TPTP Problem File: SLH0660^1.p
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%------------------------------------------------------------------------------
% 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_01314_042884__5751296_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1327 ( 488 unt; 56 typ; 0 def)
% Number of atoms : 3703 (1177 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10700 ( 393 ~; 126 |; 172 &;8315 @)
% ( 0 <=>;1694 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 291 ( 291 >; 0 *; 0 +; 0 <<)
% Number of symbols : 52 ( 51 usr; 11 con; 0-2 aty)
% Number of variables : 3639 ( 188 ^;3336 !; 115 ?;3639 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:30:21.776
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Dedekind____Real__Opreal,type,
dedekind_preal: $tType ).
thf(ty_n_t__Dedekind____Real__Oreal,type,
dedekind_real: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (51)
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Int__Oint,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Int__Oint,type,
bit_se4203085406695923979it_int: nat > int > int ).
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_Oone__class_Oone_001t__Dedekind____Real__Opreal,type,
one_on9143529541772854033_preal: dedekind_preal ).
thf(sy_c_Groups_Oone__class_Oone_001t__Dedekind____Real__Oreal,type,
one_on6069100329679821595d_real: dedekind_real ).
thf(sy_c_Groups_Oone__class_Oone_001t__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_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_Otimes__class_Otimes_001t__Dedekind____Real__Opreal,type,
times_3000655703912201937_preal: dedekind_preal > dedekind_preal > dedekind_preal ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Dedekind____Real__Oreal,type,
times_2157731159493324635d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Dedekind____Real__Oreal,type,
uminus7714077491378687647d_real: dedekind_real > dedekind_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
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_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Dedekind____Real__Oreal,type,
neg_nu8665065731627423525d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Int__Oint,type,
neg_numeral_dbl_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Dedekind____Real__Oreal,type,
neg_nu2849763295382964967d_real: dedekind_real > dedekind_real ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
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_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__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_Rings_Odivide__class_Odivide_001t__Dedekind____Real__Oreal,type,
divide9119111104558704680d_real: dedekind_real > dedekind_real > dedekind_real ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
divide_divide_int: int > int > int ).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
divide_divide_nat: nat > nat > nat ).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
dvd_dvd_nat: nat > nat > $o ).
thf(sy_v_x,type,
x: dedekind_real ).
thf(sy_v_y,type,
y: dedekind_real ).
thf(sy_v_z,type,
z: dedekind_real ).
% Relevant facts (1268)
thf(fact_0_real__mult__assoc,axiom,
! [Z1: dedekind_real,Z2: dedekind_real,Z3: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ Z1 @ Z2 ) @ Z3 )
= ( times_2157731159493324635d_real @ Z1 @ ( times_2157731159493324635d_real @ Z2 @ Z3 ) ) ) ).
% real_mult_assoc
thf(fact_1_real__mult__order,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ X )
=> ( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ Y )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( times_2157731159493324635d_real @ X @ Y ) ) ) ) ).
% real_mult_order
thf(fact_2_real__mult__commute,axiom,
( times_2157731159493324635d_real
= ( ^ [Z: dedekind_real,W: dedekind_real] : ( times_2157731159493324635d_real @ W @ Z ) ) ) ).
% real_mult_commute
thf(fact_3_mult__zero__left,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% mult_zero_left
thf(fact_4_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_5_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_6_mult__zero__right,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% mult_zero_right
thf(fact_7_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_8_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_9_mult__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% mult_eq_0_iff
thf(fact_10_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_11_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_12_mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_13_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_14_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_15_mult__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_16_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_17_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_18_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_19_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_20_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_21_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_22_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_23_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_24_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_25_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_26_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_27_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_28_zero__reorient,axiom,
! [X: dedekind_real] :
( ( zero_z580800474297136991d_real = X )
= ( X = zero_z580800474297136991d_real ) ) ).
% zero_reorient
thf(fact_29_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_30_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_31_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_32_mult_Oleft__commute,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ B @ ( times_2157731159493324635d_real @ A @ C ) )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_33_mult_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ B @ ( times_3000655703912201937_preal @ A @ C ) )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_34_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_35_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_36_mult_Ocommute,axiom,
( times_2157731159493324635d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( times_2157731159493324635d_real @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_37_mult_Ocommute,axiom,
( times_3000655703912201937_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( times_3000655703912201937_preal @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_38_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_39_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A2: int,B2: int] : ( times_times_int @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_40_mult_Oassoc,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_41_mult_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% mult.assoc
thf(fact_42_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_43_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_44_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C )
= ( times_2157731159493324635d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_45_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( times_3000655703912201937_preal @ A @ B ) @ C )
= ( times_3000655703912201937_preal @ A @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_46_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_47_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_48_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_49_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_50_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_51_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_52_mult__right__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= ( times_2157731159493324635d_real @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_53_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_54_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_55_mult__left__cancel,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ C @ A )
= ( times_2157731159493324635d_real @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_56_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_57_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_58_no__zero__divisors,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( B != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real ) ) ) ).
% no_zero_divisors
thf(fact_59_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_60_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_61_divisors__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
= zero_z580800474297136991d_real )
=> ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divisors_zero
thf(fact_62_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_63_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_64_mult__not__zero,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ B )
!= zero_z580800474297136991d_real )
=> ( ( A != zero_z580800474297136991d_real )
& ( B != zero_z580800474297136991d_real ) ) ) ).
% mult_not_zero
thf(fact_65_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_66_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_67_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_68_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_69_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_70_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_71_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_72_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_73_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_74_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_75_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_76_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_77_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_78_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_79_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_80_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_81_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_82_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_83_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_84_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_85_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_86_mult__less__iff1,axiom,
! [Z4: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z4 ) @ ( times_times_int @ Y @ Z4 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_87_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_88_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_89_order__less__imp__not__less,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_90_order__less__imp__not__less,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_91_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_92_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_93_order__less__imp__not__eq2,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_94_order__less__imp__not__eq2,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_95_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_96_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_97_order__less__imp__not__eq,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_98_order__less__imp__not__eq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_99_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_100_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_101_linorder__less__linear,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
| ( X = Y )
| ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_102_linorder__less__linear,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
| ( X = Y )
| ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_103_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_104_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_105_order__less__imp__triv,axiom,
! [X: dedekind_real,Y: dedekind_real,P: $o] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_106_order__less__imp__triv,axiom,
! [X: dedekind_preal,Y: dedekind_preal,P: $o] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_107_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_108_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_109_order__less__not__sym,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_110_order__less__not__sym,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_111_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_112_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_113_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_114_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_115_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_116_order__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_117_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_118_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_119_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_120_order__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_121_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_122_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_123_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_124_order__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_125_order__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_126_order__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_127_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_128_order__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_129_order__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_130_order__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_131_order__less__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_132_order__less__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_133_order__less__irrefl,axiom,
! [X: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ X @ X ) ).
% order_less_irrefl
thf(fact_134_order__less__irrefl,axiom,
! [X: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ X @ X ) ).
% order_less_irrefl
thf(fact_135_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_136_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_137_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_138_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_139_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_140_less__imp__neq,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_141_less__imp__neq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_142_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_143_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_144_order_Oasym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order.asym
thf(fact_145_order_Oasym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order.asym
thf(fact_146_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_147_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_148_ord__eq__less__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A = B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_149_ord__eq__less__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_150_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_151_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_152_ord__less__eq__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( B = C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_153_ord__less__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_154_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_155_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_156_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_157_antisym__conv3,axiom,
! [Y: dedekind_real,X: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ Y @ X )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_158_antisym__conv3,axiom,
! [Y: dedekind_preal,X: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ Y @ X )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_159_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_160_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_161_linorder__cases,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ( X != Y )
=> ( ord_le2991122432403439658d_real @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_162_linorder__cases,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( X != Y )
=> ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_163_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_164_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_165_dual__order_Oasym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ~ ( ord_le2991122432403439658d_real @ A @ B ) ) ).
% dual_order.asym
thf(fact_166_dual__order_Oasym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ~ ( ord_le5708704896291381698_preal @ A @ B ) ) ).
% dual_order.asym
thf(fact_167_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_168_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_169_dual__order_Oirrefl,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% dual_order.irrefl
thf(fact_170_dual__order_Oirrefl,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% dual_order.irrefl
thf(fact_171_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_172_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_173_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [N2: nat] :
( ( P3 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_174_linorder__less__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A3: dedekind_real,B3: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: dedekind_real] : ( P @ A3 @ A3 )
=> ( ! [A3: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_175_linorder__less__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A3: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: dedekind_preal] : ( P @ A3 @ A3 )
=> ( ! [A3: dedekind_preal,B3: dedekind_preal] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_176_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_177_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_178_order_Ostrict__trans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_179_order_Ostrict__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_180_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_181_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_182_not__less__iff__gr__or__eq,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X @ Y ) )
= ( ( ord_le2991122432403439658d_real @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_183_not__less__iff__gr__or__eq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( ( ord_le5708704896291381698_preal @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_184_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_185_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_186_dual__order_Ostrict__trans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_187_dual__order_Ostrict__trans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_188_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_189_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_190_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_191_order_Ostrict__implies__not__eq,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_192_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_193_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_194_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_195_dual__order_Ostrict__implies__not__eq,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_196_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_197_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_198_linorder__neqE,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( X != Y )
=> ( ~ ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2991122432403439658d_real @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_199_linorder__neqE,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( X != Y )
=> ( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_200_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_201_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_202_order__less__asym,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ~ ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% order_less_asym
thf(fact_203_order__less__asym,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ~ ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% order_less_asym
thf(fact_204_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_205_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_206_linorder__neq__iff,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( X != Y )
= ( ( ord_le2991122432403439658d_real @ X @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_207_linorder__neq__iff,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( X != Y )
= ( ( ord_le5708704896291381698_preal @ X @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_208_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_209_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_210_order__less__asym_H,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ~ ( ord_le2991122432403439658d_real @ B @ A ) ) ).
