero_real ) ).
% add.right_inverse
thf(fact_25_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_26_ab__left__minus,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= zero_z580800474297136991d_real ) ).
% ab_left_minus
thf(fact_27_ab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_left_minus
thf(fact_28_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_29_less__neg__neg,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% less_neg_neg
thf(fact_30_less__neg__neg,axiom,
! [A: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% less_neg_neg
thf(fact_31_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_32_neg__less__pos,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ).
% neg_less_pos
thf(fact_33_neg__less__pos,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_pos
thf(fact_34_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_35_neg__0__less__iff__less,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% neg_0_less_iff_less
thf(fact_36_neg__0__less__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% neg_0_less_iff_less
thf(fact_37_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_38_neg__less__0__iff__less,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_39_neg__less__0__iff__less,axiom,
! [A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% neg_less_0_iff_less
thf(fact_40_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_41_add__less__same__cancel1,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ B @ A ) @ B )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_less_same_cancel1
thf(fact_42_add__less__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel1
thf(fact_43_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_44_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_45_add__less__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_less_same_cancel2
thf(fact_46_add__less__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% add_less_same_cancel2
thf(fact_47_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_48_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_49_less__add__same__cancel1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_50_less__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel1
thf(fact_51_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_52_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_53_Y__isUb,axiom,
! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ s )
=> ( ord_le2716243287969276982d_real @ X3 @ y ) ) ).
% Y_isUb
thf(fact_54__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Y_O_A_092_060forall_062s_092_060in_062S_O_As_A_092_060le_062_AY_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y3: dedekind_real] :
~ ! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ s )
=> ( ord_le2716243287969276982d_real @ X3 @ Y3 ) ) ).
% \<open>\<And>thesis. (\<And>Y. \<forall>s\<in>S. s \<le> Y \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_55_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_56_add__right__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_57_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_58_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_59_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_60_add__left__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_61_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_62_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_63_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_64_neg__equal__iff__equal,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= ( uminus_uminus_real @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_65_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_66_add_Oinverse__inverse,axiom,
! [A: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_67_add_Oinverse__inverse,axiom,
! [A: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_68_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_69_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_70_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_71_add__le__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_72_add__le__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_73_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_74_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_75_add__le__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_76_add__le__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_77_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_78_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_79_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_80_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_81_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_82_add__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add_0
thf(fact_83_zero__eq__add__iff__both__eq__0,axiom,
! [X5: nat,Y4: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X5 @ Y4 ) )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_84_add__eq__0__iff__both__eq__0,axiom,
! [X5: nat,Y4: nat] :
( ( ( plus_plus_nat @ X5 @ Y4 )
= zero_zero_nat )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_85_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_86_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_87_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_88_add__cancel__right__right,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ A @ B ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_right
thf(fact_89_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_90_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_91_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_92_add__cancel__right__left,axiom,
! [A: real,B: real] :
( ( A
= ( plus_plus_real @ B @ A ) )
= ( B = zero_zero_real ) ) ).
% add_cancel_right_left
thf(fact_93_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_94_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_95_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_96_add__cancel__left__right,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_right
thf(fact_97_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_98_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_99_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_100_add__cancel__left__left,axiom,
! [B: real,A: real] :
( ( ( plus_plus_real @ B @ A )
= A )
= ( B = zero_zero_real ) ) ).
% add_cancel_left_left
thf(fact_101_mem__Collect__eq,axiom,
! [A: dedekind_real,P: dedekind_real > $o] :
( ( member_Dedekind_real @ A @ ( collec1873404778653292075d_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_102_Collect__mem__eq,axiom,
! [A2: set_Dedekind_real] :
( ( collec1873404778653292075d_real
@ ^ [X2: dedekind_real] : ( member_Dedekind_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_103_Collect__cong,axiom,
! [P: dedekind_real > $o,Q: dedekind_real > $o] :
( ! [X4: dedekind_real] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collec1873404778653292075d_real @ P )
= ( collec1873404778653292075d_real @ Q ) ) ) ).
% Collect_cong
thf(fact_104_double__zero__sym,axiom,
! [A: dedekind_real] :
( ( zero_z580800474297136991d_real
= ( plus_p4060926892116697567d_real @ A @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% double_zero_sym
thf(fact_105_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_106_double__zero__sym,axiom,
! [A: real] :
( ( zero_zero_real
= ( plus_plus_real @ A @ A ) )
= ( A = zero_zero_real ) ) ).
% double_zero_sym
thf(fact_107_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_108_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_109_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_110_add_Oright__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.right_neutral
thf(fact_111_add__less__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_112_add__less__cancel__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_113_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_114_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_115_add__less__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_116_add__less__cancel__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
= ( ord_less_real @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_117_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_118_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_119_neg__le__iff__le,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_120_neg__le__iff__le,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_121_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_122_neg__equal__zero,axiom,
! [A: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= A )
= ( A = zero_z580800474297136991d_real ) ) ).
% neg_equal_zero
thf(fact_123_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_124_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_125_equal__neg__zero,axiom,
! [A: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ A ) )
= ( A = zero_z580800474297136991d_real ) ) ).
% equal_neg_zero
thf(fact_126_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_127_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_128_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_129_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_130_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_131_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_132_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_133_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_134_add_Oinverse__neutral,axiom,
( ( uminus7714077491378687647d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% add.inverse_neutral
thf(fact_135_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_136_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_137_neg__less__iff__less,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_138_neg__less__iff__less,axiom,
! [B: real,A: real] :
( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
= ( ord_less_real @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_139_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_140_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_141_minus__add__distrib,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% minus_add_distrib
thf(fact_142_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_143_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_144_minus__add__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_145_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_146_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_147_add__minus__cancel,axiom,
! [A: real,B: real] :
( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_148_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_149_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ A ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_150_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_151_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_152_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_153_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_154_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_155_le__add__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( plus_p4060926892116697567d_real @ B @ A ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_156_le__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel2
thf(fact_157_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_158_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_159_le__add__same__cancel1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( plus_p4060926892116697567d_real @ A @ B ) )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_160_le__add__same__cancel1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
= ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% le_add_same_cancel1
thf(fact_161_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_162_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_163_add__le__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ B )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel2
thf(fact_164_add__le__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel2
thf(fact_165_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_166_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_167_add__le__same__cancel1,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ B @ A ) @ B )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% add_le_same_cancel1
thf(fact_168_add__le__same__cancel1,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% add_le_same_cancel1
thf(fact_169_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_170_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_171_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ A ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_172_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
= ( ord_less_real @ zero_zero_real @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_173_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_174_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_175_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
= ( ord_less_real @ A @ zero_zero_real ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_176_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_177_less__add__same__cancel2,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( plus_p4060926892116697567d_real @ B @ A ) )
= ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_178_less__add__same__cancel2,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
= ( ord_less_real @ zero_zero_real @ B ) ) ).
% less_add_same_cancel2
thf(fact_179_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_180_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_181_neg__less__eq__nonneg,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ A ) @ A )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_182_neg__less__eq__nonneg,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_183_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_184_less__eq__neg__nonpos,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% less_eq_neg_nonpos
thf(fact_185_less__eq__neg__nonpos,axiom,
! [A: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% less_eq_neg_nonpos
thf(fact_186_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_187_neg__le__0__iff__le,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ A ) @ zero_z580800474297136991d_real )
= ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_188_neg__le__0__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
= ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% neg_le_0_iff_le
thf(fact_189_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_190_neg__0__le__iff__le,axiom,
! [A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ A ) )
= ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real ) ) ).
% neg_0_le_iff_le
thf(fact_191_neg__0__le__iff__le,axiom,
! [A: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
= ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% neg_0_le_iff_le
thf(fact_192_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_193__C_K_C,axiom,
! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Y2: dedekind_real] :
( ( member_Dedekind_real @ Y2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Y2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
=> ( ord_le2716243287969276982d_real @ X3 @ ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ y @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) ).
% "*"
thf(fact_194_real__le__refl,axiom,
! [W: dedekind_real] : ( ord_le2716243287969276982d_real @ W @ W ) ).
% real_le_refl
thf(fact_195_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_196_real__le__antisym,axiom,
! [Z2: dedekind_real,W: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z2 @ W )
=> ( ( ord_le2716243287969276982d_real @ W @ Z2 )
=> ( Z2 = W ) ) ) ).
% real_le_antisym
thf(fact_197_zero__le,axiom,
! [X5: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X5 ) ).
% zero_le
thf(fact_198_le__numeral__extra_I3_J,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ).
% le_numeral_extra(3)
thf(fact_199_le__numeral__extra_I3_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% le_numeral_extra(3)
thf(fact_200_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_201_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_202_add__le__imp__le__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_203_add__le__imp__le__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_204_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_205_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_206_add__le__imp__le__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_207_add__le__imp__le__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_208_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_209_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_210_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_211_add__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_212_add__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_right_mono
thf(fact_213_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_214_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_215_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_216_add__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_217_add__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_left_mono
thf(fact_218_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_219_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_220_add__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_221_add__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_mono
thf(fact_222_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_223_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_224_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_225_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_226_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_227_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_228_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_229_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_230_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_231_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_232_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( K = L ) )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_233_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_234_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_235_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_236_le__numeral__extra_I4_J,axiom,
ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ).
% le_numeral_extra(4)
thf(fact_237_le__numeral__extra_I4_J,axiom,
ord_less_eq_real @ one_one_real @ one_one_real ).
% le_numeral_extra(4)
thf(fact_238_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_239_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_240_le__imp__neg__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ B ) @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_241_le__imp__neg__le,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% le_imp_neg_le
thf(fact_242_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_243_minus__le__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_244_minus__le__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_245_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_246_le__minus__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( ord_le2716243287969276982d_real @ B @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% le_minus_iff
thf(fact_247_le__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% le_minus_iff
thf(fact_248_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_249_add__nonpos__eq__0__iff,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ zero_z580800474297136991d_real )
=> ( ( ( plus_p4060926892116697567d_real @ X5 @ Y4 )
= zero_z580800474297136991d_real )
= ( ( X5 = zero_z580800474297136991d_real )
& ( Y4 = zero_z580800474297136991d_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_250_add__nonpos__eq__0__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ zero_zero_real )
=> ( ( ord_less_eq_real @ Y4 @ zero_zero_real )
=> ( ( ( plus_plus_real @ X5 @ Y4 )
= zero_zero_real )
= ( ( X5 = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_251_add__nonpos__eq__0__iff,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y4 @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X5 @ Y4 )
= zero_zero_nat )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_252_add__nonpos__eq__0__iff,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y4 @ zero_zero_int )
=> ( ( ( plus_plus_int @ X5 @ Y4 )
= zero_zero_int )
= ( ( X5 = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_253_add__nonneg__eq__0__iff,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ X5 )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ Y4 )
=> ( ( ( plus_p4060926892116697567d_real @ X5 @ Y4 )
= zero_z580800474297136991d_real )
= ( ( X5 = zero_z580800474297136991d_real )
& ( Y4 = zero_z580800474297136991d_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_254_add__nonneg__eq__0__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ X5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
=> ( ( ( plus_plus_real @ X5 @ Y4 )
= zero_zero_real )
= ( ( X5 = zero_zero_real )
& ( Y4 = zero_zero_real ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_255_add__nonneg__eq__0__iff,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y4 )
=> ( ( ( plus_plus_nat @ X5 @ Y4 )
= zero_zero_nat )
= ( ( X5 = zero_zero_nat )
& ( Y4 = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_256_add__nonneg__eq__0__iff,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y4 )
=> ( ( ( plus_plus_int @ X5 @ Y4 )
= zero_zero_int )
= ( ( X5 = zero_zero_int )
& ( Y4 = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_257_add__nonpos__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_258_add__nonpos__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_nonpos
thf(fact_259_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_260_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_261_add__nonneg__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_262_add__nonneg__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_263_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_264_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_265_add__increasing2,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ C )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ord_le2716243287969276982d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_266_add__increasing2,axiom,
! [C: real,B: real,A: real] :
( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ B @ A )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_267_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_268_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_269_add__decreasing2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ C @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_270_add__decreasing2,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ A @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_271_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_272_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_273_add__increasing,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_274_add__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_275_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_276_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_277_add__decreasing,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_278_add__decreasing,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ C @ B )
=> ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_279_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_280_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_281_add__less__le__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ C @ D )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_282_add__less__le__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_eq_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_283_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_284_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_285_add__le__less__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_286_add__le__less__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_287_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_288_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_289_add__mono__thms__linordered__field_I3_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( ord_le2716243287969276982d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_290_add__mono__thms__linordered__field_I3_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_eq_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_291_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_292_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_293_add__mono__thms__linordered__field_I4_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ I @ J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_294_add__mono__thms__linordered__field_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_eq_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_295_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_296_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_297_le__minus__one__simps_I2_J,axiom,
ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ one_on6069100329679821595d_real ).
% le_minus_one_simps(2)
thf(fact_298_le__minus__one__simps_I2_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% le_minus_one_simps(2)
thf(fact_299_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_300_le__minus__one__simps_I4_J,axiom,
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% le_minus_one_simps(4)
thf(fact_301_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(4)
thf(fact_302_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_303_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% real_less_def
thf(fact_304_real__add__left__mono,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z2 @ X5 ) @ ( plus_p4060926892116697567d_real @ Z2 @ Y4 ) ) ) ).