% order_less_asym'
thf(fact_211_order__less__asym_H,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ~ ( ord_le5708704896291381698_preal @ B @ A ) ) ).
% order_less_asym'
thf(fact_212_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_213_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_214_order__less__trans,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z4 )
=> ( ord_le2991122432403439658d_real @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_215_order__less__trans,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z4 )
=> ( ord_le5708704896291381698_preal @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_216_order__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_217_order__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_trans
thf(fact_218_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_219_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_220_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_221_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_222_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_223_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_224_ord__eq__less__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_225_ord__eq__less__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_226_ord__eq__less__subst,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_227_ord__eq__less__subst,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_228_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_229_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_230_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_231_ord__less__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_232_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_233_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_234_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_235_ord__less__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_236_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_237_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_238_minf_I7_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ~ ( ord_le2991122432403439658d_real @ T @ X4 ) ) ).
% minf(7)
thf(fact_239_minf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ~ ( ord_le5708704896291381698_preal @ T @ X4 ) ) ).
% minf(7)
thf(fact_240_minf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_nat @ T @ X4 ) ) ).
% minf(7)
thf(fact_241_minf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_int @ T @ X4 ) ) ).
% minf(7)
thf(fact_242_minf_I5_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( ord_le2991122432403439658d_real @ X4 @ T ) ) ).
% minf(5)
thf(fact_243_minf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( ord_le5708704896291381698_preal @ X4 @ T ) ) ).
% minf(5)
thf(fact_244_minf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_nat @ X4 @ T ) ) ).
% minf(5)
thf(fact_245_minf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_int @ X4 @ T ) ) ).
% minf(5)
thf(fact_246_minf_I4_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_247_minf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_248_minf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_249_minf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(4)
thf(fact_250_minf_I3_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_251_minf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_252_minf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_253_minf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( X4 != T ) ) ).
% minf(3)
thf(fact_254_minf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_255_minf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_256_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_257_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(2)
thf(fact_258_minf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_259_minf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_260_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_261_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z6 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% minf(1)
thf(fact_262_pinf_I7_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( ord_le2991122432403439658d_real @ T @ X4 ) ) ).
% pinf(7)
thf(fact_263_pinf_I7_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( ord_le5708704896291381698_preal @ T @ X4 ) ) ).
% pinf(7)
thf(fact_264_pinf_I7_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_nat @ T @ X4 ) ) ).
% pinf(7)
thf(fact_265_pinf_I7_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_int @ T @ X4 ) ) ).
% pinf(7)
thf(fact_266_pinf_I5_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ~ ( ord_le2991122432403439658d_real @ X4 @ T ) ) ).
% pinf(5)
thf(fact_267_pinf_I5_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ~ ( ord_le5708704896291381698_preal @ X4 @ T ) ) ).
% pinf(5)
thf(fact_268_pinf_I5_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_nat @ X4 @ T ) ) ).
% pinf(5)
thf(fact_269_pinf_I5_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_int @ X4 @ T ) ) ).
% pinf(5)
thf(fact_270_pinf_I4_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_271_pinf_I4_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_272_pinf_I4_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_273_pinf_I4_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(4)
thf(fact_274_pinf_I1_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_275_pinf_I1_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_276_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_277_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P4 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_278_pinf_I2_J,axiom,
! [P: dedekind_real > $o,P4: dedekind_real > $o,Q: dedekind_real > $o,Q2: dedekind_real > $o] :
( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_real] :
! [X2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_279_pinf_I2_J,axiom,
! [P: dedekind_preal > $o,P4: dedekind_preal > $o,Q: dedekind_preal > $o,Q2: dedekind_preal > $o] :
( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: dedekind_preal] :
! [X2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_280_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_281_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z6: int] :
! [X2: int] :
( ( ord_less_int @ Z6 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P4 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_282_pinf_I3_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_283_pinf_I3_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_284_pinf_I3_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_285_pinf_I3_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( X4 != T ) ) ).
% pinf(3)
thf(fact_286_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_real] :
~ ( ord_le2991122432403439658d_real @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_287_verit__comp__simplify1_I1_J,axiom,
! [A: dedekind_preal] :
~ ( ord_le5708704896291381698_preal @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_288_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_289_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_290_dbl__simps_I2_J,axiom,
( ( neg_nu8665065731627423525d_real @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% dbl_simps(2)
thf(fact_291_dbl__simps_I2_J,axiom,
( ( neg_numeral_dbl_int @ zero_zero_int )
= zero_zero_int ) ).
% dbl_simps(2)
thf(fact_292_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_293_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_294_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_295_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_296_order__refl,axiom,
! [X: dedekind_real] : ( ord_le2716243287969276982d_real @ X @ X ) ).
% order_refl
thf(fact_297_order__refl,axiom,
! [X: dedekind_preal] : ( ord_le5604041210740703414_preal @ X @ X ) ).
% order_refl
thf(fact_298_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_299_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_300_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_301_dual__order_Orefl,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% dual_order.refl
thf(fact_302_dual__order_Orefl,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% dual_order.refl
thf(fact_303_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_304_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_305_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_306_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_307_verit__la__disequality,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A = B )
| ~ ( ord_le2716243287969276982d_real @ A @ B )
| ~ ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% verit_la_disequality
thf(fact_308_verit__la__disequality,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A = B )
| ~ ( ord_le5604041210740703414_preal @ A @ B )
| ~ ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% verit_la_disequality
thf(fact_309_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_310_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_311_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_312_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_real] : ( ord_le2716243287969276982d_real @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_313_verit__comp__simplify1_I2_J,axiom,
! [A: dedekind_preal] : ( ord_le5604041210740703414_preal @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_314_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_315_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_316_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_317_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_318_nle__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ A @ B ) )
= ( ( ord_le5604041210740703414_preal @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_319_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_320_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_321_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_322_le__cases3,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ( ord_le2716243287969276982d_real @ X @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ Z4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ X )
=> ~ ( ord_le2716243287969276982d_real @ X @ Z4 ) )
=> ( ( ( ord_le2716243287969276982d_real @ X @ Z4 )
=> ~ ( ord_le2716243287969276982d_real @ Z4 @ Y ) )
=> ( ( ( ord_le2716243287969276982d_real @ Z4 @ Y )
=> ~ ( ord_le2716243287969276982d_real @ Y @ X ) )
=> ( ( ( ord_le2716243287969276982d_real @ Y @ Z4 )
=> ~ ( ord_le2716243287969276982d_real @ Z4 @ X ) )
=> ~ ( ( ord_le2716243287969276982d_real @ Z4 @ X )
=> ~ ( ord_le2716243287969276982d_real @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_323_le__cases3,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ X @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ Z4 ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ X )
=> ~ ( ord_le5604041210740703414_preal @ X @ Z4 ) )
=> ( ( ( ord_le5604041210740703414_preal @ X @ Z4 )
=> ~ ( ord_le5604041210740703414_preal @ Z4 @ Y ) )
=> ( ( ( ord_le5604041210740703414_preal @ Z4 @ Y )
=> ~ ( ord_le5604041210740703414_preal @ Y @ X ) )
=> ( ( ( ord_le5604041210740703414_preal @ Y @ Z4 )
=> ~ ( ord_le5604041210740703414_preal @ Z4 @ X ) )
=> ~ ( ( ord_le5604041210740703414_preal @ Z4 @ X )
=> ~ ( ord_le5604041210740703414_preal @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_324_le__cases3,axiom,
! [X: num,Y: num,Z4: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z4 ) )
=> ( ( ( ord_less_eq_num @ X @ Z4 )
=> ~ ( ord_less_eq_num @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_num @ Z4 @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z4 )
=> ~ ( ord_less_eq_num @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_num @ Z4 @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_325_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_326_le__cases3,axiom,
! [X: int,Y: int,Z4: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z4 ) )
=> ( ( ( ord_less_eq_int @ X @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z4 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z4 )
=> ~ ( ord_less_eq_int @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z4 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_327_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z7: dedekind_real] : ( Y4 = Z7 ) )
= ( ^ [X5: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y5 )
& ( ord_le2716243287969276982d_real @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_328_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z7: dedekind_preal] : ( Y4 = Z7 ) )
= ( ^ [X5: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y5 )
& ( ord_le5604041210740703414_preal @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_329_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z7: num] : ( Y4 = Z7 ) )
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_330_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z7: nat] : ( Y4 = Z7 ) )
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_331_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z7: int] : ( Y4 = Z7 ) )
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_332_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_333_ord__eq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A = B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_334_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_335_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_336_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_337_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_338_ord__le__eq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( B = C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_339_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_340_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_341_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_342_order__antisym,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_343_order__antisym,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_344_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_345_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_346_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_347_order_Otrans,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2716243287969276982d_real @ A @ C ) ) ) ).
% order.trans
thf(fact_348_order_Otrans,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5604041210740703414_preal @ A @ C ) ) ) ).