% real_add_left_mono
thf(fact_305_add__strict__increasing2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_306_add__strict__increasing2,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_307_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_308_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_309_add__strict__increasing,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_310_add__strict__increasing,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_311_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_312_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_313_add__pos__nonneg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_314_add__pos__nonneg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_eq_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_315_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_316_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_317_add__nonpos__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_nonpos_neg
thf(fact_318_add__nonpos__neg,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_nonpos_neg
thf(fact_319_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_320_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_321_add__nonneg__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_322_add__nonneg__pos,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_323_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_324_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_325_add__neg__nonpos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2716243287969276982d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_neg_nonpos
thf(fact_326_add__neg__nonpos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_eq_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_nonpos
thf(fact_327_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_328_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_329_le__minus__one__simps_I1_J,axiom,
ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ zero_z580800474297136991d_real ).
% le_minus_one_simps(1)
thf(fact_330_le__minus__one__simps_I1_J,axiom,
ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% le_minus_one_simps(1)
thf(fact_331_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_332_le__minus__one__simps_I3_J,axiom,
~ ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% le_minus_one_simps(3)
thf(fact_333_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% le_minus_one_simps(3)
thf(fact_334_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_335_zero__reorient,axiom,
! [X5: dedekind_real] :
( ( zero_z580800474297136991d_real = X5 )
= ( X5 = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_336_zero__reorient,axiom,
! [X5: nat] :
( ( zero_zero_nat = X5 )
= ( X5 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_337_zero__reorient,axiom,
! [X5: int] :
( ( zero_zero_int = X5 )
= ( X5 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_338_zero__reorient,axiom,
! [X5: real] :
( ( zero_zero_real = X5 )
= ( X5 = zero_zero_real ) ) ).
% zero_reorient
thf(fact_339_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_340_add__right__imp__eq,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_341_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_342_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_343_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_344_add__left__imp__eq,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_345_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_346_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_347_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_348_add_Oleft__commute,axiom,
! [B: real,A: real,C: real] :
( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.left_commute
thf(fact_349_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_350_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_351_add_Ocommute,axiom,
( plus_p4060926892116697567d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] : ( plus_p4060926892116697567d_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_352_add_Ocommute,axiom,
( plus_plus_real
= ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_353_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_354_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% add.commute
thf(fact_355_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_356_add_Oright__cancel,axiom,
! [B: real,A: real,C: real] :
( ( ( plus_plus_real @ B @ A )
= ( plus_plus_real @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_357_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_358_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_359_add_Oleft__cancel,axiom,
! [A: real,B: real,C: real] :
( ( ( plus_plus_real @ A @ B )
= ( plus_plus_real @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_360_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_361_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_362_add_Oassoc,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% add.assoc
thf(fact_363_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_364_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_365_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_366_group__cancel_Oadd2,axiom,
! [B3: real,K: real,B: real,A: real] :
( ( B3
= ( plus_plus_real @ K @ B ) )
=> ( ( plus_plus_real @ A @ B3 )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_367_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_368_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_369_group__cancel_Oadd1,axiom,
! [A2: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( plus_p4060926892116697567d_real @ A2 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_370_group__cancel_Oadd1,axiom,
! [A2: real,K: real,A: real,B: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( plus_plus_real @ A2 @ B )
= ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_371_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_372_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_373_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p4060926892116697567d_real @ I @ K )
= ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_374_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_real @ I @ K )
= ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_375_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_376_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_377_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_378_is__num__normalize_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_379_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_380_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_381_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: real,B: real,C: real] :
( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
= ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_382_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_383_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_384_one__reorient,axiom,
! [X5: dedekind_real] :
( ( one_on6069100329679821595d_real = X5 )
= ( X5 = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_385_one__reorient,axiom,
! [X5: nat] :
( ( one_one_nat = X5 )
= ( X5 = one_one_nat ) ) ).
% one_reorient
thf(fact_386_one__reorient,axiom,
! [X5: real] :
( ( one_one_real = X5 )
= ( X5 = one_one_real ) ) ).
% one_reorient
thf(fact_387_one__reorient,axiom,
! [X5: int] :
( ( one_one_int = X5 )
= ( X5 = one_one_int ) ) ).
% one_reorient
thf(fact_388_minus__equation__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( uminus7714077491378687647d_real @ A )
= B )
= ( ( uminus7714077491378687647d_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_389_minus__equation__iff,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( uminus_uminus_real @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_390_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_391_equation__minus__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( uminus7714077491378687647d_real @ B ) )
= ( B
= ( uminus7714077491378687647d_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_392_equation__minus__iff,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% equation_minus_iff
thf(fact_393_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_394_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_395_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_396_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_397_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_398_less__numeral__extra_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ zero_z580800474297136991d_real ) ).
% less_numeral_extra(3)
thf(fact_399_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_400_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_401_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_402_add_Ogroup__left__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_403_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_404_add_Ogroup__left__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_405_add_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.comm_neutral
thf(fact_406_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_407_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_408_add_Ocomm__neutral,axiom,
! [A: real] :
( ( plus_plus_real @ A @ zero_zero_real )
= A ) ).
% add.comm_neutral
thf(fact_409_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_410_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_411_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_412_comm__monoid__add__class_Oadd__0,axiom,
! [A: real] :
( ( plus_plus_real @ zero_zero_real @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_413_add__less__imp__less__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_414_add__less__imp__less__right,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_415_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_416_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_417_add__less__imp__less__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_418_add__less__imp__less__left,axiom,
! [C: real,A: real,B: real] :
( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
=> ( ord_less_real @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_419_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_420_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_421_add__strict__right__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_422_add__strict__right__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_423_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_424_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_425_add__strict__left__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ C @ A ) @ ( plus_p4060926892116697567d_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_426_add__strict__left__mono,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_427_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_428_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_429_add__strict__mono,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real,D: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ C @ D )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ C ) @ ( plus_p4060926892116697567d_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_430_add__strict__mono,axiom,
! [A: real,B: real,C: real,D: real] :
( ( ord_less_real @ A @ B )
=> ( ( ord_less_real @ C @ D )
=> ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_431_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_432_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_433_add__mono__thms__linordered__field_I1_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( K = L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_434_add__mono__thms__linordered__field_I1_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( K = L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_435_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_436_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_437_add__mono__thms__linordered__field_I2_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( I = J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_438_add__mono__thms__linordered__field_I2_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( I = J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_439_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_440_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_441_add__mono__thms__linordered__field_I5_J,axiom,
! [I: dedekind_real,J: dedekind_real,K: dedekind_real,L: dedekind_real] :
( ( ( ord_le2991122432403439658d_real @ I @ J )
& ( ord_le2991122432403439658d_real @ K @ L ) )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ I @ K ) @ ( plus_p4060926892116697567d_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_442_add__mono__thms__linordered__field_I5_J,axiom,
! [I: real,J: real,K: real,L: real] :
( ( ( ord_less_real @ I @ J )
& ( ord_less_real @ K @ L ) )
=> ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_443_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_444_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_445_less__numeral__extra_I4_J,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ) ).
% less_numeral_extra(4)
thf(fact_446_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_447_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_448_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_449_minus__less__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ A ) @ B )
= ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_450_minus__less__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
= ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_451_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_452_less__minus__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( uminus7714077491378687647d_real @ B ) )
= ( ord_le2991122432403439658d_real @ B @ ( uminus7714077491378687647d_real @ A ) ) ) ).
% less_minus_iff
thf(fact_453_less__minus__iff,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
= ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% less_minus_iff
thf(fact_454_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_455_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_456_add_Oinverse__distrib__swap,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_457_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_458_group__cancel_Oneg1,axiom,
! [A2: dedekind_real,K: dedekind_real,A: dedekind_real] :
( ( A2
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( uminus7714077491378687647d_real @ A2 )
= ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ K ) @ ( uminus7714077491378687647d_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_459_group__cancel_Oneg1,axiom,
! [A2: real,K: real,A: real] :
( ( A2
= ( plus_plus_real @ K @ A ) )
=> ( ( uminus_uminus_real @ A2 )
= ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_460_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_461_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_462_is__num__normalize_I8_J,axiom,
! [A: real,B: real] :
( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
= ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_463_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_464_one__neq__neg__one,axiom,
( one_on6069100329679821595d_real
!= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% one_neq_neg_one
thf(fact_465_one__neq__neg__one,axiom,
( one_one_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% one_neq_neg_one
thf(fact_466_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_467_pos__add__strict,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ B @ ( plus_p4060926892116697567d_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_468_pos__add__strict,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_469_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_470_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_471_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_472_add__pos__pos,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ A )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ B )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_473_add__pos__pos,axiom,
! [A: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_474_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_475_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_476_add__neg__neg,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ B @ zero_z580800474297136991d_real )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ zero_z580800474297136991d_real ) ) ) ).
% add_neg_neg
thf(fact_477_add__neg__neg,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% add_neg_neg
thf(fact_478_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_479_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_480_less__numeral__extra_I1_J,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% less_numeral_extra(1)
thf(fact_481_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_482_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_483_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_484_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_485_neg__eq__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( ( uminus_uminus_real @ A )
= B )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_486_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_487_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_488_eq__neg__iff__add__eq__0,axiom,
! [A: real,B: real] :
( ( A
= ( uminus_uminus_real @ B ) )
= ( ( plus_plus_real @ A @ B )
= zero_zero_real ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_489_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_490_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_491_add_Oinverse__unique,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
=> ( ( uminus_uminus_real @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_492_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_493_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_494_ab__group__add__class_Oab__left__minus,axiom,
! [A: real] :
( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
= zero_zero_real ) ).
% ab_group_add_class.ab_left_minus
thf(fact_495_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% 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__eq__0__iff,axiom,
! [A: real,B: real] :
( ( ( plus_plus_real @ A @ B )
= zero_zero_real )
= ( B
= ( uminus_uminus_real @ A ) ) ) ).
% add_eq_0_iff
thf(fact_498_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_499_zero__neq__neg__one,axiom,
( zero_z580800474297136991d_real
!= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% zero_neq_neg_one
thf(fact_500_zero__neq__neg__one,axiom,
( zero_zero_real
!= ( uminus_uminus_real @ one_one_real ) ) ).
% zero_neq_neg_one
thf(fact_501_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_502_less__minus__one__simps_I2_J,axiom,
ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ one_on6069100329679821595d_real ).
% less_minus_one_simps(2)
thf(fact_503_less__minus__one__simps_I2_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% less_minus_one_simps(2)
thf(fact_504_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_505_less__minus__one__simps_I4_J,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% less_minus_one_simps(4)
thf(fact_506_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(4)
thf(fact_507_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_508_less__minus__one__simps_I1_J,axiom,
ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ zero_z580800474297136991d_real ).
% less_minus_one_simps(1)
thf(fact_509_less__minus__one__simps_I1_J,axiom,
ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% less_minus_one_simps(1)
thf(fact_510_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_511_less__minus__one__simps_I3_J,axiom,
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% less_minus_one_simps(3)
thf(fact_512_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% less_minus_one_simps(3)
thf(fact_513_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_514_bdd__above_OI,axiom,
! [A2: set_real,M2: real] :
( ! [X4: real] :
( ( member_real @ X4 @ A2 )
=> ( ord_less_eq_real @ X4 @ M2 ) )
=> ( condit7084745239686109512e_real @ A2 ) ) ).
% bdd_above.I
thf(fact_515_bdd__above_OI,axiom,
! [A2: set_num,M2: num] :
( ! [X4: num] :
( ( member_num @ X4 @ A2 )
=> ( ord_less_eq_num @ X4 @ M2 ) )
=> ( condit7995530295919666870ve_num @ A2 ) ) ).
% bdd_above.I
thf(fact_516_bdd__above_OI,axiom,
! [A2: set_nat,M2: nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ord_less_eq_nat @ X4 @ M2 ) )
=> ( condit2214826472909112428ve_nat @ A2 ) ) ).
% bdd_above.I
thf(fact_517_bdd__above_OI,axiom,
! [A2: set_int,M2: int] :
( ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( ord_less_eq_int @ X4 @ M2 ) )
=> ( condit2212336002400062152ve_int @ A2 ) ) ).
% bdd_above.I
thf(fact_518_bdd__above_OI,axiom,
! [A2: set_Dedekind_real,M2: dedekind_real] :
( ! [X4: dedekind_real] :
( ( member_Dedekind_real @ X4 @ A2 )
=> ( ord_le2716243287969276982d_real @ X4 @ M2 ) )
=> ( condit7337920006933021298d_real @ A2 ) ) ).
% bdd_above.I
thf(fact_519_inf_Obounded__iff,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real,C: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) )
= ( ( ord_le5071027967042688620d_real @ A @ B )
& ( ord_le5071027967042688620d_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_520_inf_Obounded__iff,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( inf_in6372492281993387880d_real @ B @ C ) )
= ( ( ord_le2716243287969276982d_real @ A @ B )
& ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_521_inf_Obounded__iff,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
= ( ( ord_less_eq_real @ A @ B )
& ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_522_inf_Obounded__iff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
= ( ( ord_less_eq_nat @ A @ B )
& ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_523_inf_Obounded__iff,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.bounded_iff
thf(fact_524_le__inf__iff,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) )
= ( ( ord_le5071027967042688620d_real @ X5 @ Y4 )
& ( ord_le5071027967042688620d_real @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_525_le__inf__iff,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ ( inf_in6372492281993387880d_real @ Y4 @ Z2 ) )
= ( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
& ( ord_le2716243287969276982d_real @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_526_le__inf__iff,axiom,
! [X5: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X5 @ ( inf_inf_real @ Y4 @ Z2 ) )
= ( ( ord_less_eq_real @ X5 @ Y4 )
& ( ord_less_eq_real @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_527_le__inf__iff,axiom,
! [X5: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y4 @ Z2 ) )
= ( ( ord_less_eq_nat @ X5 @ Y4 )
& ( ord_less_eq_nat @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_528_le__inf__iff,axiom,
! [X5: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y4 @ Z2 ) )
= ( ( ord_less_eq_int @ X5 @ Y4 )
& ( ord_less_eq_int @ X5 @ Z2 ) ) ) ).
% le_inf_iff
thf(fact_529_double__eq__0__iff,axiom,
! [A: dedekind_real] :
( ( ( plus_p4060926892116697567d_real @ A @ A )
= zero_z580800474297136991d_real )
= ( A = zero_z580800474297136991d_real ) ) ).
% double_eq_0_iff
thf(fact_530_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_531_double__eq__0__iff,axiom,
! [A: real] :
( ( ( plus_plus_real @ A @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% double_eq_0_iff
thf(fact_532_zero__less__two,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real ) ).
% zero_less_two
thf(fact_533_zero__less__two,axiom,
ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% zero_less_two
thf(fact_534_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_535_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_536_field__le__epsilon,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ! [E: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ E )
=> ( ord_le2716243287969276982d_real @ X5 @ ( plus_p4060926892116697567d_real @ Y4 @ E ) ) )
=> ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ).
% field_le_epsilon
thf(fact_537_field__le__epsilon,axiom,
! [X5: real,Y4: real] :
( ! [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
=> ( ord_less_eq_real @ X5 @ ( plus_plus_real @ Y4 @ E ) ) )
=> ( ord_less_eq_real @ X5 @ Y4 ) ) ).
% field_le_epsilon
thf(fact_538_add__mono1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ A @ one_on6069100329679821595d_real ) @ ( plus_p4060926892116697567d_real @ B @ one_on6069100329679821595d_real ) ) ) ).
% add_mono1
thf(fact_539_add__mono1,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% add_mono1
thf(fact_540_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_541_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_542_less__add__one,axiom,
! [A: dedekind_real] : ( ord_le2991122432403439658d_real @ A @ ( plus_p4060926892116697567d_real @ A @ one_on6069100329679821595d_real ) ) ).
% less_add_one
thf(fact_543_less__add__one,axiom,
! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% less_add_one
thf(fact_544_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_545_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_546_not__one__less__zero,axiom,
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_less_zero
thf(fact_547_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_548_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_549_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_550_inf_Oidem,axiom,
! [A: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ A @ A )
= A ) ).
% inf.idem
thf(fact_551_inf__idem,axiom,
! [X5: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X5 @ X5 )
= X5 ) ).
% inf_idem
thf(fact_552_inf_Oleft__idem,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ A @ ( inf_in841446065189194142d_real @ A @ B ) )
= ( inf_in841446065189194142d_real @ A @ B ) ) ).
% inf.left_idem
thf(fact_553_inf__left__idem,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) )
= ( inf_in841446065189194142d_real @ X5 @ Y4 ) ) ).
% inf_left_idem
thf(fact_554_inf_Oright__idem,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ B )
= ( inf_in841446065189194142d_real @ A @ B ) ) ).
% inf.right_idem
thf(fact_555_inf__right__idem,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ Y4 )
= ( inf_in841446065189194142d_real @ X5 @ Y4 ) ) ).
% inf_right_idem
thf(fact_556_bdd__above__mono,axiom,
! [B3: set_Dedekind_real,A2: set_Dedekind_real] :
( ( condit7337920006933021298d_real @ B3 )
=> ( ( ord_le5071027967042688620d_real @ A2 @ B3 )
=> ( condit7337920006933021298d_real @ A2 ) ) ) ).
% bdd_above_mono
thf(fact_557_real__trans__lemma,axiom,
! [X5: dedekind_preal,V: dedekind_preal,U2: dedekind_preal,Y4: dedekind_preal,V2: dedekind_preal,U3: dedekind_preal,X22: dedekind_real,V22: dedekind_real,U22: dedekind_real,Y22: dedekind_real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V ) @ ( plus_p3173629198307831117_preal @ U2 @ Y4 ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_p4060926892116697567d_real @ X22 @ V22 )
= ( plus_p4060926892116697567d_real @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) ) ) ) ) ).
% real_trans_lemma
thf(fact_558_real__trans__lemma,axiom,
! [X5: dedekind_preal,V: dedekind_preal,U2: dedekind_preal,Y4: dedekind_preal,V2: dedekind_preal,U3: dedekind_preal,X22: real,V22: real,U22: real,Y22: real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V ) @ ( plus_p3173629198307831117_preal @ U2 @ Y4 ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_real @ X22 @ V22 )
= ( plus_plus_real @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) ) ) ) ) ).
% real_trans_lemma
thf(fact_559_real__trans__lemma,axiom,
! [X5: dedekind_preal,V: dedekind_preal,U2: dedekind_preal,Y4: dedekind_preal,V2: dedekind_preal,U3: dedekind_preal,X22: num,V22: num,U22: num,Y22: num] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V ) @ ( plus_p3173629198307831117_preal @ U2 @ Y4 ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_num @ X22 @ V22 )
= ( plus_plus_num @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) ) ) ) ) ).
% real_trans_lemma
thf(fact_560_real__trans__lemma,axiom,
! [X5: dedekind_preal,V: dedekind_preal,U2: dedekind_preal,Y4: dedekind_preal,V2: dedekind_preal,U3: dedekind_preal,X22: nat,V22: nat,U22: nat,Y22: nat] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V ) @ ( plus_p3173629198307831117_preal @ U2 @ Y4 ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_nat @ X22 @ V22 )
= ( plus_plus_nat @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) ) ) ) ) ).
% real_trans_lemma
thf(fact_561_real__trans__lemma,axiom,
! [X5: dedekind_preal,V: dedekind_preal,U2: dedekind_preal,Y4: dedekind_preal,V2: dedekind_preal,U3: dedekind_preal,X22: int,V22: int,U22: int,Y22: int] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V ) @ ( plus_p3173629198307831117_preal @ U2 @ Y4 ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U2 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ V ) )
=> ( ( ( plus_plus_int @ X22 @ V22 )
= ( plus_plus_int @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X5 @ V2 ) @ ( plus_p3173629198307831117_preal @ U3 @ Y4 ) ) ) ) ) ).
% real_trans_lemma
thf(fact_562_linordered__field__no__lb,axiom,
! [X3: dedekind_real] :
? [Y5: dedekind_real] : ( ord_le2991122432403439658d_real @ Y5 @ X3 ) ).
% linordered_field_no_lb
thf(fact_563_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y5: real] : ( ord_less_real @ Y5 @ X3 ) ).
% linordered_field_no_lb
thf(fact_564_linordered__field__no__ub,axiom,
! [X3: dedekind_real] :
? [X_1: dedekind_real] : ( ord_le2991122432403439658d_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_565_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_566_linorder__neqE__linordered__idom,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( X5 != Y4 )
=> ( ~ ( ord_le2991122432403439658d_real @ X5 @ Y4 )
=> ( ord_le2991122432403439658d_real @ Y4 @ X5 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_567_linorder__neqE__linordered__idom,axiom,
! [X5: real,Y4: real] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_real @ X5 @ Y4 )
=> ( ord_less_real @ Y4 @ X5 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_568_linorder__neqE__linordered__idom,axiom,
! [X5: int,Y4: int] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_int @ X5 @ Y4 )
=> ( ord_less_int @ Y4 @ X5 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_569_ex__gt__or__lt,axiom,
! [A: real] :
? [B4: real] :
( ( ord_less_real @ A @ B4 )
| ( ord_less_real @ B4 @ A ) ) ).
% ex_gt_or_lt
thf(fact_570_inf__sup__aci_I4_J,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) )
= ( inf_in841446065189194142d_real @ X5 @ Y4 ) ) ).
% inf_sup_aci(4)
thf(fact_571_inf__sup__aci_I3_J,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) )
= ( inf_in841446065189194142d_real @ Y4 @ ( inf_in841446065189194142d_real @ X5 @ Z2 ) ) ) ).
% inf_sup_aci(3)
thf(fact_572_inf__sup__aci_I2_J,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ Z2 )
= ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) ) ) ).
% inf_sup_aci(2)
thf(fact_573_inf__sup__aci_I1_J,axiom,
( inf_in841446065189194142d_real
= ( ^ [X2: set_Dedekind_real,Y2: set_Dedekind_real] : ( inf_in841446065189194142d_real @ Y2 @ X2 ) ) ) ).
% inf_sup_aci(1)
thf(fact_574_inf_Oassoc,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real,C: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ C )
= ( inf_in841446065189194142d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) ) ) ).
% inf.assoc
thf(fact_575_inf__assoc,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ Z2 )
= ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) ) ) ).
% inf_assoc
thf(fact_576_inf_Ocommute,axiom,
( inf_in841446065189194142d_real
= ( ^ [A3: set_Dedekind_real,B2: set_Dedekind_real] : ( inf_in841446065189194142d_real @ B2 @ A3 ) ) ) ).
% inf.commute
thf(fact_577_inf__commute,axiom,
( inf_in841446065189194142d_real
= ( ^ [X2: set_Dedekind_real,Y2: set_Dedekind_real] : ( inf_in841446065189194142d_real @ Y2 @ X2 ) ) ) ).
% inf_commute
thf(fact_578_boolean__algebra__cancel_Oinf1,axiom,
! [A2: set_Dedekind_real,K: set_Dedekind_real,A: set_Dedekind_real,B: set_Dedekind_real] :
( ( A2
= ( inf_in841446065189194142d_real @ K @ A ) )
=> ( ( inf_in841446065189194142d_real @ A2 @ B )
= ( inf_in841446065189194142d_real @ K @ ( inf_in841446065189194142d_real @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_579_boolean__algebra__cancel_Oinf2,axiom,
! [B3: set_Dedekind_real,K: set_Dedekind_real,B: set_Dedekind_real,A: set_Dedekind_real] :
( ( B3
= ( inf_in841446065189194142d_real @ K @ B ) )
=> ( ( inf_in841446065189194142d_real @ A @ B3 )
= ( inf_in841446065189194142d_real @ K @ ( inf_in841446065189194142d_real @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_580_inf_Oleft__commute,axiom,
! [B: set_Dedekind_real,A: set_Dedekind_real,C: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ B @ ( inf_in841446065189194142d_real @ A @ C ) )
= ( inf_in841446065189194142d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) ) ) ).
% inf.left_commute
thf(fact_581_inf__left__commute,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) )
= ( inf_in841446065189194142d_real @ Y4 @ ( inf_in841446065189194142d_real @ X5 @ Z2 ) ) ) ).
% inf_left_commute
thf(fact_582_complete__interval,axiom,
! [A: real,B: real,P: real > $o] :
( ( ord_less_real @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: real] :
( ( ord_less_eq_real @ A @ C3 )
& ( ord_less_eq_real @ C3 @ B )
& ! [X3: real] :
( ( ( ord_less_eq_real @ A @ X3 )
& ( ord_less_real @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D2: real] :
( ! [X4: real] :
( ( ( ord_less_eq_real @ A @ X4 )
& ( ord_less_real @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_real @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_583_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D2: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_584_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D2: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D2 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_585_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_586_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_587_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_588_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_589_inf__sup__ord_I2_J,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_590_inf__sup__ord_I2_J,axiom,
! [X5: dedekind_real,Y4: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_591_inf__sup__ord_I2_J,axiom,
! [X5: real,Y4: real] : ( ord_less_eq_real @ ( inf_inf_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_592_inf__sup__ord_I2_J,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_593_inf__sup__ord_I2_J,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ Y4 ) ).
% inf_sup_ord(2)
thf(fact_594_inf__sup__ord_I1_J,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_595_inf__sup__ord_I1_J,axiom,
! [X5: dedekind_real,Y4: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_596_inf__sup__ord_I1_J,axiom,
! [X5: real,Y4: real] : ( ord_less_eq_real @ ( inf_inf_real @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_597_inf__sup__ord_I1_J,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_598_inf__sup__ord_I1_J,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ X5 ) ).
% inf_sup_ord(1)
thf(fact_599_inf__le1,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_600_inf__le1,axiom,
! [X5: dedekind_real,Y4: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_601_inf__le1,axiom,
! [X5: real,Y4: real] : ( ord_less_eq_real @ ( inf_inf_real @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_602_inf__le1,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_603_inf__le1,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ X5 ) ).
% inf_le1
thf(fact_604_inf__le2,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_605_inf__le2,axiom,
! [X5: dedekind_real,Y4: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_606_inf__le2,axiom,
! [X5: real,Y4: real] : ( ord_less_eq_real @ ( inf_inf_real @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_607_inf__le2,axiom,
! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_608_inf__le2,axiom,
! [X5: int,Y4: int] : ( ord_less_eq_int @ ( inf_inf_int @ X5 @ Y4 ) @ Y4 ) ).