% order.trans
thf(fact_349_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_350_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_351_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_352_order__trans,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z4 )
=> ( ord_le2716243287969276982d_real @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_353_order__trans,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z4 )
=> ( ord_le5604041210740703414_preal @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_354_order__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z4 )
=> ( ord_less_eq_num @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_355_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_356_order__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_eq_int @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_357_linorder__wlog,axiom,
! [P: dedekind_real > dedekind_real > $o,A: dedekind_real,B: dedekind_real] :
( ! [A3: dedekind_real,B3: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: dedekind_real,B3: dedekind_real] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_358_linorder__wlog,axiom,
! [P: dedekind_preal > dedekind_preal > $o,A: dedekind_preal,B: dedekind_preal] :
( ! [A3: dedekind_preal,B3: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: dedekind_preal,B3: dedekind_preal] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_359_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A3: num,B3: num] :
( ( ord_less_eq_num @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: num,B3: num] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_360_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_361_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_362_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z7: dedekind_real] : ( Y4 = Z7 ) )
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ( ord_le2716243287969276982d_real @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_363_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z7: dedekind_preal] : ( Y4 = Z7 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_364_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: num,Z7: num] : ( Y4 = Z7 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_365_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z7: nat] : ( Y4 = Z7 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_366_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: int,Z7: int] : ( Y4 = Z7 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_367_dual__order_Oantisym,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_368_dual__order_Oantisym,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_369_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_370_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_371_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_372_dual__order_Otrans,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2716243287969276982d_real @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_373_dual__order_Otrans,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5604041210740703414_preal @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_374_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_375_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_376_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_377_antisym,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_378_antisym,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_379_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_380_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_381_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_382_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_real,Z7: dedekind_real] : ( Y4 = Z7 ) )
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ( ord_le2716243287969276982d_real @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_383_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: dedekind_preal,Z7: dedekind_preal] : ( Y4 = Z7 ) )
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_384_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: num,Z7: num] : ( Y4 = Z7 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_385_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: nat,Z7: nat] : ( Y4 = Z7 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_386_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y4: int,Z7: int] : ( Y4 = Z7 ) )
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_387_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_388_order__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_389_order__subst1,axiom,
! [A: dedekind_real,F: num > dedekind_real,B: num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_390_order__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_391_order__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_392_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_393_order__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_394_order__subst1,axiom,
! [A: dedekind_preal,F: num > dedekind_preal,B: num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X2: num,Y2: num] :
( ( ord_less_eq_num @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_395_order__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_396_order__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_eq_int @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_398_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_399_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_400_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_401_order__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_402_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_403_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_404_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_405_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_406_order__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_407_order__eq__refl,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( X = Y )
=> ( ord_le2716243287969276982d_real @ X @ Y ) ) ).
% order_eq_refl
thf(fact_408_order__eq__refl,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( X = Y )
=> ( ord_le5604041210740703414_preal @ X @ Y ) ) ).
% order_eq_refl
thf(fact_409_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_410_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_411_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_412_linorder__linear,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
| ( ord_le2716243287969276982d_real @ Y @ X ) ) ).
% linorder_linear
thf(fact_413_linorder__linear,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
| ( ord_le5604041210740703414_preal @ Y @ X ) ) ).
% linorder_linear
thf(fact_414_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_415_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_416_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_417_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_418_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_419_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_420_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_421_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( A
= ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_422_ord__eq__le__subst,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_423_ord__eq__le__subst,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_424_ord__eq__le__subst,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_425_ord__eq__le__subst,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_426_ord__eq__le__subst,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( A
= ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_427_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_428_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_429_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_430_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_431_ord__le__eq__subst,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_432_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2716243287969276982d_real @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_433_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5604041210740703414_preal @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_434_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_435_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_436_ord__le__eq__subst,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_437_linorder__le__cases,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_438_linorder__le__cases,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_439_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_440_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_441_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_442_order__antisym__conv,axiom,
! [Y: dedekind_real,X: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X )
=> ( ( ord_le2716243287969276982d_real @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_443_order__antisym__conv,axiom,
! [Y: dedekind_preal,X: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X )
=> ( ( ord_le5604041210740703414_preal @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_444_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_445_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_446_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_447_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_real,A4: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ B4 @ A4 ) )
= ( ord_le2991122432403439658d_real @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_448_verit__comp__simplify1_I3_J,axiom,
! [B4: dedekind_preal,A4: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ B4 @ A4 ) )
= ( ord_le5708704896291381698_preal @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_449_verit__comp__simplify1_I3_J,axiom,
! [B4: num,A4: num] :
( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
= ( ord_less_num @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_450_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_451_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A4: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
= ( ord_less_int @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_452_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_453_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ C2 ) )
=> ( P @ X4 ) )
& ! [D2: int] :
( ! [X2: int] :
( ( ( ord_less_eq_int @ A @ X2 )
& ( ord_less_int @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_int @ D2 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_454_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_455_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_456_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_457_leD,axiom,
! [Y: dedekind_real,X: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Y @ X )
=> ~ ( ord_le2991122432403439658d_real @ X @ Y ) ) ).
% leD
thf(fact_458_leD,axiom,
! [Y: dedekind_preal,X: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ Y @ X )
=> ~ ( ord_le5708704896291381698_preal @ X @ Y ) ) ).
% leD
thf(fact_459_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_460_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_461_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_462_leI,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ Y @ X ) ) ).
% leI
thf(fact_463_leI,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ Y @ X ) ) ).
% leI
thf(fact_464_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_465_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_466_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_467_nless__le,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ A @ B ) )
= ( ~ ( ord_le2716243287969276982d_real @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_468_nless__le,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ A @ B ) )
= ( ~ ( ord_le5604041210740703414_preal @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_469_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_470_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_471_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_472_antisym__conv1,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ~ ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ( ord_le2716243287969276982d_real @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_473_antisym__conv1,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ~ ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5604041210740703414_preal @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_474_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_475_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_476_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_477_antisym__conv2,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ( ~ ( ord_le2991122432403439658d_real @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_478_antisym__conv2,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_479_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_480_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_481_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_482_less__le__not__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y5 )
& ~ ( ord_le2716243287969276982d_real @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_483_less__le__not__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X5: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y5 )
& ~ ( ord_le5604041210740703414_preal @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_484_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ~ ( ord_less_eq_num @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_485_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_486_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ~ ( ord_less_eq_int @ Y5 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_487_not__le__imp__less,axiom,
! [Y: dedekind_real,X: dedekind_real] :
( ~ ( ord_le2716243287969276982d_real @ Y @ X )
=> ( ord_le2991122432403439658d_real @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_488_not__le__imp__less,axiom,
! [Y: dedekind_preal,X: dedekind_preal] :
( ~ ( ord_le5604041210740703414_preal @ Y @ X )
=> ( ord_le5708704896291381698_preal @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_489_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_490_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_491_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_492_order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_493_order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_494_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_495_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_496_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_497_order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_498_order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_499_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_500_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_501_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_502_order_Ostrict__trans1,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_503_order_Ostrict__trans1,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_504_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_505_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_506_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_507_order_Ostrict__trans2,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ord_le2991122432403439658d_real @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_508_order_Ostrict__trans2,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ord_le5708704896291381698_preal @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_509_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_510_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_511_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_512_order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A2 @ B2 )
& ~ ( ord_le2716243287969276982d_real @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_513_order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A2 @ B2 )
& ~ ( ord_le5604041210740703414_preal @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_514_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_515_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_516_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
& ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_517_dual__order_Oorder__iff__strict,axiom,
( ord_le2716243287969276982d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_518_dual__order_Oorder__iff__strict,axiom,
( ord_le5604041210740703414_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_519_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_520_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_521_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_522_dual__order_Ostrict__iff__order,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_523_dual__order_Ostrict__iff__order,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_524_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_525_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_526_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_527_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B @ A )
=> ( ( ord_le2991122432403439658d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_528_dual__order_Ostrict__trans1,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B @ A )
=> ( ( ord_le5708704896291381698_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_529_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_530_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_531_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_532_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_real,A: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ( ord_le2716243287969276982d_real @ C @ B )
=> ( ord_le2991122432403439658d_real @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_533_dual__order_Ostrict__trans2,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ( ord_le5604041210740703414_preal @ C @ B )
=> ( ord_le5708704896291381698_preal @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_534_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_535_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_536_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_537_dual__order_Ostrict__iff__not,axiom,
( ord_le2991122432403439658d_real
= ( ^ [B2: dedekind_real,A2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ B2 @ A2 )
& ~ ( ord_le2716243287969276982d_real @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_538_dual__order_Ostrict__iff__not,axiom,
( ord_le5708704896291381698_preal
= ( ^ [B2: dedekind_preal,A2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ B2 @ A2 )
& ~ ( ord_le5604041210740703414_preal @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_539_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_540_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_541_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
& ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_542_order_Ostrict__implies__order,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ord_le2716243287969276982d_real @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_543_order_Ostrict__implies__order,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_544_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_545_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_546_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_547_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( ord_le2991122432403439658d_real @ B @ A )
=> ( ord_le2716243287969276982d_real @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_548_dual__order_Ostrict__implies__order,axiom,
! [B: dedekind_preal,A: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ B @ A )
=> ( ord_le5604041210740703414_preal @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_549_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_550_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_551_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_552_order__le__less,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X5: dedekind_real,Y5: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_553_order__le__less,axiom,
( ord_le5604041210740703414_preal
= ( ^ [X5: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_554_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_num @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_555_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_nat @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_556_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_int @ X5 @ Y5 )
| ( X5 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_557_order__less__le,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_558_order__less__le,axiom,
( ord_le5708704896291381698_preal
= ( ^ [X5: dedekind_preal,Y5: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_559_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y5: num] :
( ( ord_less_eq_num @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_560_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y5: nat] :
( ( ord_less_eq_nat @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_561_order__less__le,axiom,
( ord_less_int
= ( ^ [X5: int,Y5: int] :
( ( ord_less_eq_int @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_562_linorder__not__le,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2716243287969276982d_real @ X @ Y ) )
= ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% linorder_not_le
thf(fact_563_linorder__not__le,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5604041210740703414_preal @ X @ Y ) )
= ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% linorder_not_le
thf(fact_564_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_565_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_566_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_567_linorder__not__less,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ~ ( ord_le2991122432403439658d_real @ X @ Y ) )
= ( ord_le2716243287969276982d_real @ Y @ X ) ) ).