% inf_le2
thf(fact_609_le__infE,axiom,
! [X5: set_Dedekind_real,A: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X5 @ ( inf_in841446065189194142d_real @ A @ B ) )
=> ~ ( ( ord_le5071027967042688620d_real @ X5 @ A )
=> ~ ( ord_le5071027967042688620d_real @ X5 @ B ) ) ) ).
% le_infE
thf(fact_610_le__infE,axiom,
! [X5: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ ( inf_in6372492281993387880d_real @ A @ B ) )
=> ~ ( ( ord_le2716243287969276982d_real @ X5 @ A )
=> ~ ( ord_le2716243287969276982d_real @ X5 @ B ) ) ) ).
% le_infE
thf(fact_611_le__infE,axiom,
! [X5: real,A: real,B: real] :
( ( ord_less_eq_real @ X5 @ ( inf_inf_real @ A @ B ) )
=> ~ ( ( ord_less_eq_real @ X5 @ A )
=> ~ ( ord_less_eq_real @ X5 @ B ) ) ) ).
% le_infE
thf(fact_612_le__infE,axiom,
! [X5: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B ) )
=> ~ ( ( ord_less_eq_nat @ X5 @ A )
=> ~ ( ord_less_eq_nat @ X5 @ B ) ) ) ).
% le_infE
thf(fact_613_le__infE,axiom,
! [X5: int,A: int,B: int] :
( ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A @ B ) )
=> ~ ( ( ord_less_eq_int @ X5 @ A )
=> ~ ( ord_less_eq_int @ X5 @ B ) ) ) ).
% le_infE
thf(fact_614_le__infI,axiom,
! [X5: set_Dedekind_real,A: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X5 @ A )
=> ( ( ord_le5071027967042688620d_real @ X5 @ B )
=> ( ord_le5071027967042688620d_real @ X5 @ ( inf_in841446065189194142d_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_615_le__infI,axiom,
! [X5: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ A )
=> ( ( ord_le2716243287969276982d_real @ X5 @ B )
=> ( ord_le2716243287969276982d_real @ X5 @ ( inf_in6372492281993387880d_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_616_le__infI,axiom,
! [X5: real,A: real,B: real] :
( ( ord_less_eq_real @ X5 @ A )
=> ( ( ord_less_eq_real @ X5 @ B )
=> ( ord_less_eq_real @ X5 @ ( inf_inf_real @ A @ B ) ) ) ) ).
% le_infI
thf(fact_617_le__infI,axiom,
! [X5: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ X5 @ A )
=> ( ( ord_less_eq_nat @ X5 @ B )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ A @ B ) ) ) ) ).
% le_infI
thf(fact_618_le__infI,axiom,
! [X5: int,A: int,B: int] :
( ( ord_less_eq_int @ X5 @ A )
=> ( ( ord_less_eq_int @ X5 @ B )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ A @ B ) ) ) ) ).
% le_infI
thf(fact_619_inf__mono,axiom,
! [A: set_Dedekind_real,C: set_Dedekind_real,B: set_Dedekind_real,D: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ C )
=> ( ( ord_le5071027967042688620d_real @ B @ D )
=> ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ ( inf_in841446065189194142d_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_620_inf__mono,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real,D: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ C )
=> ( ( ord_le2716243287969276982d_real @ B @ D )
=> ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ ( inf_in6372492281993387880d_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_621_inf__mono,axiom,
! [A: real,C: real,B: real,D: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ( ord_less_eq_real @ B @ D )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ ( inf_inf_real @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_622_inf__mono,axiom,
! [A: nat,C: nat,B: nat,D: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_623_inf__mono,axiom,
! [A: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ( ord_less_eq_int @ B @ D )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_624_le__infI1,axiom,
! [A: set_Dedekind_real,X5: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ X5 )
=> ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_625_le__infI1,axiom,
! [A: dedekind_real,X5: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ X5 )
=> ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_626_le__infI1,axiom,
! [A: real,X5: real,B: real] :
( ( ord_less_eq_real @ A @ X5 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_627_le__infI1,axiom,
! [A: nat,X5: nat,B: nat] :
( ( ord_less_eq_nat @ A @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_628_le__infI1,axiom,
! [A: int,X5: int,B: int] :
( ( ord_less_eq_int @ A @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X5 ) ) ).
% le_infI1
thf(fact_629_le__infI2,axiom,
! [B: set_Dedekind_real,X5: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ B @ X5 )
=> ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_630_le__infI2,axiom,
! [B: dedekind_real,X5: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ X5 )
=> ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_631_le__infI2,axiom,
! [B: real,X5: real,A: real] :
( ( ord_less_eq_real @ B @ X5 )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_632_le__infI2,axiom,
! [B: nat,X5: nat,A: nat] :
( ( ord_less_eq_nat @ B @ X5 )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_633_le__infI2,axiom,
! [B: int,X5: int,A: int] :
( ( ord_less_eq_int @ B @ X5 )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X5 ) ) ).
% le_infI2
thf(fact_634_inf_OorderE,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ B )
=> ( A
= ( inf_in841446065189194142d_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_635_inf_OorderE,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( A
= ( inf_in6372492281993387880d_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_636_inf_OorderE,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( A
= ( inf_inf_real @ A @ B ) ) ) ).
% inf.orderE
thf(fact_637_inf_OorderE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( A
= ( inf_inf_nat @ A @ B ) ) ) ).
% inf.orderE
thf(fact_638_inf_OorderE,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( A
= ( inf_inf_int @ A @ B ) ) ) ).
% inf.orderE
thf(fact_639_inf_OorderI,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( A
= ( inf_in841446065189194142d_real @ A @ B ) )
=> ( ord_le5071027967042688620d_real @ A @ B ) ) ).
% inf.orderI
thf(fact_640_inf_OorderI,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A
= ( inf_in6372492281993387880d_real @ A @ B ) )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% inf.orderI
thf(fact_641_inf_OorderI,axiom,
! [A: real,B: real] :
( ( A
= ( inf_inf_real @ A @ B ) )
=> ( ord_less_eq_real @ A @ B ) ) ).
% inf.orderI
thf(fact_642_inf_OorderI,axiom,
! [A: nat,B: nat] :
( ( A
= ( inf_inf_nat @ A @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% inf.orderI
thf(fact_643_inf_OorderI,axiom,
! [A: int,B: int] :
( ( A
= ( inf_inf_int @ A @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% inf.orderI
thf(fact_644_inf__unique,axiom,
! [F: set_Dedekind_real > set_Dedekind_real > set_Dedekind_real,X5: set_Dedekind_real,Y4: set_Dedekind_real] :
( ! [X4: set_Dedekind_real,Y5: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: set_Dedekind_real,Y5: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: set_Dedekind_real,Y5: set_Dedekind_real,Z3: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X4 @ Y5 )
=> ( ( ord_le5071027967042688620d_real @ X4 @ Z3 )
=> ( ord_le5071027967042688620d_real @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in841446065189194142d_real @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_645_inf__unique,axiom,
! [F: dedekind_real > dedekind_real > dedekind_real,X5: dedekind_real,Y4: dedekind_real] :
( ! [X4: dedekind_real,Y5: dedekind_real] : ( ord_le2716243287969276982d_real @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: dedekind_real,Y5: dedekind_real] : ( ord_le2716243287969276982d_real @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: dedekind_real,Y5: dedekind_real,Z3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X4 @ Y5 )
=> ( ( ord_le2716243287969276982d_real @ X4 @ Z3 )
=> ( ord_le2716243287969276982d_real @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_in6372492281993387880d_real @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_646_inf__unique,axiom,
! [F: real > real > real,X5: real,Y4: real] :
( ! [X4: real,Y5: real] : ( ord_less_eq_real @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: real,Y5: real] : ( ord_less_eq_real @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: real,Y5: real,Z3: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
=> ( ( ord_less_eq_real @ X4 @ Z3 )
=> ( ord_less_eq_real @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_real @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_647_inf__unique,axiom,
! [F: nat > nat > nat,X5: nat,Y4: nat] :
( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: nat,Y5: nat] : ( ord_less_eq_nat @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: nat,Y5: nat,Z3: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
=> ( ( ord_less_eq_nat @ X4 @ Z3 )
=> ( ord_less_eq_nat @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_648_inf__unique,axiom,
! [F: int > int > int,X5: int,Y4: int] :
( ! [X4: int,Y5: int] : ( ord_less_eq_int @ ( F @ X4 @ Y5 ) @ X4 )
=> ( ! [X4: int,Y5: int] : ( ord_less_eq_int @ ( F @ X4 @ Y5 ) @ Y5 )
=> ( ! [X4: int,Y5: int,Z3: int] :
( ( ord_less_eq_int @ X4 @ Y5 )
=> ( ( ord_less_eq_int @ X4 @ Z3 )
=> ( ord_less_eq_int @ X4 @ ( F @ Y5 @ Z3 ) ) ) )
=> ( ( inf_inf_int @ X5 @ Y4 )
= ( F @ X5 @ Y4 ) ) ) ) ) ).
% inf_unique
thf(fact_649_le__iff__inf,axiom,
( ord_le5071027967042688620d_real
= ( ^ [X2: set_Dedekind_real,Y2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_650_le__iff__inf,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X2: dedekind_real,Y2: dedekind_real] :
( ( inf_in6372492281993387880d_real @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_651_le__iff__inf,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( inf_inf_real @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_652_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y2: nat] :
( ( inf_inf_nat @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_653_le__iff__inf,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y2: int] :
( ( inf_inf_int @ X2 @ Y2 )
= X2 ) ) ) ).
% le_iff_inf
thf(fact_654_inf_Oabsorb1,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ B )
=> ( ( inf_in841446065189194142d_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_655_inf_Oabsorb1,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( inf_in6372492281993387880d_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_656_inf_Oabsorb1,axiom,
! [A: real,B: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( inf_inf_real @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_657_inf_Oabsorb1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_658_inf_Oabsorb1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb1
thf(fact_659_inf_Oabsorb2,axiom,
! [B: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ B @ A )
=> ( ( inf_in841446065189194142d_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_660_inf_Oabsorb2,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( inf_in6372492281993387880d_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_661_inf_Oabsorb2,axiom,
! [B: real,A: real] :
( ( ord_less_eq_real @ B @ A )
=> ( ( inf_inf_real @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_662_inf_Oabsorb2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_663_inf_Oabsorb2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb2
thf(fact_664_inf__absorb1,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X5 @ Y4 )
=> ( ( inf_in841446065189194142d_real @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_665_inf__absorb1,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( inf_in6372492281993387880d_real @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_666_inf__absorb1,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ( inf_inf_real @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_667_inf__absorb1,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_668_inf__absorb1,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( inf_inf_int @ X5 @ Y4 )
= X5 ) ) ).
% inf_absorb1
thf(fact_669_inf__absorb2,axiom,
! [Y4: set_Dedekind_real,X5: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ Y4 @ X5 )
=> ( ( inf_in841446065189194142d_real @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_670_inf__absorb2,axiom,
! [Y4: dedekind_real,X5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ( ( inf_in6372492281993387880d_real @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_671_inf__absorb2,axiom,
! [Y4: real,X5: real] :
( ( ord_less_eq_real @ Y4 @ X5 )
=> ( ( inf_inf_real @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_672_inf__absorb2,axiom,
! [Y4: nat,X5: nat] :
( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( ( inf_inf_nat @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_673_inf__absorb2,axiom,
! [Y4: int,X5: int] :
( ( ord_less_eq_int @ Y4 @ X5 )
=> ( ( inf_inf_int @ X5 @ Y4 )
= Y4 ) ) ).
% inf_absorb2
thf(fact_674_inf_OboundedE,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real,C: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) )
=> ~ ( ( ord_le5071027967042688620d_real @ A @ B )
=> ~ ( ord_le5071027967042688620d_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_675_inf_OboundedE,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( inf_in6372492281993387880d_real @ B @ C ) )
=> ~ ( ( ord_le2716243287969276982d_real @ A @ B )
=> ~ ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_676_inf_OboundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_eq_real @ A @ B )
=> ~ ( ord_less_eq_real @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_677_inf_OboundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_eq_nat @ A @ B )
=> ~ ( ord_less_eq_nat @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_678_inf_OboundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_eq_int @ A @ B )
=> ~ ( ord_less_eq_int @ A @ C ) ) ) ).
% inf.boundedE
thf(fact_679_inf_OboundedI,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real,C: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ B )
=> ( ( ord_le5071027967042688620d_real @ A @ C )
=> ( ord_le5071027967042688620d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_680_inf_OboundedI,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ A @ C )
=> ( ord_le2716243287969276982d_real @ A @ ( inf_in6372492281993387880d_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_681_inf_OboundedI,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ A @ ( inf_inf_real @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_682_inf_OboundedI,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_683_inf_OboundedI,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).
% inf.boundedI
thf(fact_684_inf__greatest,axiom,
! [X5: set_Dedekind_real,Y4: set_Dedekind_real,Z2: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ X5 @ Y4 )
=> ( ( ord_le5071027967042688620d_real @ X5 @ Z2 )
=> ( ord_le5071027967042688620d_real @ X5 @ ( inf_in841446065189194142d_real @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_685_inf__greatest,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ X5 @ Z2 )
=> ( ord_le2716243287969276982d_real @ X5 @ ( inf_in6372492281993387880d_real @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_686_inf__greatest,axiom,
! [X5: real,Y4: real,Z2: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ( ord_less_eq_real @ X5 @ Z2 )
=> ( ord_less_eq_real @ X5 @ ( inf_inf_real @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_687_inf__greatest,axiom,
! [X5: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ( ord_less_eq_nat @ X5 @ ( inf_inf_nat @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_688_inf__greatest,axiom,
! [X5: int,Y4: int,Z2: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ X5 @ Z2 )
=> ( ord_less_eq_int @ X5 @ ( inf_inf_int @ Y4 @ Z2 ) ) ) ) ).
% inf_greatest
thf(fact_689_inf_Oorder__iff,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A3: set_Dedekind_real,B2: set_Dedekind_real] :
( A3
= ( inf_in841446065189194142d_real @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_690_inf_Oorder__iff,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( A3
= ( inf_in6372492281993387880d_real @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_691_inf_Oorder__iff,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( A3
= ( inf_inf_real @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_692_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( A3
= ( inf_inf_nat @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_693_inf_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( A3
= ( inf_inf_int @ A3 @ B2 ) ) ) ) ).
% inf.order_iff
thf(fact_694_inf_Ocobounded1,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_695_inf_Ocobounded1,axiom,
! [A: dedekind_real,B: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_696_inf_Ocobounded1,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_697_inf_Ocobounded1,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_698_inf_Ocobounded1,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).
% inf.cobounded1
thf(fact_699_inf_Ocobounded2,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] : ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_700_inf_Ocobounded2,axiom,
! [A: dedekind_real,B: dedekind_real] : ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_701_inf_Ocobounded2,axiom,
! [A: real,B: real] : ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_702_inf_Ocobounded2,axiom,
! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_703_inf_Ocobounded2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).
% inf.cobounded2
thf(fact_704_inf_Oabsorb__iff1,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A3: set_Dedekind_real,B2: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_705_inf_Oabsorb__iff1,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( inf_in6372492281993387880d_real @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_706_inf_Oabsorb__iff1,axiom,
( ord_less_eq_real
= ( ^ [A3: real,B2: real] :
( ( inf_inf_real @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_707_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( inf_inf_nat @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_708_inf_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B2: int] :
( ( inf_inf_int @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb_iff1
thf(fact_709_inf_Oabsorb__iff2,axiom,
( ord_le5071027967042688620d_real
= ( ^ [B2: set_Dedekind_real,A3: set_Dedekind_real] :
( ( inf_in841446065189194142d_real @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_710_inf_Oabsorb__iff2,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B2: dedekind_real,A3: dedekind_real] :
( ( inf_in6372492281993387880d_real @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_711_inf_Oabsorb__iff2,axiom,
( ord_less_eq_real
= ( ^ [B2: real,A3: real] :
( ( inf_inf_real @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_712_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( inf_inf_nat @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_713_inf_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A3: int] :
( ( inf_inf_int @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb_iff2
thf(fact_714_inf_OcoboundedI1,axiom,
! [A: set_Dedekind_real,C: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ A @ C )
=> ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_715_inf_OcoboundedI1,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ C )
=> ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_716_inf_OcoboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_eq_real @ A @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_717_inf_OcoboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_718_inf_OcoboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI1
thf(fact_719_inf_OcoboundedI2,axiom,
! [B: set_Dedekind_real,C: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ B @ C )
=> ( ord_le5071027967042688620d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_720_inf_OcoboundedI2,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_721_inf_OcoboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_722_inf_OcoboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_723_inf_OcoboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.coboundedI2
thf(fact_724_less__infI1,axiom,
! [A: set_Dedekind_real,X5: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ A @ X5 )
=> ( ord_le1729950913864802912d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_725_less__infI1,axiom,
! [A: dedekind_real,X5: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ X5 )
=> ( ord_le2991122432403439658d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_726_less__infI1,axiom,
! [A: real,X5: real,B: real] :
( ( ord_less_real @ A @ X5 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_727_less__infI1,axiom,
! [A: nat,X5: nat,B: nat] :
( ( ord_less_nat @ A @ X5 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_728_less__infI1,axiom,
! [A: int,X5: int,B: int] :
( ( ord_less_int @ A @ X5 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X5 ) ) ).
% less_infI1
thf(fact_729_less__infI2,axiom,
! [B: set_Dedekind_real,X5: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ B @ X5 )
=> ( ord_le1729950913864802912d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_730_less__infI2,axiom,
! [B: dedekind_real,X5: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ X5 )
=> ( ord_le2991122432403439658d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_731_less__infI2,axiom,
! [B: real,X5: real,A: real] :
( ( ord_less_real @ B @ X5 )
=> ( ord_less_real @ ( inf_inf_real @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_732_less__infI2,axiom,
! [B: nat,X5: nat,A: nat] :
( ( ord_less_nat @ B @ X5 )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_733_less__infI2,axiom,
! [B: int,X5: int,A: int] :
( ( ord_less_int @ B @ X5 )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X5 ) ) ).
% less_infI2
thf(fact_734_inf_Oabsorb3,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ A @ B )
=> ( ( inf_in841446065189194142d_real @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_735_inf_Oabsorb3,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( inf_in6372492281993387880d_real @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_736_inf_Oabsorb3,axiom,
! [A: real,B: real] :
( ( ord_less_real @ A @ B )
=> ( ( inf_inf_real @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_737_inf_Oabsorb3,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( inf_inf_nat @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_738_inf_Oabsorb3,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ( inf_inf_int @ A @ B )
= A ) ) ).
% inf.absorb3
thf(fact_739_inf_Oabsorb4,axiom,
! [B: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ B @ A )
=> ( ( inf_in841446065189194142d_real @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_740_inf_Oabsorb4,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( inf_in6372492281993387880d_real @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_741_inf_Oabsorb4,axiom,
! [B: real,A: real] :
( ( ord_less_real @ B @ A )
=> ( ( inf_inf_real @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_742_inf_Oabsorb4,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( inf_inf_nat @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_743_inf_Oabsorb4,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ( inf_inf_int @ A @ B )
= B ) ) ).
% inf.absorb4
thf(fact_744_inf_Ostrict__boundedE,axiom,
! [A: set_Dedekind_real,B: set_Dedekind_real,C: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ A @ ( inf_in841446065189194142d_real @ B @ C ) )
=> ~ ( ( ord_le1729950913864802912d_real @ A @ B )
=> ~ ( ord_le1729950913864802912d_real @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_745_inf_Ostrict__boundedE,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( inf_in6372492281993387880d_real @ B @ C ) )
=> ~ ( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_746_inf_Ostrict__boundedE,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_real @ A @ ( inf_inf_real @ B @ C ) )
=> ~ ( ( ord_less_real @ A @ B )
=> ~ ( ord_less_real @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_747_inf_Ostrict__boundedE,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C ) )
=> ~ ( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_748_inf_Ostrict__boundedE,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ ( inf_inf_int @ B @ C ) )
=> ~ ( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ A @ C ) ) ) ).
% inf.strict_boundedE
thf(fact_749_inf_Ostrict__order__iff,axiom,
( ord_le1729950913864802912d_real
= ( ^ [A3: set_Dedekind_real,B2: set_Dedekind_real] :
( ( A3
= ( inf_in841446065189194142d_real @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_750_inf_Ostrict__order__iff,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A3: dedekind_real,B2: dedekind_real] :
( ( A3
= ( inf_in6372492281993387880d_real @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_751_inf_Ostrict__order__iff,axiom,
( ord_less_real
= ( ^ [A3: real,B2: real] :
( ( A3
= ( inf_inf_real @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_752_inf_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( A3
= ( inf_inf_nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_753_inf_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [A3: int,B2: int] :
( ( A3
= ( inf_inf_int @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% inf.strict_order_iff
thf(fact_754_inf_Ostrict__coboundedI1,axiom,
! [A: set_Dedekind_real,C: set_Dedekind_real,B: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ A @ C )
=> ( ord_le1729950913864802912d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_755_inf_Ostrict__coboundedI1,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ C )
=> ( ord_le2991122432403439658d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_756_inf_Ostrict__coboundedI1,axiom,
! [A: real,C: real,B: real] :
( ( ord_less_real @ A @ C )
=> ( ord_less_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_757_inf_Ostrict__coboundedI1,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ A @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_758_inf_Ostrict__coboundedI1,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI1
thf(fact_759_inf_Ostrict__coboundedI2,axiom,
! [B: set_Dedekind_real,C: set_Dedekind_real,A: set_Dedekind_real] :
( ( ord_le1729950913864802912d_real @ B @ C )
=> ( ord_le1729950913864802912d_real @ ( inf_in841446065189194142d_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_760_inf_Ostrict__coboundedI2,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ ( inf_in6372492281993387880d_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_761_inf_Ostrict__coboundedI2,axiom,
! [B: real,C: real,A: real] :
( ( ord_less_real @ B @ C )
=> ( ord_less_real @ ( inf_inf_real @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_762_inf_Ostrict__coboundedI2,axiom,
! [B: nat,C: nat,A: nat] :
( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_763_inf_Ostrict__coboundedI2,axiom,
! [B: int,C: int,A: int] :
( ( ord_less_int @ B @ C )
=> ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).
% inf.strict_coboundedI2
thf(fact_764_bdd__above_Ounfold,axiom,
( condit7084745239686109512e_real
= ( ^ [A4: set_real] :
? [M3: real] :
! [X2: real] :
( ( member_real @ X2 @ A4 )
=> ( ord_less_eq_real @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_765_bdd__above_Ounfold,axiom,
( condit7995530295919666870ve_num
= ( ^ [A4: set_num] :
? [M3: num] :
! [X2: num] :
( ( member_num @ X2 @ A4 )
=> ( ord_less_eq_num @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_766_bdd__above_Ounfold,axiom,
( condit2214826472909112428ve_nat
= ( ^ [A4: set_nat] :
? [M3: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_767_bdd__above_Ounfold,axiom,
( condit2212336002400062152ve_int
= ( ^ [A4: set_int] :
? [M3: int] :
! [X2: int] :
( ( member_int @ X2 @ A4 )
=> ( ord_less_eq_int @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_768_bdd__above_Ounfold,axiom,
( condit7337920006933021298d_real
= ( ^ [A4: set_Dedekind_real] :
? [M3: dedekind_real] :
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ A4 )
=> ( ord_le2716243287969276982d_real @ X2 @ M3 ) ) ) ) ).
% bdd_above.unfold
thf(fact_769_bdd__above_OE,axiom,
! [A2: set_real] :
( ( condit7084745239686109512e_real @ A2 )
=> ~ ! [M4: real] :
~ ! [X3: real] :
( ( member_real @ X3 @ A2 )
=> ( ord_less_eq_real @ X3 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_770_bdd__above_OE,axiom,
! [A2: set_num] :
( ( condit7995530295919666870ve_num @ A2 )
=> ~ ! [M4: num] :
~ ! [X3: num] :
( ( member_num @ X3 @ A2 )
=> ( ord_less_eq_num @ X3 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_771_bdd__above_OE,axiom,
! [A2: set_nat] :
( ( condit2214826472909112428ve_nat @ A2 )
=> ~ ! [M4: nat] :
~ ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( ord_less_eq_nat @ X3 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_772_bdd__above_OE,axiom,
! [A2: set_int] :
( ( condit2212336002400062152ve_int @ A2 )
=> ~ ! [M4: int] :
~ ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ( ord_less_eq_int @ X3 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_773_bdd__above_OE,axiom,
! [A2: set_Dedekind_real] :
( ( condit7337920006933021298d_real @ A2 )
=> ~ ! [M4: dedekind_real] :
~ ! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3 @ A2 )
=> ( ord_le2716243287969276982d_real @ X3 @ M4 ) ) ) ).
% bdd_above.E
thf(fact_774_bdd__above__Int1,axiom,
! [A2: set_Dedekind_real,B3: set_Dedekind_real] :
( ( condit7337920006933021298d_real @ A2 )
=> ( condit7337920006933021298d_real @ ( inf_in841446065189194142d_real @ A2 @ B3 ) ) ) ).
% bdd_above_Int1
thf(fact_775_bdd__above__Int2,axiom,
! [B3: set_Dedekind_real,A2: set_Dedekind_real] :
( ( condit7337920006933021298d_real @ B3 )
=> ( condit7337920006933021298d_real @ ( inf_in841446065189194142d_real @ A2 @ B3 ) ) ) ).
% bdd_above_Int2
thf(fact_776_zero__less__one__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one_class.zero_le_one
thf(fact_777_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_778_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_779_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_780_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_781_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_782_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_783_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_784_not__one__le__zero,axiom,
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ zero_z580800474297136991d_real ) ).
% not_one_le_zero
thf(fact_785_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_786_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_787_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_788_add__less__zeroD,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ ( plus_p4060926892116697567d_real @ X5 @ Y4 ) @ zero_z580800474297136991d_real )
=> ( ( ord_le2991122432403439658d_real @ X5 @ zero_z580800474297136991d_real )
| ( ord_le2991122432403439658d_real @ Y4 @ zero_z580800474297136991d_real ) ) ) ).