% linorder_not_less
thf(fact_568_linorder__not__less,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ~ ( ord_le5708704896291381698_preal @ X @ Y ) )
= ( ord_le5604041210740703414_preal @ Y @ X ) ) ).
% linorder_not_less
thf(fact_569_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_570_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_571_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_572_order__less__imp__le,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_573_order__less__imp__le,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ord_le5604041210740703414_preal @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_574_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_575_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_576_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_577_order__le__neq__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( A != B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_578_order__le__neq__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( A != B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_579_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_580_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_581_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_582_order__neq__le__trans,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != B )
=> ( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ord_le2991122432403439658d_real @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_583_order__neq__le__trans,axiom,
! [A: dedekind_preal,B: dedekind_preal] :
( ( A != B )
=> ( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5708704896291381698_preal @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_584_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_585_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_586_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_587_order__le__less__trans,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ( ord_le2991122432403439658d_real @ Y @ Z4 )
=> ( ord_le2991122432403439658d_real @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_588_order__le__less__trans,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ Y @ Z4 )
=> ( ord_le5708704896291381698_preal @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_589_order__le__less__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z4 )
=> ( ord_less_num @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_590_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_591_order__le__less__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_le_less_trans
thf(fact_592_order__less__le__trans,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X @ Y )
=> ( ( ord_le2716243287969276982d_real @ Y @ Z4 )
=> ( ord_le2991122432403439658d_real @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_593_order__less__le__trans,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X @ Y )
=> ( ( ord_le5604041210740703414_preal @ Y @ Z4 )
=> ( ord_le5708704896291381698_preal @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_594_order__less__le__trans,axiom,
! [X: num,Y: num,Z4: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z4 )
=> ( ord_less_num @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_595_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_596_order__less__le__trans,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z4 )
=> ( ord_less_int @ X @ Z4 ) ) ) ).
% order_less_le_trans
thf(fact_597_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_598_order__le__less__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_599_order__le__less__subst1,axiom,
! [A: dedekind_real,F: nat > dedekind_real,B: nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_600_order__le__less__subst1,axiom,
! [A: dedekind_real,F: int > dedekind_real,B: int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_601_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_602_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_603_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: nat > dedekind_preal,B: nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_604_order__le__less__subst1,axiom,
! [A: dedekind_preal,F: int > dedekind_preal,B: int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_605_order__le__less__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le2991122432403439658d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_606_order__le__less__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_le5708704896291381698_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_607_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_608_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_609_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_610_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > nat,C: nat] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_611_order__le__less__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > int,C: int] :
( ( ord_le2716243287969276982d_real @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_612_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le2991122432403439658d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_613_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5708704896291381698_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_614_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_615_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > nat,C: nat] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_616_order__le__less__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > int,C: int] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_617_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_real > dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_618_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_real > dedekind_preal,B: dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_619_order__less__le__subst1,axiom,
! [A: num,F: dedekind_real > num,B: dedekind_real,C: dedekind_real] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_620_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_real > nat,B: dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_621_order__less__le__subst1,axiom,
! [A: int,F: dedekind_real > int,B: dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le2716243287969276982d_real @ B @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_622_order__less__le__subst1,axiom,
! [A: dedekind_real,F: dedekind_preal > dedekind_real,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le2716243287969276982d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_623_order__less__le__subst1,axiom,
! [A: dedekind_preal,F: dedekind_preal > dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_le5604041210740703414_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_624_order__less__le__subst1,axiom,
! [A: num,F: dedekind_preal > num,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_625_order__less__le__subst1,axiom,
! [A: nat,F: dedekind_preal > nat,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_626_order__less__le__subst1,axiom,
! [A: int,F: dedekind_preal > int,B: dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_le5604041210740703414_preal @ B @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X2 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_627_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_real,C: dedekind_real] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_628_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_real,C: dedekind_real] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_629_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_real,C: dedekind_real] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_630_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_real,C: dedekind_real] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le2716243287969276982d_real @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le2991122432403439658d_real @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le2991122432403439658d_real @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_631_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > dedekind_preal,C: dedekind_preal] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_632_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > dedekind_preal,C: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_633_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > dedekind_preal,C: dedekind_preal] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_634_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > dedekind_preal,C: dedekind_preal] :
( ( ord_less_int @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ ( F @ B ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_le5708704896291381698_preal @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le5708704896291381698_preal @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_635_order__less__le__subst2,axiom,
! [A: dedekind_real,B: dedekind_real,F: dedekind_real > num,C: num] :
( ( ord_le2991122432403439658d_real @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_real,Y2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_636_order__less__le__subst2,axiom,
! [A: dedekind_preal,B: dedekind_preal,F: dedekind_preal > num,C: num] :
( ( ord_le5708704896291381698_preal @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X2: dedekind_preal,Y2: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X2 @ Y2 )
=> ( ord_less_num @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_637_linorder__le__less__linear,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
| ( ord_le2991122432403439658d_real @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_638_linorder__le__less__linear,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
| ( ord_le5708704896291381698_preal @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_639_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_640_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_641_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_642_order__le__imp__less__or__eq,axiom,
! [X: dedekind_real,Y: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ( ord_le2991122432403439658d_real @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_643_order__le__imp__less__or__eq,axiom,
! [X: dedekind_preal,Y: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ X @ Y )
=> ( ( ord_le5708704896291381698_preal @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_644_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_645_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_646_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_647_minf_I8_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ~ ( ord_le2716243287969276982d_real @ T @ X4 ) ) ).
% minf(8)
thf(fact_648_minf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ~ ( ord_le5604041210740703414_preal @ T @ X4 ) ) ).
% minf(8)
thf(fact_649_minf_I8_J,axiom,
! [T: num] :
? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ~ ( ord_less_eq_num @ T @ X4 ) ) ).
% minf(8)
thf(fact_650_minf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% minf(8)
thf(fact_651_minf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% minf(8)
thf(fact_652_minf_I6_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ X4 @ Z5 )
=> ( ord_le2716243287969276982d_real @ X4 @ T ) ) ).
% minf(6)
thf(fact_653_minf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ X4 @ Z5 )
=> ( ord_le5604041210740703414_preal @ X4 @ T ) ) ).
% minf(6)
thf(fact_654_minf_I6_J,axiom,
! [T: num] :
? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ X4 @ Z5 )
=> ( ord_less_eq_num @ X4 @ T ) ) ).
% minf(6)
thf(fact_655_minf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z5 )
=> ( ord_less_eq_nat @ X4 @ T ) ) ).
% minf(6)
thf(fact_656_minf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z5 )
=> ( ord_less_eq_int @ X4 @ T ) ) ).
% minf(6)
thf(fact_657_pinf_I8_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ( ord_le2716243287969276982d_real @ T @ X4 ) ) ).
% pinf(8)
thf(fact_658_pinf_I8_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ( ord_le5604041210740703414_preal @ T @ X4 ) ) ).
% pinf(8)
thf(fact_659_pinf_I8_J,axiom,
! [T: num] :
? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ( ord_less_eq_num @ T @ X4 ) ) ).
% pinf(8)
thf(fact_660_pinf_I8_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ( ord_less_eq_nat @ T @ X4 ) ) ).
% pinf(8)
thf(fact_661_pinf_I8_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ( ord_less_eq_int @ T @ X4 ) ) ).
% pinf(8)
thf(fact_662_pinf_I6_J,axiom,
! [T: dedekind_real] :
? [Z5: dedekind_real] :
! [X4: dedekind_real] :
( ( ord_le2991122432403439658d_real @ Z5 @ X4 )
=> ~ ( ord_le2716243287969276982d_real @ X4 @ T ) ) ).
% pinf(6)
thf(fact_663_pinf_I6_J,axiom,
! [T: dedekind_preal] :
? [Z5: dedekind_preal] :
! [X4: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ Z5 @ X4 )
=> ~ ( ord_le5604041210740703414_preal @ X4 @ T ) ) ).
% pinf(6)
thf(fact_664_pinf_I6_J,axiom,
! [T: num] :
? [Z5: num] :
! [X4: num] :
( ( ord_less_num @ Z5 @ X4 )
=> ~ ( ord_less_eq_num @ X4 @ T ) ) ).
% pinf(6)
thf(fact_665_pinf_I6_J,axiom,
! [T: nat] :
? [Z5: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z5 @ X4 )
=> ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% pinf(6)
thf(fact_666_pinf_I6_J,axiom,
! [T: int] :
? [Z5: int] :
! [X4: int] :
( ( ord_less_int @ Z5 @ X4 )
=> ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% pinf(6)
thf(fact_667_real__less__def,axiom,
( ord_le2991122432403439658d_real
= ( ^ [X5: dedekind_real,Y5: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X5 @ Y5 )
& ( X5 != Y5 ) ) ) ) ).
% real_less_def
thf(fact_668_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_669_mult__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 @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_670_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_671_mult__mono_H,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 @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_672_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_673_split__mult__pos__le,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 @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_674_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_675_mult__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 @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_676_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_677_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_678_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_679_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_680_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_681_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_682_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_683_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_684_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_685_mult__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 @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_686_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_687_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_688_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_689_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_690_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_691_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_692_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_693_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_694_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_695_mult__le__cancel__iff1,axiom,
! [Z4: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z4 ) @ ( times_times_int @ Y @ Z4 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_696_mult__le__cancel__iff2,axiom,
! [Z4: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z4 @ X ) @ ( times_times_int @ Z4 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_697_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_698_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_699_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_700_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_701_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_702_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_703_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_704_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_705_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_706_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_707_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_708_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_709_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_710_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_711_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_712_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_713_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_714_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_715_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_716_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_717_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_718_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_719_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_720_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_721_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_722_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_723_mult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% mult_1
thf(fact_724_mult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% mult_1
thf(fact_725_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_726_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_727_mult_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.right_neutral
thf(fact_728_mult_Oright__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.right_neutral
thf(fact_729_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_730_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_731_mult__cancel__right2,axiom,
! [A: dedekind_real,C: dedekind_real] :
( ( ( times_2157731159493324635d_real @ A @ C )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right2
thf(fact_732_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_733_mult__cancel__right1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_right1
thf(fact_734_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_735_mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real] :
( ( ( times_2157731159493324635d_real @ C @ A )
= C )
= ( ( C = zero_z580800474297136991d_real )
| ( A = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left2
thf(fact_736_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_737_mult__cancel__left1,axiom,
! [C: dedekind_real,B: dedekind_real] :
( ( C
= ( times_2157731159493324635d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( B = one_on6069100329679821595d_real ) ) ) ).