% add_less_zeroD
thf(fact_789_add__less__zeroD,axiom,
! [X5: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X5 @ Y4 ) @ zero_zero_real )
=> ( ( ord_less_real @ X5 @ zero_zero_real )
| ( ord_less_real @ Y4 @ zero_zero_real ) ) ) ).
% add_less_zeroD
thf(fact_790_add__less__zeroD,axiom,
! [X5: int,Y4: int] :
( ( ord_less_int @ ( plus_plus_int @ X5 @ Y4 ) @ zero_zero_int )
=> ( ( ord_less_int @ X5 @ zero_zero_int )
| ( ord_less_int @ Y4 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_791_zero__less__one,axiom,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ one_on6069100329679821595d_real ).
% zero_less_one
thf(fact_792_zero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one
thf(fact_793_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_794_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_795_Int__subset__iff,axiom,
! [C4: set_Dedekind_real,A2: set_Dedekind_real,B3: set_Dedekind_real] :
( ( ord_le5071027967042688620d_real @ C4 @ ( inf_in841446065189194142d_real @ A2 @ B3 ) )
= ( ( ord_le5071027967042688620d_real @ C4 @ A2 )
& ( ord_le5071027967042688620d_real @ C4 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_796_IntI,axiom,
! [C: dedekind_real,A2: set_Dedekind_real,B3: set_Dedekind_real] :
( ( member_Dedekind_real @ C @ A2 )
=> ( ( member_Dedekind_real @ C @ B3 )
=> ( member_Dedekind_real @ C @ ( inf_in841446065189194142d_real @ A2 @ B3 ) ) ) ) ).
% IntI
thf(fact_797_Int__iff,axiom,
! [C: dedekind_real,A2: set_Dedekind_real,B3: set_Dedekind_real] :
( ( member_Dedekind_real @ C @ ( inf_in841446065189194142d_real @ A2 @ B3 ) )
= ( ( member_Dedekind_real @ C @ A2 )
& ( member_Dedekind_real @ C @ B3 ) ) ) ).
% Int_iff
thf(fact_798_dbl__dec__simps_I2_J,axiom,
( ( neg_nu7286324685880884523d_real @ zero_z580800474297136991d_real )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% dbl_dec_simps(2)
thf(fact_799_dbl__dec__simps_I2_J,axiom,
( ( neg_nu6075765906172075777c_real @ zero_zero_real )
= ( uminus_uminus_real @ one_one_real ) ) ).
% dbl_dec_simps(2)
thf(fact_800_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_801_verit__minus__simplify_I4_J,axiom,
! [B: dedekind_real] :
( ( uminus7714077491378687647d_real @ ( uminus7714077491378687647d_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_802_verit__minus__simplify_I4_J,axiom,
! [B: real] :
( ( uminus_uminus_real @ ( uminus_uminus_real @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_803_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_804_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_805_dual__order_Orefl,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% dual_order.refl
thf(fact_806_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_807_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_808_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_809_order__refl,axiom,
! [X5: dedekind_real] : ( ord_le2716243287969276982d_real @ X5 @ X5 ) ).
% order_refl
thf(fact_810_order__refl,axiom,
! [X5: real] : ( ord_less_eq_real @ X5 @ X5 ) ).
% order_refl
thf(fact_811_order__refl,axiom,
! [X5: num] : ( ord_less_eq_num @ X5 @ X5 ) ).
% order_refl
thf(fact_812_order__refl,axiom,
! [X5: nat] : ( ord_less_eq_nat @ X5 @ X5 ) ).
% order_refl
thf(fact_813_order__refl,axiom,
! [X5: int] : ( ord_less_eq_int @ X5 @ X5 ) ).
% order_refl
thf(fact_814__092_060open_062_092_060forall_062s_092_060in_062_123z_O_A_092_060exists_062x_092_060in_062S_O_Az_A_061_Ax_A_L_A_N_AX_A_L_A1_125_A_092_060inter_062_A_123x_O_A0_A_060_Ax_125_O_As_A_060_AY_A_L_A_N_AX_A_L_A2_092_060close_062,axiom,
! [X3: dedekind_real] :
( ( member_Dedekind_real @ X3
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [Y2: dedekind_real] :
( ( member_Dedekind_real @ Y2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ Y2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
=> ( ord_le2991122432403439658d_real @ X3 @ ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ y @ ( uminus7714077491378687647d_real @ x ) ) @ ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ) ) ).
% \<open>\<forall>s\<in>{z. \<exists>x\<in>S. z = x + - X + 1} \<inter> {x. 0 < x}. s < Y + - X + 2\<close>
thf(fact_815_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numera6440378960330123505d_real @ M )
= ( numera6440378960330123505d_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_816_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_817_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_int @ M )
= ( numeral_numeral_int @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_818_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_real @ M )
= ( numeral_numeral_real @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_819_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_820_subsetI,axiom,
! [A2: set_Dedekind_real,B3: set_Dedekind_real] :
( ! [X4: dedekind_real] :
( ( member_Dedekind_real @ X4 @ A2 )
=> ( member_Dedekind_real @ X4 @ B3 ) )
=> ( ord_le5071027967042688620d_real @ A2 @ B3 ) ) ).
% subsetI
thf(fact_821_preal__add__less__cancel__right,axiom,
! [R: dedekind_preal,T2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T2 ) @ ( plus_p3173629198307831117_preal @ S2 @ T2 ) )
= ( ord_le5708704896291381698_preal @ R @ S2 ) ) ).
% preal_add_less_cancel_right
thf(fact_822_preal__add__less__cancel__left,axiom,
! [T2: dedekind_preal,R: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T2 @ R ) @ ( plus_p3173629198307831117_preal @ T2 @ S2 ) )
= ( ord_le5708704896291381698_preal @ R @ S2 ) ) ).
% preal_add_less_cancel_left
thf(fact_823_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2716243287969276982d_real @ ( numera6440378960330123505d_real @ M ) @ ( numera6440378960330123505d_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_824_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_825_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_826_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_827_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le2991122432403439658d_real @ ( numera6440378960330123505d_real @ M ) @ ( numera6440378960330123505d_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_828_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_829_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_830_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_831_add__numeral__left,axiom,
! [V: num,W: num,Z2: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ V ) @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ W ) @ Z2 ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_832_add__numeral__left,axiom,
! [V: num,W: num,Z2: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_833_add__numeral__left,axiom,
! [V: num,W: num,Z2: int] :
( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_834_add__numeral__left,axiom,
! [V: num,W: num,Z2: real] :
( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z2 ) ) ).
% add_numeral_left
thf(fact_835_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M ) @ ( numera6440378960330123505d_real @ N ) )
= ( numera6440378960330123505d_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_836_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_837_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_838_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_839_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_840_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_841_neg__numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( M = N ) ) ).
% neg_numeral_eq_iff
thf(fact_842_preal__add__le__cancel__left,axiom,
! [T2: dedekind_preal,R: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T2 @ R ) @ ( plus_p3173629198307831117_preal @ T2 @ S2 ) )
= ( ord_le5604041210740703414_preal @ R @ S2 ) ) ).
% preal_add_le_cancel_left
thf(fact_843_preal__add__le__cancel__right,axiom,
! [R: dedekind_preal,T2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R @ T2 ) @ ( plus_p3173629198307831117_preal @ S2 @ T2 ) )
= ( ord_le5604041210740703414_preal @ R @ S2 ) ) ).
% preal_add_le_cancel_right
thf(fact_844_dbl__dec__simps_I3_J,axiom,
( ( neg_nu7286324685880884523d_real @ one_on6069100329679821595d_real )
= one_on6069100329679821595d_real ) ).
% dbl_dec_simps(3)
thf(fact_845_dbl__dec__simps_I3_J,axiom,
( ( neg_nu6075765906172075777c_real @ one_one_real )
= one_one_real ) ).
% dbl_dec_simps(3)
thf(fact_846_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_847_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_848_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_849_neg__numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_eq_num @ N @ M ) ) ).
% neg_numeral_le_iff
thf(fact_850_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_851_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_852_neg__numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( ord_less_num @ N @ M ) ) ).
% neg_numeral_less_iff
thf(fact_853_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_on6069100329679821595d_real
= ( numera6440378960330123505d_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_854_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_855_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_int
= ( numeral_numeral_int @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_856_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_real
= ( numeral_numeral_real @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_857_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numera6440378960330123505d_real @ N )
= one_on6069100329679821595d_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_858_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_859_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_int @ N )
= one_one_int )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_860_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_real @ N )
= one_one_real )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_861_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( uminus7714077491378687647d_real @ ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ M ) @ ( numera6440378960330123505d_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_862_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_863_add__neg__numeral__simps_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_864_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_le2716243287969276982d_real @ ( numera6440378960330123505d_real @ N ) @ one_on6069100329679821595d_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_865_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_866_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_867_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_868_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_869_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_870_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_871_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_872_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_873_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_874_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N ) )
= ( numeral_numeral_int @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_875_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N ) )
= ( numeral_numeral_real @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_876_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_877_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_878_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_int @ ( numeral_numeral_int @ N ) @ one_one_int )
= ( numeral_numeral_int @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_879_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_real @ ( numeral_numeral_real @ N ) @ one_one_real )
= ( numeral_numeral_real @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_880_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_881_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N ) )
= ( uminus_uminus_real @ one_one_real ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_882_numeral__eq__neg__one__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N ) )
= ( uminus_uminus_int @ one_one_int ) )
= ( N = one ) ) ).
% numeral_eq_neg_one_iff
thf(fact_883_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_884_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_real @ one_one_real )
= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_885_neg__one__eq__numeral__iff,axiom,
! [N: num] :
( ( ( uminus_uminus_int @ one_one_int )
= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
= ( N = one ) ) ).
% neg_one_eq_numeral_iff
thf(fact_886_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_887_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_888_not__neg__one__le__neg__numeral__iff,axiom,
! [M: num] :
( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
= ( M != one ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_889_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_890_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_891_neg__numeral__less__neg__one__iff,axiom,
! [M: num] :
( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
= ( M != one ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_892_one__add__one,axiom,
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ one_on6069100329679821595d_real )
= ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_893_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_894_one__add__one,axiom,
( ( plus_plus_int @ one_one_int @ one_one_int )
= ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_895_one__add__one,axiom,
( ( plus_plus_real @ one_one_real @ one_one_real )
= ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_896_add__neg__numeral__special_I9_J,axiom,
( ( plus_p4060926892116697567d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) )
= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_897_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
= ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_898_add__neg__numeral__special_I9_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_899_in__mono,axiom,
! [A2: set_Dedekind_real,B3: set_Dedekind_real,X5: dedekind_real] :
( ( ord_le5071027967042688620d_real @ A2 @ B3 )
=> ( ( member_Dedekind_real @ X5 @ A2 )
=> ( member_Dedekind_real @ X5 @ B3 ) ) ) ).
% in_mono
thf(fact_900_subsetD,axiom,
! [A2: set_Dedekind_real,B3: set_Dedekind_real,C: dedekind_real] :
( ( ord_le5071027967042688620d_real @ A2 @ B3 )
=> ( ( member_Dedekind_real @ C @ A2 )
=> ( member_Dedekind_real @ C @ B3 ) ) ) ).
% subsetD
thf(fact_901_subset__eq,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A4: set_Dedekind_real,B5: set_Dedekind_real] :
! [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ A4 )
=> ( member_Dedekind_real @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_902_subset__iff,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A4: set_Dedekind_real,B5: set_Dedekind_real] :
! [T3: dedekind_real] :
( ( member_Dedekind_real @ T3 @ A4 )
=> ( member_Dedekind_real @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_903_Collect__mono,axiom,
! [P: dedekind_real > $o,Q: dedekind_real > $o] :
( ! [X4: dedekind_real] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( ord_le5071027967042688620d_real @ ( collec1873404778653292075d_real @ P ) @ ( collec1873404778653292075d_real @ Q ) ) ) ).
% Collect_mono
thf(fact_904_le__num__One__iff,axiom,
! [X5: num] :
( ( ord_less_eq_num @ X5 @ one )
= ( X5 = one ) ) ).
% le_num_One_iff
thf(fact_905_Collect__subset,axiom,
! [A2: set_Dedekind_real,P: dedekind_real > $o] :
( ord_le5071027967042688620d_real
@ ( collec1873404778653292075d_real
@ ^ [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ A2 )
& ( P @ X2 ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_906_less__eq__set__def,axiom,
( ord_le5071027967042688620d_real
= ( ^ [A4: set_Dedekind_real,B5: set_Dedekind_real] :
( ord_le60669685330047769real_o
@ ^ [X2: dedekind_real] : ( member_Dedekind_real @ X2 @ A4 )
@ ^ [X2: dedekind_real] : ( member_Dedekind_real @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_907_Collect__mono__iff,axiom,
! [P: dedekind_real > $o,Q: dedekind_real > $o] :
( ( ord_le5071027967042688620d_real @ ( collec1873404778653292075d_real @ P ) @ ( collec1873404778653292075d_real @ Q ) )
= ( ! [X2: dedekind_real] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_908_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,Y22: 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 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X1 @ Y22 ) @ ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ) ).
% real_le_lemma
thf(fact_909_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_910_numeral__Bit0,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bit0 @ N ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ ( numera6440378960330123505d_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_911_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_912_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_Bit0
thf(fact_913_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_Bit0
thf(fact_914_numeral__One,axiom,
( ( numera6440378960330123505d_real @ one )
= one_on6069100329679821595d_real ) ).