% mult_cancel_left1
thf(fact_738_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_739_real__le__antisym,axiom,
! [Z4: dedekind_real,W2: dedekind_real] :
( ( ord_le2716243287969276982d_real @ Z4 @ W2 )
=> ( ( ord_le2716243287969276982d_real @ W2 @ Z4 )
=> ( Z4 = W2 ) ) ) ).
% real_le_antisym
thf(fact_740_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_741_real__le__refl,axiom,
! [W2: dedekind_real] : ( ord_le2716243287969276982d_real @ W2 @ W2 ) ).
% real_le_refl
thf(fact_742_one__reorient,axiom,
! [X: dedekind_real] :
( ( one_on6069100329679821595d_real = X )
= ( X = one_on6069100329679821595d_real ) ) ).
% one_reorient
thf(fact_743_one__reorient,axiom,
! [X: dedekind_preal] :
( ( one_on9143529541772854033_preal = X )
= ( X = one_on9143529541772854033_preal ) ) ).
% one_reorient
thf(fact_744_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_745_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_746_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_747_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_748_zero__neq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% zero_neq_one
thf(fact_749_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_750_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_751_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_752_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_753_mult_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% mult.comm_neutral
thf(fact_754_mult_Ocomm__neutral,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ one_on9143529541772854033_preal )
= A ) ).
% mult.comm_neutral
thf(fact_755_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_756_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_757_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_758_comm__monoid__mult__class_Omult__1,axiom,
! [A: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_759_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_760_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_761_real__zero__not__eq__one,axiom,
zero_z580800474297136991d_real != one_on6069100329679821595d_real ).
% real_zero_not_eq_one
thf(fact_762_real__mult__1,axiom,
! [Z4: dedekind_real] :
( ( times_2157731159493324635d_real @ one_on6069100329679821595d_real @ Z4 )
= Z4 ) ).
% real_mult_1
thf(fact_763_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_764_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_765_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_766_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_767_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_768_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_769_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_770_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_771_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_772_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_773_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_774_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_775_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_776_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_777_real__mult__inverse__left__ex,axiom,
! [X: dedekind_real] :
( ( X != zero_z580800474297136991d_real )
=> ~ ! [Y2: dedekind_real] :
( ( times_2157731159493324635d_real @ Y2 @ X )
!= one_on6069100329679821595d_real ) ) ).
% real_mult_inverse_left_ex
thf(fact_778_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_779_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_780_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_781_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_782_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_783_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_784_dbl__inc__simps_I2_J,axiom,
( ( neg_nu2849763295382964967d_real @ zero_z580800474297136991d_real )
= one_on6069100329679821595d_real ) ).
% dbl_inc_simps(2)
thf(fact_785_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_786_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_787_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_788_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_789_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_790_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_791_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_792_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_793_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_794_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_795_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_796_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_797_add_Oright__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.right_neutral
thf(fact_798_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_799_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_800_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_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_813_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_814_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_815_add__0,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add_0
thf(fact_816_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_817_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_818_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_819_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_820_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_821_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_822_divide__eq__0__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= zero_z580800474297136991d_real )
= ( ( A = zero_z580800474297136991d_real )
| ( B = zero_z580800474297136991d_real ) ) ) ).
% divide_eq_0_iff
thf(fact_823_divide__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ C @ A )
= ( divide9119111104558704680d_real @ C @ B ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_left
thf(fact_824_divide__cancel__right,axiom,
! [A: dedekind_real,C: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ C )
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( C = zero_z580800474297136991d_real )
| ( A = B ) ) ) ).
% divide_cancel_right
thf(fact_825_division__ring__divide__zero,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% division_ring_divide_zero
thf(fact_826_div__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ zero_z580800474297136991d_real @ A )
= zero_z580800474297136991d_real ) ).
% div_0
thf(fact_827_div__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% div_0
thf(fact_828_div__0,axiom,
! [A: int] :
( ( divide_divide_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% div_0
thf(fact_829_div__by__0,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ zero_z580800474297136991d_real )
= zero_z580800474297136991d_real ) ).
% div_by_0
thf(fact_830_div__by__0,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% div_by_0
thf(fact_831_div__by__0,axiom,
! [A: int] :
( ( divide_divide_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% div_by_0
thf(fact_832_times__divide__eq__left,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( times_2157731159493324635d_real @ ( divide9119111104558704680d_real @ B @ C ) @ A )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ B @ A ) @ C ) ) ).
% times_divide_eq_left
thf(fact_833_divide__divide__eq__left,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ A @ B ) @ C )
= ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% divide_divide_eq_left
thf(fact_834_divide__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ B ) ) ).
% divide_divide_eq_right
thf(fact_835_times__divide__eq__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( divide9119111104558704680d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ C ) ) ).
% times_divide_eq_right
thf(fact_836_div__by__1,axiom,
! [A: dedekind_real] :
( ( divide9119111104558704680d_real @ A @ one_on6069100329679821595d_real )
= A ) ).
% div_by_1
thf(fact_837_div__by__1,axiom,
! [A: nat] :
( ( divide_divide_nat @ A @ one_one_nat )
= A ) ).
% div_by_1
thf(fact_838_div__by__1,axiom,
! [A: int] :
( ( divide_divide_int @ A @ one_one_int )
= A ) ).
% div_by_1
thf(fact_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_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_859_mult__divide__mult__cancel__left__if,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( C = zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= zero_z580800474297136991d_real ) )
& ( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_860_nonzero__mult__divide__mult__cancel__left,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_861_nonzero__mult__divide__mult__cancel__left2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ C @ A ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_862_nonzero__mult__divide__mult__cancel__right,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_863_nonzero__mult__divide__mult__cancel__right2,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ C @ B ) )
= ( divide9119111104558704680d_real @ A @ B ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_864_nonzero__mult__div__cancel__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_865_nonzero__mult__div__cancel__left,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_866_nonzero__mult__div__cancel__left,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
= B ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_867_nonzero__mult__div__cancel__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_868_nonzero__mult__div__cancel__right,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_869_nonzero__mult__div__cancel__right,axiom,
! [B: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
= A ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_870_div__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% div_self
thf(fact_871_div__self,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
=> ( ( divide_divide_nat @ A @ A )
= one_one_nat ) ) ).
% div_self
thf(fact_872_div__self,axiom,
! [A: int] :
( ( A != zero_zero_int )
=> ( ( divide_divide_int @ A @ A )
= one_one_int ) ) ).
% div_self
thf(fact_873_divide__self__if,axiom,
! [A: dedekind_real] :
( ( ( A = zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= zero_z580800474297136991d_real ) )
& ( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ) ).
% divide_self_if
thf(fact_874_divide__self,axiom,
! [A: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ A )
= one_on6069100329679821595d_real ) ) ).
% divide_self
thf(fact_875_one__eq__divide__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( one_on6069100329679821595d_real
= ( divide9119111104558704680d_real @ A @ B ) )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% one_eq_divide_iff
thf(fact_876_divide__eq__1__iff,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( ( B != zero_z580800474297136991d_real )
& ( A = B ) ) ) ).
% divide_eq_1_iff
thf(fact_877_nonzero__divide__mult__cancel__left,axiom,
! [A: dedekind_real,B: dedekind_real] :
( ( A != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ B ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_878_nonzero__divide__mult__cancel__right,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( divide9119111104558704680d_real @ B @ ( times_2157731159493324635d_real @ A @ B ) )
= ( divide9119111104558704680d_real @ one_on6069100329679821595d_real @ A ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_879_real__trans__lemma,axiom,
! [X: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: dedekind_real,V22: dedekind_real,U22: dedekind_real,Y22: dedekind_real] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_p4060926892116697567d_real @ X22 @ V22 )
= ( plus_p4060926892116697567d_real @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_880_real__trans__lemma,axiom,
! [X: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: dedekind_preal,V22: dedekind_preal,U22: dedekind_preal,Y22: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_p3173629198307831117_preal @ X22 @ V22 )
= ( plus_p3173629198307831117_preal @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_881_real__trans__lemma,axiom,
! [X: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: num,V22: num,U22: num,Y22: num] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_plus_num @ X22 @ V22 )
= ( plus_plus_num @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_882_real__trans__lemma,axiom,
! [X: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: nat,V22: nat,U22: nat,Y22: nat] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_plus_nat @ X22 @ V22 )
= ( plus_plus_nat @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_883_real__trans__lemma,axiom,
! [X: dedekind_preal,V: dedekind_preal,U: dedekind_preal,Y: dedekind_preal,V2: dedekind_preal,U2: dedekind_preal,X22: int,V22: int,U22: int,Y22: int] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V ) @ ( plus_p3173629198307831117_preal @ U @ Y ) )
=> ( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ U @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ V ) )
=> ( ( ( plus_plus_int @ X22 @ V22 )
= ( plus_plus_int @ U22 @ Y22 ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ X @ V2 ) @ ( plus_p3173629198307831117_preal @ U2 @ Y ) ) ) ) ) ).