% numeral_One
thf(fact_915_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_916_numeral__One,axiom,
( ( numeral_numeral_int @ one )
= one_one_int ) ).
% numeral_One
thf(fact_917_numeral__One,axiom,
( ( numeral_numeral_real @ one )
= one_one_real ) ).
% numeral_One
thf(fact_918_numeral__code_I2_J,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ ( bit0 @ N ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ N ) @ ( numera6440378960330123505d_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_919_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_code(2)
thf(fact_920_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_int @ ( bit0 @ N ) )
= ( plus_plus_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_code(2)
thf(fact_921_numeral__code_I2_J,axiom,
! [N: num] :
( ( numeral_numeral_real @ ( bit0 @ N ) )
= ( plus_plus_real @ ( numeral_numeral_real @ N ) @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_code(2)
thf(fact_922_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_923_preal__add__right__less__cancel,axiom,
! [R: dedekind_preal,T2: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T2 ) @ ( plus_p3173629198307831117_preal @ S2 @ T2 ) )
=> ( ord_le5708704896291381698_preal @ R @ S2 ) ) ).
% preal_add_right_less_cancel
thf(fact_924_preal__add__left__less__cancel,axiom,
! [T2: dedekind_preal,R: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T2 @ R ) @ ( plus_p3173629198307831117_preal @ T2 @ S2 ) )
=> ( ord_le5708704896291381698_preal @ R @ S2 ) ) ).
% preal_add_left_less_cancel
thf(fact_925_preal__self__less__add__left,axiom,
! [R: dedekind_preal,S2: dedekind_preal] : ( ord_le5708704896291381698_preal @ R @ ( plus_p3173629198307831117_preal @ R @ S2 ) ) ).
% preal_self_less_add_left
thf(fact_926_preal__add__right__cancel,axiom,
! [R: dedekind_preal,T2: dedekind_preal,S2: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R @ T2 )
= ( plus_p3173629198307831117_preal @ S2 @ T2 ) )
=> ( R = S2 ) ) ).
% preal_add_right_cancel
thf(fact_927_preal__add__less2__mono2,axiom,
! [R: dedekind_preal,S2: dedekind_preal,T2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T2 @ R ) @ ( plus_p3173629198307831117_preal @ T2 @ S2 ) ) ) ).
% preal_add_less2_mono2
thf(fact_928_preal__add__less2__mono1,axiom,
! [R: dedekind_preal,S2: dedekind_preal,T2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S2 )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T2 ) @ ( plus_p3173629198307831117_preal @ S2 @ T2 ) ) ) ).
% preal_add_less2_mono1
thf(fact_929_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_930_preal__trans__lemma,axiom,
! [X5: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y4: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X5 @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y4 ) )
=> ( ( ( plus_p3173629198307831117_preal @ X5 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y4 ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_931_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X2: dedekind_preal,Y2: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y2 @ X2 ) ) ) ).
% preal_add_commute
thf(fact_932_preal__add__assoc,axiom,
! [X5: dedekind_preal,Y4: dedekind_preal,Z2: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X5 @ Y4 ) @ Z2 )
= ( plus_p3173629198307831117_preal @ X5 @ ( plus_p3173629198307831117_preal @ Y4 @ Z2 ) ) ) ).
% preal_add_assoc
thf(fact_933_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_934_uminus__numeral__One,axiom,
( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ one ) )
= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% uminus_numeral_One
thf(fact_935_uminus__numeral__One,axiom,
( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
= ( uminus_uminus_real @ one_one_real ) ) ).
% uminus_numeral_One
thf(fact_936_uminus__numeral__One,axiom,
( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% uminus_numeral_One
thf(fact_937_zero__neq__numeral,axiom,
! [N: num] :
( zero_z580800474297136991d_real
!= ( numera6440378960330123505d_real @ N ) ) ).
% zero_neq_numeral
thf(fact_938_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_939_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( numeral_numeral_int @ N ) ) ).
% zero_neq_numeral
thf(fact_940_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( numeral_numeral_real @ N ) ) ).
% zero_neq_numeral
thf(fact_941_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numera6440378960330123505d_real @ M )
!= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_942_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_real @ M )
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_943_numeral__neq__neg__numeral,axiom,
! [M: num,N: num] :
( ( numeral_numeral_int @ M )
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_944_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) )
!= ( numera6440378960330123505d_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_945_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
!= ( numeral_numeral_real @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_946_neg__numeral__neq__numeral,axiom,
! [M: num,N: num] :
( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
!= ( numeral_numeral_int @ N ) ) ).
% neg_numeral_neq_numeral
thf(fact_947_zero__le__numeral,axiom,
! [N: num] : ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( numera6440378960330123505d_real @ N ) ) ).
% zero_le_numeral
thf(fact_948_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_le_numeral
thf(fact_949_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_950_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_le_numeral
thf(fact_951_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_le2716243287969276982d_real @ ( numera6440378960330123505d_real @ N ) @ zero_z580800474297136991d_real ) ).
% not_numeral_le_zero
thf(fact_952_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_le_zero
thf(fact_953_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_954_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_le_zero
thf(fact_955_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_le2991122432403439658d_real @ ( numera6440378960330123505d_real @ N ) @ zero_z580800474297136991d_real ) ).
% not_numeral_less_zero
thf(fact_956_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_957_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% not_numeral_less_zero
thf(fact_958_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% not_numeral_less_zero
thf(fact_959_zero__less__numeral,axiom,
! [N: num] : ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( numera6440378960330123505d_real @ N ) ) ).
% zero_less_numeral
thf(fact_960_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_961_zero__less__numeral,axiom,
! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% zero_less_numeral
thf(fact_962_zero__less__numeral,axiom,
! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% zero_less_numeral
thf(fact_963_one__le__numeral,axiom,
! [N: num] : ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ N ) ) ).
% one_le_numeral
thf(fact_964_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% one_le_numeral
thf(fact_965_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_966_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% one_le_numeral
thf(fact_967_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_le2991122432403439658d_real @ ( numera6440378960330123505d_real @ N ) @ one_on6069100329679821595d_real ) ).
% not_numeral_less_one
thf(fact_968_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_969_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% not_numeral_less_one
thf(fact_970_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% not_numeral_less_one
thf(fact_971_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( numera6440378960330123505d_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_972_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_973_neg__numeral__le__numeral,axiom,
! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_le_numeral
thf(fact_974_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le2716243287969276982d_real @ ( numera6440378960330123505d_real @ M ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_975_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_976_not__numeral__le__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_977_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_z580800474297136991d_real
!= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_978_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_979_zero__neq__neg__numeral,axiom,
! [N: num] :
( zero_zero_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% zero_neq_neg_numeral
thf(fact_980_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_le2991122432403439658d_real @ ( numera6440378960330123505d_real @ M ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_981_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_982_not__numeral__less__neg__numeral,axiom,
! [M: num,N: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_983_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( numera6440378960330123505d_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_984_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_985_neg__numeral__less__numeral,axiom,
! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% neg_numeral_less_numeral
thf(fact_986_one__plus__numeral__commute,axiom,
! [X5: num] :
( ( plus_p4060926892116697567d_real @ one_on6069100329679821595d_real @ ( numera6440378960330123505d_real @ X5 ) )
= ( plus_p4060926892116697567d_real @ ( numera6440378960330123505d_real @ X5 ) @ one_on6069100329679821595d_real ) ) ).
% one_plus_numeral_commute
thf(fact_987_one__plus__numeral__commute,axiom,
! [X5: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X5 ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X5 ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_988_one__plus__numeral__commute,axiom,
! [X5: num] :
( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X5 ) )
= ( plus_plus_int @ ( numeral_numeral_int @ X5 ) @ one_one_int ) ) ).
% one_plus_numeral_commute
thf(fact_989_one__plus__numeral__commute,axiom,
! [X5: num] :
( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X5 ) )
= ( plus_plus_real @ ( numeral_numeral_real @ X5 ) @ one_one_real ) ) ).
% one_plus_numeral_commute
thf(fact_990_one__neq__neg__numeral,axiom,
! [N: num] :
( one_on6069100329679821595d_real
!= ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_991_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_real
!= ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_992_one__neq__neg__numeral,axiom,
! [N: num] :
( one_one_int
!= ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% one_neq_neg_numeral
thf(fact_993_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numera6440378960330123505d_real @ N )
!= ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% numeral_neq_neg_one
thf(fact_994_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_real @ N )
!= ( uminus_uminus_real @ one_one_real ) ) ).
% numeral_neq_neg_one
thf(fact_995_numeral__neq__neg__one,axiom,
! [N: num] :
( ( numeral_numeral_int @ N )
!= ( uminus_uminus_int @ one_one_int ) ) ).
% numeral_neq_neg_one
thf(fact_996_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_le2716243287969276982d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_997_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_998_not__zero__le__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_999_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) @ zero_z580800474297136991d_real ) ).
% neg_numeral_le_zero
thf(fact_1000_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_le_zero
thf(fact_1001_neg__numeral__le__zero,axiom,
! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_le_zero
thf(fact_1002_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1003_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1004_not__zero__less__neg__numeral,axiom,
! [N: num] :
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1005_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ N ) ) @ zero_z580800474297136991d_real ) ).
% neg_numeral_less_zero
thf(fact_1006_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% neg_numeral_less_zero
thf(fact_1007_neg__numeral__less__zero,axiom,
! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% neg_numeral_less_zero
thf(fact_1008_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_le2716243287969276982d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1009_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1010_not__one__le__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_le_neg_numeral
thf(fact_1011_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_le2716243287969276982d_real @ ( numera6440378960330123505d_real @ M ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% not_numeral_le_neg_one
thf(fact_1012_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_le_neg_one
thf(fact_1013_not__numeral__le__neg__one,axiom,
! [M: num] :
~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_le_neg_one
thf(fact_1014_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_1015_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% neg_numeral_le_neg_one
thf(fact_1016_neg__numeral__le__neg__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% neg_numeral_le_neg_one
thf(fact_1017_neg__one__le__numeral,axiom,
! [M: num] : ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( numera6440378960330123505d_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_1018_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_le_numeral
thf(fact_1019_neg__one__le__numeral,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_le_numeral
thf(fact_1020_neg__numeral__le__one,axiom,
! [M: num] : ( ord_le2716243287969276982d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ one_on6069100329679821595d_real ) ).
% neg_numeral_le_one
thf(fact_1021_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_le_one
thf(fact_1022_neg__numeral__le__one,axiom,
! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_le_one
thf(fact_1023_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1024_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1025_not__neg__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_1026_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_le2991122432403439658d_real @ one_on6069100329679821595d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1027_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1028_not__one__less__neg__numeral,axiom,
! [M: num] :
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% not_one_less_neg_numeral
thf(fact_1029_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_le2991122432403439658d_real @ ( numera6440378960330123505d_real @ M ) @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) ) ).
% not_numeral_less_neg_one
thf(fact_1030_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% not_numeral_less_neg_one
thf(fact_1031_not__numeral__less__neg__one,axiom,
! [M: num] :
~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% not_numeral_less_neg_one
thf(fact_1032_neg__one__less__numeral,axiom,
! [M: num] : ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ one_on6069100329679821595d_real ) @ ( numera6440378960330123505d_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_1033_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% neg_one_less_numeral
thf(fact_1034_neg__one__less__numeral,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% neg_one_less_numeral
thf(fact_1035_neg__numeral__less__one,axiom,
! [M: num] : ( ord_le2991122432403439658d_real @ ( uminus7714077491378687647d_real @ ( numera6440378960330123505d_real @ M ) ) @ one_on6069100329679821595d_real ) ).
% neg_numeral_less_one
thf(fact_1036_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% neg_numeral_less_one
thf(fact_1037_neg__numeral__less__one,axiom,
! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% neg_numeral_less_one
thf(fact_1038_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1039_verit__comp__simplify1_I2_J,axiom,
! [A: real] : ( ord_less_eq_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1040_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1041_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1042_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_1043_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_1044_nle__le,axiom,
! [A: real,B: real] :
( ( ~ ( ord_less_eq_real @ A @ B ) )
= ( ( ord_less_eq_real @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1045_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_1046_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_1047_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_1048_le__cases3,axiom,
! [X5: dedekind_real,Y4: dedekind_real,Z2: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ Z2 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ~ ( ord_le2716243287969276982d_real @ X5 @ Z2 ) )
=> ( ( ( ord_le2716243287969276982d_real @ X5 @ Z2 )
=> ~ ( ord_le2716243287969276982d_real @ Z2 @ Y4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z2 @ Y4 )
=> ~ ( ord_le2716243287969276982d_real @ Y4 @ X5 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y4 @ Z2 )
=> ~ ( ord_le2716243287969276982d_real @ Z2 @ X5 ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z2 @ X5 )
=> ~ ( ord_le2716243287969276982d_real @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1049_le__cases3,axiom,
! [X5: real,Y4: real,Z2: real] :
( ( ( ord_less_eq_real @ X5 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_real @ X5 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_real @ Z2 @ Y4 )
=> ~ ( ord_less_eq_real @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_real @ Y4 @ Z2 )
=> ~ ( ord_less_eq_real @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_real @ Z2 @ X5 )
=> ~ ( ord_less_eq_real @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1050_le__cases3,axiom,
! [X5: num,Y4: num,Z2: num] :
( ( ( ord_less_eq_num @ X5 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_num @ X5 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_num @ Z2 @ Y4 )
=> ~ ( ord_less_eq_num @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_num @ Y4 @ Z2 )
=> ~ ( ord_less_eq_num @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_num @ Z2 @ X5 )
=> ~ ( ord_less_eq_num @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1051_le__cases3,axiom,
! [X5: nat,Y4: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X5 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X5 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y4 )
=> ~ ( ord_less_eq_nat @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_nat @ Y4 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1052_le__cases3,axiom,
! [X5: int,Y4: int,Z2: int] :
( ( ( ord_less_eq_int @ X5 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X5 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y4 ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y4 )
=> ~ ( ord_less_eq_int @ Y4 @ X5 ) )
=> ( ( ( ord_less_eq_int @ Y4 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X5 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ Y4 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1053_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: dedekind_real,Z4: dedekind_real] : ( Y6 = Z4 ) )
= ( ^ [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
& ( ord_le2716243287969276982d_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1054_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z4: real] : ( Y6 = Z4 ) )
= ( ^ [X2: real,Y2: real] :
( ( ord_less_eq_real @ X2 @ Y2 )
& ( ord_less_eq_real @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1055_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z4: num] : ( Y6 = Z4 ) )
= ( ^ [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
& ( ord_less_eq_num @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1056_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
= ( ^ [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
& ( ord_less_eq_nat @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1057_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: int,Z4: int] : ( Y6 = Z4 ) )
= ( ^ [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
& ( ord_less_eq_int @ Y2 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1058_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_1059_ord__eq__le__trans,axiom,
! [A: real,B: real,C: real] :
( ( A = B )
=> ( ( ord_less_eq_real @ B @ C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1060_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_1061_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_1062_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_1063_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_1064_ord__le__eq__trans,axiom,
! [A: real,B: real,C: real] :
( ( ord_less_eq_real @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_real @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1065_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_1066_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_1067_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_1068_order__antisym,axiom,
! [X5: dedekind_real,Y4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y4 )
=> ( ( ord_le2716243287969276982d_real @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_1069_order__antisym,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ X5 @ Y4 )
=> ( ( ord_less_eq_real @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_1070_order__antisym,axiom,
! [X5: num,Y4: num] :
( ( ord_less_eq_num @ X5 @ Y4 )
=> ( ( ord_less_eq_num @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_1071_order__antisym,axiom,
! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ( ord_less_eq_nat @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_1072_order__antisym,axiom,
! [X5: int,Y4: int] :
( ( ord_less_eq_int @ X5 @ Y4 )
=> ( ( ord_less_eq_int @ Y4 @ X5 )
=> ( X5 = Y4 ) ) ) ).
% order_antisym
thf(fact_1073_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1074_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_1075_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_1076_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_1077_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_1078_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_1079_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_1080_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_1081_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_1082_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_1083_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1084_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1085_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_1086_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1087_pos2,axiom,
ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% pos2
thf(fact_1088_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1089_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_1090_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1091_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_1092_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1093_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1094_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1095_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_1096_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_1097_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_eq_nat @ M5 @ N3 )
& ( M5 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_1098_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K2 )
=> ~ ( P @ I2 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1099_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_1100_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
| ( M5 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1101_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_1102_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_1103_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1104_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1105_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_1106_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1107_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_1108_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1109_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_1110_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_1111_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_1112_less__not__refl3,axiom,
! [S2: nat,T2: nat] :
( ( ord_less_nat @ S2 @ T2 )
=> ( S2 != T2 ) ) ).
% less_not_refl3
thf(fact_1113_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1114_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_1115_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( P @ M6 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_1116_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_1117_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
& ~ ( P @ M6 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_1118_linorder__neqE__nat,axiom,
! [X5: nat,Y4: nat] :
( ( X5 != Y4 )
=> ( ~ ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ Y4 @ X5 ) ) ) ).
% linorder_neqE_nat
thf(fact_1119_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1120_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1121_add__less__mono,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_less_mono
thf(fact_1122_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1123_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1124_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_1125_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_1126_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_1127_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1128_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_1129_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1130_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M7: nat,N2: nat] :
( ( ord_less_nat @ M7 @ N2 )
=> ( ord_less_nat @ ( F @ M7 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1131_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1132_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_1133_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_1134_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1135_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1136_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1137_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_1138_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_1139_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1140_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1141_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1142_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus_nat @ A5 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1143_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I2: nat] :
( ( ord_less_nat @ K2 @ I2 )
=> ( P @ I2 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_1144_semiring__norm_I26_J,axiom,
( ( bitM @ one )
= one ) ).
% semiring_norm(26)
thf(fact_1145_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_1146_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus_num @ ( bitM @ N ) @ one )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_1147_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_1148_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1149_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1150_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1151_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1152_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_1153_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1154_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_1155_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1156_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1157_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1158_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_1159_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_1160_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_1161_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_1162_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_1163_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_1164_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1165_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_1166_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1167_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_1168_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_1169_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_1170_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_1171_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_1172_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_1173_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1174_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1175_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1176_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1177_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_1178_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1179_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1180_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1181_num_Oexhaust,axiom,
! [Y4: num] :
( ( Y4 != one )
=> ( ! [X23: num] :
( Y4
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y4
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1182_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1183_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_1184_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1185_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1186_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1187_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1188_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1189_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1190_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1191_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1192_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1193_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_1194_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_1195_less__add__left,axiom,
! [R: dedekind_preal,S2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S2 )
=> ( ( plus_p3173629198307831117_preal @ R @ ( minus_7336623429200594941_preal @ S2 @ R ) )
= S2 ) ) ).
% less_add_left
thf(fact_1196_real__le__eq__diff,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X2: dedekind_real,Y2: dedekind_real] : ( ord_le2716243287969276982d_real @ ( minus_5539002012860128047d_real @ X2 @ Y2 ) @ zero_z580800474297136991d_real ) ) ) ).
% real_le_eq_diff
thf(fact_1197_real__diff__def,axiom,
( minus_5539002012860128047d_real
= ( ^ [R2: dedekind_real,S3: dedekind_real] : ( plus_p4060926892116697567d_real @ R2 @ ( uminus7714077491378687647d_real @ S3 ) ) ) ) ).
% real_diff_def
thf(fact_1198_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
=> ( P @ D3 ) ) ) ) ).
% nat_diff_split
thf(fact_1199_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D3: nat] :
( ( A
= ( plus_plus_nat @ B @ D3 ) )
& ~ ( P @ D3 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1200_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1201_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_1202_nat__dvd__1__iff__1,axiom,
! [M: nat] :
( ( dvd_dvd_nat @ M @ one_one_nat )
= ( M = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1203_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
= ( ( ord_less_nat @ N @ M )
| ( dvd_dvd_nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_1204_num__induct,axiom,
! [P: num > $o,X5: num] :
( ( P @ one )
=> ( ! [X4: num] :
( ( P @ X4 )
=> ( P @ ( inc @ X4 ) ) )
=> ( P @ X5 ) ) ) ).
% num_induct
thf(fact_1205_add__inc,axiom,
! [X5: num,Y4: num] :
( ( plus_plus_num @ X5 @ ( inc @ Y4 ) )
= ( inc @ ( plus_plus_num @ X5 @ Y4 ) ) ) ).
% add_inc
thf(fact_1206_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_nat @ M @ N )
=> ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_1207_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( dvd_dvd_nat @ M @ N )
=> ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% dvd_pos_nat
thf(fact_1208_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1209_inc_Osimps_I2_J,axiom,
! [X5: num] :
( ( inc @ ( bit0 @ X5 ) )
= ( bit1 @ X5 ) ) ).
% inc.simps(2)
thf(fact_1210_inc_Osimps_I3_J,axiom,
! [X5: num] :
( ( inc @ ( bit1 @ X5 ) )
= ( bit0 @ ( inc @ X5 ) ) ) ).
% inc.simps(3)
thf(fact_1211_add__One,axiom,
! [X5: num] :
( ( plus_plus_num @ X5 @ one )
= ( inc @ X5 ) ) ).
% add_One
thf(fact_1212_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_1213_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_1214_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd_nat @ K @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_1215_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% odd_pos
thf(fact_1216_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% not_exp_less_eq_0_int
thf(fact_1217_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
= ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_1218_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_1219_realpow__pos__nth,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N )
= A ) ) ) ) ).
% realpow_pos_nth
thf(fact_1220_realpow__pos__nth__unique,axiom,
! [N: nat,A: real] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_real @ zero_zero_real @ A )
=> ? [X4: real] :
( ( ord_less_real @ zero_zero_real @ X4 )
& ( ( power_power_real @ X4 @ N )
= A )
& ! [Y: real] :
( ( ( ord_less_real @ zero_zero_real @ Y )
& ( ( power_power_real @ Y @ N )
= A ) )
=> ( Y = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1221_realpow__square__minus__le,axiom,
! [U2: real,X5: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% realpow_square_minus_le
thf(fact_1222_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
= ( ord_less_int @ K @ zero_zero_int ) ) ).
% unset_bit_negative_int_iff
thf(fact_1223_nat__zero__less__power__iff,axiom,
! [X5: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X5 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X5 )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1224_real__add__minus__iff,axiom,
! [X5: real,A: real] :
( ( ( plus_plus_real @ X5 @ ( uminus_uminus_real @ A ) )
= zero_zero_real )
= ( X5 = A ) ) ).
% real_add_minus_iff
thf(fact_1225_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1226_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_1227_minus__real__def,axiom,
( minus_minus_real
= ( ^ [X2: real,Y2: real] : ( plus_plus_real @ X2 @ ( uminus_uminus_real @ Y2 ) ) ) ) ).
% minus_real_def
thf(fact_1228_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1229_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1230_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1231_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1232_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1233_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N ) )
=> ( ( dvd_dvd_int @ K @ N )
=> ( dvd_dvd_int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_1234_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq_int @ zero_zero_int @ M )
=> ( ( ord_less_eq_int @ zero_zero_int @ N )
=> ( ( dvd_dvd_int @ M @ N )
=> ( ( dvd_dvd_int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1235_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ord_less_int @ M @ N )
=> ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_1236_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd_int @ Z2 @ N )
=> ( ( ord_less_int @ zero_zero_int @ N )
=> ( ord_less_eq_int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1237_uminus__dvd__conv_I2_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int,T3: int] : ( dvd_dvd_int @ D3 @ ( uminus_uminus_int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_1238_uminus__dvd__conv_I1_J,axiom,
( dvd_dvd_int
= ( ^ [D3: int] : ( dvd_dvd_int @ ( uminus_uminus_int @ D3 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_1239_real__arch__pow__inv,axiom,
! [Y4: real,X5: real] :
( ( ord_less_real @ zero_zero_real @ Y4 )
=> ( ( ord_less_real @ X5 @ one_one_real )
=> ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X5 @ N2 ) @ Y4 ) ) ) ).
% real_arch_pow_inv
thf(fact_1240_real__arch__pow,axiom,
! [X5: real,Y4: real] :
( ( ord_less_real @ one_one_real @ X5 )
=> ? [N2: nat] : ( ord_less_real @ Y4 @ ( power_power_real @ X5 @ N2 ) ) ) ).
% real_arch_pow
thf(fact_1241_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_1242_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1243_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1244_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1245_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1246_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1247_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I3: int] :
( ( ord_less_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1248_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq_int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1249_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1250_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1251_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_1252_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_1253_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1254_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1255_real__0__le__add__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X5 @ Y4 ) )
= ( ord_less_eq_real @ ( uminus_uminus_real @ X5 ) @ Y4 ) ) ).
% real_0_le_add_iff
thf(fact_1256_real__add__le__0__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_eq_real @ ( plus_plus_real @ X5 @ Y4 ) @ zero_zero_real )
= ( ord_less_eq_real @ Y4 @ ( uminus_uminus_real @ X5 ) ) ) ).
% real_add_le_0_iff
thf(fact_1257_real__add__less__0__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_real @ ( plus_plus_real @ X5 @ Y4 ) @ zero_zero_real )
= ( ord_less_real @ Y4 @ ( uminus_uminus_real @ X5 ) ) ) ).
% real_add_less_0_iff
thf(fact_1258_real__0__less__add__iff,axiom,
! [X5: real,Y4: real] :
( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X5 @ Y4 ) )
= ( ord_less_real @ ( uminus_uminus_real @ X5 ) @ Y4 ) ) ).
% real_0_less_add_iff
thf(fact_1259_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1260_less__exp,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% less_exp
thf(fact_1261_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_1262_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_1263_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_1264_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
=> ( ( ord_less_nat @ one_one_nat @ I )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1265_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
% Conjectures (2)
thf(conj_0,hypothesis,
( ? [U4: dedekind_real] :
( member_Dedekind_real @ U4
@ ( inf_in841446065189194142d_real
@ ( collec1873404778653292075d_real
@ ^ [Z: dedekind_real] :
? [X2: dedekind_real] :
( ( member_Dedekind_real @ X2 @ s )
& ( Z
= ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X2 @ ( uminus7714077491378687647d_real @ x ) ) @ one_on6069100329679821595d_real ) ) ) )
@ ( collec1873404778653292075d_real @ ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real ) ) ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------