% real_trans_lemma
thf(fact_884_dbl__inc__def,axiom,
( neg_nu2849763295382964967d_real
= ( ^ [X5: dedekind_real] : ( plus_p4060926892116697567d_real @ ( plus_p4060926892116697567d_real @ X5 @ X5 ) @ one_on6069100329679821595d_real ) ) ) ).
% dbl_inc_def
thf(fact_885_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X5: int] : ( plus_plus_int @ ( plus_plus_int @ X5 @ X5 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_886_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_887_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_888_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_889_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_890_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_p3173629198307831117_preal @ I @ K )
= ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_891_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_892_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_893_group__cancel_Oadd1,axiom,
! [A5: dedekind_real,K: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( A5
= ( plus_p4060926892116697567d_real @ K @ A ) )
=> ( ( plus_p4060926892116697567d_real @ A5 @ B )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_894_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_895_group__cancel_Oadd1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_896_group__cancel_Oadd2,axiom,
! [B5: dedekind_real,K: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( B5
= ( plus_p4060926892116697567d_real @ K @ B ) )
=> ( ( plus_p4060926892116697567d_real @ A @ B5 )
= ( plus_p4060926892116697567d_real @ K @ ( plus_p4060926892116697567d_real @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_897_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_898_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_899_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_900_add_Oassoc,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.assoc
thf(fact_901_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_902_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_903_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_904_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_905_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_906_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_907_add_Ocommute,axiom,
( plus_p4060926892116697567d_real
= ( ^ [A2: dedekind_real,B2: dedekind_real] : ( plus_p4060926892116697567d_real @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_908_add_Ocommute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [A2: dedekind_preal,B2: dedekind_preal] : ( plus_p3173629198307831117_preal @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_909_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_910_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_911_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_912_add_Oleft__commute,axiom,
! [B: dedekind_preal,A: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ B @ ( plus_p3173629198307831117_preal @ A @ C ) )
= ( plus_p3173629198307831117_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add.left_commute
thf(fact_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_add__eq__exists,axiom,
! [A: dedekind_real,B: dedekind_real] :
? [X2: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ X2 )
= B ) ).
% add_eq_exists
thf(fact_922_add__eq__exists,axiom,
! [A: int,B: int] :
? [X2: int] :
( ( plus_plus_int @ A @ X2 )
= B ) ).
% add_eq_exists
thf(fact_923_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_924_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_925_add__divide__distrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ C ) @ ( divide9119111104558704680d_real @ B @ C ) ) ) ).
% add_divide_distrib
thf(fact_926_add__divide__eq__if__simps_I2_J,axiom,
! [Z4: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z4 = zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ Z4 ) @ B )
= B ) )
& ( ( Z4 != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ A @ Z4 ) @ B )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ A @ ( times_2157731159493324635d_real @ B @ Z4 ) ) @ Z4 ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_927_add__divide__eq__if__simps_I1_J,axiom,
! [Z4: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( ( Z4 = zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ A @ ( divide9119111104558704680d_real @ B @ Z4 ) )
= A ) )
& ( ( Z4 != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ A @ ( divide9119111104558704680d_real @ B @ Z4 ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ Z4 ) @ B ) @ Z4 ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_928_add__frac__eq,axiom,
! [Y: dedekind_real,Z4: dedekind_real,X: dedekind_real,W2: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( Z4 != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X @ Y ) @ ( divide9119111104558704680d_real @ W2 @ Z4 ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ X @ Z4 ) @ ( times_2157731159493324635d_real @ W2 @ Y ) ) @ ( times_2157731159493324635d_real @ Y @ Z4 ) ) ) ) ) ).
% add_frac_eq
thf(fact_929_add__frac__num,axiom,
! [Y: dedekind_real,X: dedekind_real,Z4: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X @ Y ) @ Z4 )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X @ ( times_2157731159493324635d_real @ Z4 @ Y ) ) @ Y ) ) ) ).
% add_frac_num
thf(fact_930_add__num__frac,axiom,
! [Y: dedekind_real,Z4: dedekind_real,X: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ Z4 @ ( divide9119111104558704680d_real @ X @ Y ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X @ ( times_2157731159493324635d_real @ Z4 @ Y ) ) @ Y ) ) ) ).
% add_num_frac
thf(fact_931_add__divide__eq__iff,axiom,
! [Z4: dedekind_real,X: dedekind_real,Y: dedekind_real] :
( ( Z4 != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ X @ ( divide9119111104558704680d_real @ Y @ Z4 ) )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ X @ Z4 ) @ Y ) @ Z4 ) ) ) ).
% add_divide_eq_iff
thf(fact_932_divide__add__eq__iff,axiom,
! [Z4: dedekind_real,X: dedekind_real,Y: dedekind_real] :
( ( Z4 != zero_z580800474297136991d_real )
=> ( ( plus_p4060926892116697567d_real @ ( divide9119111104558704680d_real @ X @ Z4 ) @ Y )
= ( divide9119111104558704680d_real @ ( plus_p4060926892116697567d_real @ X @ ( times_2157731159493324635d_real @ Y @ Z4 ) ) @ Z4 ) ) ) ).
% divide_add_eq_iff
thf(fact_933_times__divide__times__eq,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real,W2: dedekind_real] :
( ( times_2157731159493324635d_real @ ( divide9119111104558704680d_real @ X @ Y ) @ ( divide9119111104558704680d_real @ Z4 @ W2 ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ X @ Z4 ) @ ( times_2157731159493324635d_real @ Y @ W2 ) ) ) ).
% times_divide_times_eq
thf(fact_934_divide__divide__times__eq,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real,W2: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ X @ Y ) @ ( divide9119111104558704680d_real @ Z4 @ W2 ) )
= ( divide9119111104558704680d_real @ ( times_2157731159493324635d_real @ X @ W2 ) @ ( times_2157731159493324635d_real @ Y @ Z4 ) ) ) ).
% divide_divide_times_eq
thf(fact_935_divide__divide__eq__left_H,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( divide9119111104558704680d_real @ ( divide9119111104558704680d_real @ A @ B ) @ C )
= ( divide9119111104558704680d_real @ A @ ( times_2157731159493324635d_real @ C @ B ) ) ) ).
% divide_divide_eq_left'
thf(fact_936_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( K = L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_937_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_938_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_939_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( I = J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_940_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_941_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_942_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: dedekind_preal,J: dedekind_preal,K: dedekind_preal,L: dedekind_preal] :
( ( ( ord_le5604041210740703414_preal @ I @ J )
& ( ord_le5604041210740703414_preal @ K @ L ) )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ I @ K ) @ ( plus_p3173629198307831117_preal @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_943_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_944_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_945_add__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal,D: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ( ord_le5604041210740703414_preal @ C @ D )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ D ) ) ) ) ).
% add_mono
thf(fact_946_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_947_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_948_add__left__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ C @ A ) @ ( plus_p3173629198307831117_preal @ C @ B ) ) ) ).
% add_left_mono
thf(fact_949_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_950_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_951_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_952_add__right__mono,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ A @ B )
=> ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ A @ C ) @ ( plus_p3173629198307831117_preal @ B @ C ) ) ) ).
% add_right_mono
thf(fact_953_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_954_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_955_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_956_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_957_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_958_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_959_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_960_verit__sum__simplify,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% verit_sum_simplify
thf(fact_961_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_962_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_963_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_964_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_965_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_966_add_Ocomm__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ A @ zero_z580800474297136991d_real )
= A ) ).
% add.comm_neutral
thf(fact_967_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_968_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_969_add_Ogroup__left__neutral,axiom,
! [A: dedekind_real] :
( ( plus_p4060926892116697567d_real @ zero_z580800474297136991d_real @ A )
= A ) ).
% add.group_left_neutral
thf(fact_970_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_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_979_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_980_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_981_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_982_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_983_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_984_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_985_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_986_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_987_ring__class_Oring__distribs_I2_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_988_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_989_ring__class_Oring__distribs_I1_J,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_990_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_991_comm__semiring__class_Odistrib,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_992_comm__semiring__class_Odistrib,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ C ) @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_993_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_994_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_995_distrib__left,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ A @ ( plus_p4060926892116697567d_real @ B @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ B ) @ ( times_2157731159493324635d_real @ A @ C ) ) ) ).
% distrib_left
thf(fact_996_distrib__left,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ A @ ( plus_p3173629198307831117_preal @ B @ C ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ B ) @ ( times_3000655703912201937_preal @ A @ C ) ) ) ).
% distrib_left
thf(fact_997_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_998_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_999_distrib__right,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ C )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ C ) @ ( times_2157731159493324635d_real @ B @ C ) ) ) ).
% distrib_right
thf(fact_1000_distrib__right,axiom,
! [A: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ C )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ C ) @ ( times_3000655703912201937_preal @ B @ C ) ) ) ).
% distrib_right
thf(fact_1001_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_1002_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_1003_combine__common__factor,axiom,
! [A: dedekind_real,E: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ A @ E ) @ ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ B @ E ) @ C ) )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1004_combine__common__factor,axiom,
! [A: dedekind_preal,E: dedekind_preal,B: dedekind_preal,C: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ A @ E ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ B @ E ) @ C ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1005_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1006_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_1007_real__add__mult__distrib,axiom,
! [Z1: dedekind_real,Z2: dedekind_real,W2: dedekind_real] :
( ( times_2157731159493324635d_real @ ( plus_p4060926892116697567d_real @ Z1 @ Z2 ) @ W2 )
= ( plus_p4060926892116697567d_real @ ( times_2157731159493324635d_real @ Z1 @ W2 ) @ ( times_2157731159493324635d_real @ Z2 @ W2 ) ) ) ).
% real_add_mult_distrib
thf(fact_1008_real__add__left__mono,axiom,
! [X: dedekind_real,Y: dedekind_real,Z4: dedekind_real] :
( ( ord_le2716243287969276982d_real @ X @ Y )
=> ( ord_le2716243287969276982d_real @ ( plus_p4060926892116697567d_real @ Z4 @ X ) @ ( plus_p4060926892116697567d_real @ Z4 @ Y ) ) ) ).
% real_add_left_mono
thf(fact_1009_dbl__def,axiom,
( neg_nu8665065731627423525d_real
= ( ^ [X5: dedekind_real] : ( plus_p4060926892116697567d_real @ X5 @ X5 ) ) ) ).
% dbl_def
thf(fact_1010_dbl__def,axiom,
( neg_numeral_dbl_int
= ( ^ [X5: int] : ( plus_plus_int @ X5 @ X5 ) ) ) ).
% dbl_def
thf(fact_1011_frac__eq__eq,axiom,
! [Y: dedekind_real,Z4: dedekind_real,X: dedekind_real,W2: dedekind_real] :
( ( Y != zero_z580800474297136991d_real )
=> ( ( Z4 != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ X @ Y )
= ( divide9119111104558704680d_real @ W2 @ Z4 ) )
= ( ( times_2157731159493324635d_real @ X @ Z4 )
= ( times_2157731159493324635d_real @ W2 @ Y ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1012_divide__eq__eq,axiom,
! [B: dedekind_real,C: dedekind_real,A: dedekind_real] :
( ( ( divide9119111104558704680d_real @ B @ C )
= A )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( B
= ( times_2157731159493324635d_real @ A @ C ) ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% divide_eq_eq
thf(fact_1013_eq__divide__eq,axiom,
! [A: dedekind_real,B: dedekind_real,C: dedekind_real] :
( ( A
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( ( C != zero_z580800474297136991d_real )
=> ( ( times_2157731159493324635d_real @ A @ C )
= B ) )
& ( ( C = zero_z580800474297136991d_real )
=> ( A = zero_z580800474297136991d_real ) ) ) ) ).
% eq_divide_eq
thf(fact_1014_divide__eq__imp,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( B
= ( times_2157731159493324635d_real @ A @ C ) )
=> ( ( divide9119111104558704680d_real @ B @ C )
= A ) ) ) ).
% divide_eq_imp
thf(fact_1015_eq__divide__imp,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( times_2157731159493324635d_real @ A @ C )
= B )
=> ( A
= ( divide9119111104558704680d_real @ B @ C ) ) ) ) ).
% eq_divide_imp
thf(fact_1016_nonzero__divide__eq__eq,axiom,
! [C: dedekind_real,B: dedekind_real,A: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ B @ C )
= A )
= ( B
= ( times_2157731159493324635d_real @ A @ C ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1017_nonzero__eq__divide__eq,axiom,
! [C: dedekind_real,A: dedekind_real,B: dedekind_real] :
( ( C != zero_z580800474297136991d_real )
=> ( ( A
= ( divide9119111104558704680d_real @ B @ C ) )
= ( ( times_2157731159493324635d_real @ A @ C )
= B ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1018_right__inverse__eq,axiom,
! [B: dedekind_real,A: dedekind_real] :
( ( B != zero_z580800474297136991d_real )
=> ( ( ( divide9119111104558704680d_real @ A @ B )
= one_on6069100329679821595d_real )
= ( A = B ) ) ) ).
% right_inverse_eq
thf(fact_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_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_1027_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_1028_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_1029_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_1030_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_1031_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1032_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_1033_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1034_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_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_1043_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_1044_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_1045_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_1046_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_1047_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_1048_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1049_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_1050_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_1051_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1052_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1068_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1069_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_1070_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_1071_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1072_div__mult__self4,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1073_div__mult__self4,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self4
thf(fact_1074_div__mult__self3,axiom,
! [B: nat,C: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1075_div__mult__self3,axiom,
! [B: int,C: int,A: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self3
thf(fact_1076_div__mult__self2,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1077_div__mult__self2,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self2
thf(fact_1078_div__mult__self1,axiom,
! [B: nat,A: nat,C: nat] :
( ( B != zero_zero_nat )
=> ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
= ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1079_div__mult__self1,axiom,
! [B: int,A: int,C: int] :
( ( B != zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
= ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% div_mult_self1
thf(fact_1080_div__mult__mult1,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
= ( divide_divide_nat @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1081_div__mult__mult1,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult1
thf(fact_1082_div__mult__mult2,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ A @ B ) ) ) ).
% div_mult_mult2
thf(fact_1083_preal__add__less__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_left
thf(fact_1084_preal__add__less__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_less_cancel_right
thf(fact_1085_preal__add__le__cancel__right,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_right
thf(fact_1086_preal__add__le__cancel__left,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5604041210740703414_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
= ( ord_le5604041210740703414_preal @ R @ S ) ) ).
% preal_add_le_cancel_left
thf(fact_1087_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_1088_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_1089_preal__add__less2__mono1,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) ) ) ).
% preal_add_less2_mono1
thf(fact_1090_preal__add__less2__mono2,axiom,
! [R: dedekind_preal,S: dedekind_preal,T: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) ) ) ).
% preal_add_less2_mono2
thf(fact_1091_preal__add__mult__distrib,axiom,
! [X: dedekind_preal,Y: dedekind_preal,W2: dedekind_preal] :
( ( times_3000655703912201937_preal @ ( plus_p3173629198307831117_preal @ X @ Y ) @ W2 )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ W2 ) @ ( times_3000655703912201937_preal @ Y @ W2 ) ) ) ).
% preal_add_mult_distrib
thf(fact_1092_preal__add__mult__distrib2,axiom,
! [W2: dedekind_preal,X: dedekind_preal,Y: dedekind_preal] :
( ( times_3000655703912201937_preal @ W2 @ ( plus_p3173629198307831117_preal @ X @ Y ) )
= ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ W2 @ X ) @ ( times_3000655703912201937_preal @ W2 @ Y ) ) ) ).
% preal_add_mult_distrib2
thf(fact_1093_preal__self__less__add__left,axiom,
! [R: dedekind_preal,S: dedekind_preal] : ( ord_le5708704896291381698_preal @ R @ ( plus_p3173629198307831117_preal @ R @ S ) ) ).
% preal_self_less_add_left
thf(fact_1094_preal__add__left__less__cancel,axiom,
! [T: dedekind_preal,R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ T @ R ) @ ( plus_p3173629198307831117_preal @ T @ S ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_left_less_cancel
thf(fact_1095_real__mult__congruent2__lemma,axiom,
! [X1: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal,Y1: dedekind_preal,X: dedekind_preal,Y: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) )
=> ( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ X1 ) @ ( times_3000655703912201937_preal @ Y @ Y1 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ Y22 ) @ ( times_3000655703912201937_preal @ Y @ X22 ) ) )
= ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ X22 ) @ ( times_3000655703912201937_preal @ Y @ Y22 ) ) @ ( plus_p3173629198307831117_preal @ ( times_3000655703912201937_preal @ X @ Y1 ) @ ( times_3000655703912201937_preal @ Y @ X1 ) ) ) ) ) ).
% real_mult_congruent2_lemma
thf(fact_1096_preal__add__right__less__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ ( plus_p3173629198307831117_preal @ R @ T ) @ ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( ord_le5708704896291381698_preal @ R @ S ) ) ).
% preal_add_right_less_cancel
thf(fact_1097_preal__add__right__cancel,axiom,
! [R: dedekind_preal,T: dedekind_preal,S: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ R @ T )
= ( plus_p3173629198307831117_preal @ S @ T ) )
=> ( R = S ) ) ).
% preal_add_right_cancel
thf(fact_1098_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_1099_preal__trans__lemma,axiom,
! [X: dedekind_preal,Y1: dedekind_preal,X1: dedekind_preal,Y: dedekind_preal,Y22: dedekind_preal,X22: dedekind_preal] :
( ( ( plus_p3173629198307831117_preal @ X @ Y1 )
= ( plus_p3173629198307831117_preal @ X1 @ Y ) )
=> ( ( ( plus_p3173629198307831117_preal @ X @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y ) )
=> ( ( plus_p3173629198307831117_preal @ X1 @ Y22 )
= ( plus_p3173629198307831117_preal @ X22 @ Y1 ) ) ) ) ).
% preal_trans_lemma
thf(fact_1100_preal__add__commute,axiom,
( plus_p3173629198307831117_preal
= ( ^ [X5: dedekind_preal,Y5: dedekind_preal] : ( plus_p3173629198307831117_preal @ Y5 @ X5 ) ) ) ).
% preal_add_commute
thf(fact_1101_preal__add__assoc,axiom,
! [X: dedekind_preal,Y: dedekind_preal,Z4: dedekind_preal] :
( ( plus_p3173629198307831117_preal @ ( plus_p3173629198307831117_preal @ X @ Y ) @ Z4 )
= ( plus_p3173629198307831117_preal @ X @ ( plus_p3173629198307831117_preal @ Y @ Z4 ) ) ) ).
% preal_add_assoc
thf(fact_1102_preal__mult__1,axiom,
! [Z4: dedekind_preal] :
( ( times_3000655703912201937_preal @ one_on9143529541772854033_preal @ Z4 )
= Z4 ) ).
% preal_mult_1
thf(fact_1103_real__less__sum__gt__zero,axiom,
! [W2: dedekind_real,S: dedekind_real] :
( ( ord_le2991122432403439658d_real @ W2 @ S )
=> ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W2 ) ) ) ) ).
% real_less_sum_gt_zero
thf(fact_1104_real__sum__gt__zero__less,axiom,
! [S: dedekind_real,W2: dedekind_real] :
( ( ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ ( plus_p4060926892116697567d_real @ S @ ( uminus7714077491378687647d_real @ W2 ) ) )
=> ( ord_le2991122432403439658d_real @ W2 @ S ) ) ).
% real_sum_gt_zero_less
thf(fact_1105_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_1106_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_1107_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one )
=> ( ! [X2: num] :
( ( P @ X2 )
=> ( P @ ( inc @ X2 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_1108_add__One,axiom,
! [X: num] :
( ( plus_plus_num @ X @ one )
= ( inc @ X ) ) ).
% add_One
thf(fact_1109_add__inc,axiom,
! [X: num,Y: num] :
( ( plus_plus_num @ X @ ( inc @ Y ) )
= ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).
% add_inc
thf(fact_1110_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_1111_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_1112_mult__inc,axiom,
! [X: num,Y: num] :
( ( times_times_num @ X @ ( inc @ Y ) )
= ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).
% mult_inc
thf(fact_1113_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_1114_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1115_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1116_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_1117_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_1118_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1119_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ one_one_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1120_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ( divide_divide_nat @ M @ N )
= M )
= ( N = one_one_nat ) ) ) ).
% div_eq_dividend_iff
thf(fact_1121_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_1122_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_1123_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_1124_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_1125_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_1126_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_1127_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_1128_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_1129_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_1130_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1131_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1132_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_1133_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_1134_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ zero_zero_int )
=> ( ( P @ ( uminus_uminus_int @ one_one_int ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2 != zero_zero_int )
=> ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus_int @ one_one_int ) )
=> ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_1135_pos__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ B @ A ) ) ) ).
% pos_zdiv_mult_2
thf(fact_1136_neg__zdiv__mult__2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
= ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).
% neg_zdiv_mult_2
thf(fact_1137_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1138_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1139_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_1140_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1141_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1142_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1143_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1144_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_1145_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1146_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_1147_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1148_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1149_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1150_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_1151_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_1152_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1153_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1154_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_1155_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_1156_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_1157_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1158_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_1159_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1160_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_1161_less__eq__div__iff__mult__less__eq,axiom,
! [Q3: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ Q3 )
=> ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
= ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1162_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1163_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1164_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ N @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1165_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1166_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1167_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1168_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_1169_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1170_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_1171_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1172_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_1173_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1174_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_1175_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1176_inc_Osimps_I1_J,axiom,
( ( inc @ one )
= ( bit0 @ one ) ) ).
% inc.simps(1)
thf(fact_1177_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_1178_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_1179_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1180_zle__add1__eq__le,axiom,
! [W2: int,Z4: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z4 @ one_one_int ) )
= ( ord_less_eq_int @ W2 @ Z4 ) ) ).
% zle_add1_eq_le
thf(fact_1181_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ K @ L )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_pos_pos_trivial
thf(fact_1182_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ( ( ord_less_int @ L @ K )
=> ( ( divide_divide_int @ K @ L )
= zero_zero_int ) ) ) ).
% div_neg_neg_trivial
thf(fact_1183_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
=> ( ( divide_divide_int @ K @ L )
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1184_div__eq__minus1,axiom,
! [B: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
= ( uminus_uminus_int @ one_one_int ) ) ) ).
% div_eq_minus1
thf(fact_1185_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( ( M = one_one_int )
& ( N = one_one_int ) )
| ( ( M
= ( uminus_uminus_int @ one_one_int ) )
& ( N
= ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1186_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times_int @ M @ N )
= one_one_int )
=> ( ( M = one_one_int )
| ( M
= ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1187_odd__nonzero,axiom,
! [Z4: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1188_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less_int @ zero_zero_int @ M )
=> ( ( ( times_times_int @ M @ N )
= one_one_int )
= ( ( M = one_one_int )
& ( N = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1189_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1190_zless__add1__eq,axiom,
! [W2: int,Z4: int] :
( ( ord_less_int @ W2 @ ( plus_plus_int @ Z4 @ one_one_int ) )
= ( ( ord_less_int @ W2 @ Z4 )
| ( W2 = Z4 ) ) ) ).
% zless_add1_eq
thf(fact_1191_odd__less__0__iff,axiom,
! [Z4: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z4 ) @ Z4 ) @ zero_zero_int )
= ( ord_less_int @ Z4 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1192_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less_int @ zero_zero_int @ X )
=> ( ( ord_less_int @ one_one_int @ K )
=> ( ord_less_int @ ( divide_divide_int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1193_int__one__le__iff__zero__less,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ one_one_int @ Z4 )
= ( ord_less_int @ zero_zero_int @ Z4 ) ) ).
% int_one_le_iff_zero_less
thf(fact_1194_zless__imp__add1__zle,axiom,
! [W2: int,Z4: int] :
( ( ord_less_int @ W2 @ Z4 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z4 ) ) ).
% zless_imp_add1_zle
thf(fact_1195_le__imp__0__less,axiom,
! [Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z4 ) ) ) ).
% le_imp_0_less
thf(fact_1196_add1__zle__eq,axiom,
! [W2: int,Z4: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z4 )
= ( ord_less_int @ W2 @ Z4 ) ) ).
% add1_zle_eq
thf(fact_1197_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1198_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_1199_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P4: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_1200_nonneg1__imp__zdiv__pos__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1201_pos__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1202_neg__imp__zdiv__nonneg__iff,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1203_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
= ( ord_less_eq_int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1204_div__nonpos__pos__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1205_div__nonneg__neg__le0,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1206_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
= ( ( K = zero_zero_int )
| ( L = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ K )
& ( ord_less_eq_int @ zero_zero_int @ L ) )
| ( ( ord_less_int @ K @ zero_zero_int )
& ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% div_int_pos_iff
thf(fact_1207_zdiv__zmult2__eq,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
= ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1208_zdiv__mono2__neg,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1209_zdiv__mono1__neg,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1210_int__div__pos__eq,axiom,
! [A: int,B: int,Q3: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ R )
=> ( ( ord_less_int @ R @ B )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1211_int__div__neg__eq,axiom,
! [A: int,B: int,Q3: int,R: int] :
( ( A
= ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R ) )
=> ( ( ord_less_eq_int @ R @ zero_zero_int )
=> ( ( ord_less_int @ B @ R )
=> ( ( divide_divide_int @ A @ B )
= Q3 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1212_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide_int @ I @ K )
= zero_zero_int )
= ( ( K = zero_zero_int )
| ( ( ord_less_eq_int @ zero_zero_int @ I )
& ( ord_less_int @ I @ K ) )
| ( ( ord_less_eq_int @ I @ zero_zero_int )
& ( ord_less_int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1213_zdiv__mono2,axiom,
! [A: int,B4: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B4 )
=> ( ( ord_less_eq_int @ B4 @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1214_zdiv__mono1,axiom,
! [A: int,A4: int,B: int] :
( ( ord_less_eq_int @ A @ A4 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).
% zdiv_mono1
thf(fact_1215_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide_int @ N @ K ) )
= ( ( ( K = zero_zero_int )
=> ( P @ zero_zero_int ) )
& ( ( ord_less_int @ zero_zero_int @ K )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
& ( ord_less_int @ J3 @ K )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) )
& ( ( ord_less_int @ K @ zero_zero_int )
=> ! [I4: int,J3: int] :
( ( ( ord_less_int @ K @ J3 )
& ( ord_less_eq_int @ J3 @ zero_zero_int )
& ( N
= ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
=> ( P @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1216_incr__mult__lemma,axiom,
! [D: int,P: int > $o,K: int] :
( ( ord_less_int @ zero_zero_int @ D )
=> ( ! [X2: int] :
( ( P @ X2 )
=> ( P @ ( plus_plus_int @ X2 @ D ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1217_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less_eq_int @ A5 @ B5 )
=> ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
=> ( ord_less_eq_int @ ( divide_divide_int @ B5 @ N ) @ ( divide_divide_int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1218_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_1219_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_1220_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_1221_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_1222_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_1223_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_1224_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_1225_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_1226_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_1227_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_1228_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_1229_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_1230_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_1231_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1232_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_1233_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_1234_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_1235_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_1236_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1237_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1238_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_1239_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
= ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1240_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_1241_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M4: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M4 ) )
=> ~ ! [M3: nat] :
( ( P @ M3 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1242_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1243_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1244_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1245_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_1246_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_1247_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_1248_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_1249_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1250_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_1251_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_1252_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_1253_zle__diff1__eq,axiom,
! [W2: int,Z4: int] :
( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z4 @ one_one_int ) )
= ( ord_less_int @ W2 @ Z4 ) ) ).
% zle_diff1_eq
thf(fact_1254_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
=> ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
= ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_1255_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_less__add__left,axiom,
! [R: dedekind_preal,S: dedekind_preal] :
( ( ord_le5708704896291381698_preal @ R @ S )
=> ( ( plus_p3173629198307831117_preal @ R @ ( minus_7336623429200594941_preal @ S @ R ) )
= S ) ) ).
% less_add_left
thf(fact_1263_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_1264_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_1265_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_1266_real__le__eq__diff,axiom,
( ord_le2716243287969276982d_real
= ( ^ [X5: dedekind_real,Y5: dedekind_real] : ( ord_le2716243287969276982d_real @ ( minus_5539002012860128047d_real @ X5 @ Y5 ) @ zero_z580800474297136991d_real ) ) ) ).
% real_le_eq_diff
thf(fact_1267_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
% Conjectures (3)
thf(conj_0,hypothesis,
ord_le2991122432403439658d_real @ zero_z580800474297136991d_real @ z ).
thf(conj_1,hypothesis,
ord_le2991122432403439658d_real @ x @ y ).
thf(conj_2,conjecture,
ord_le2991122432403439658d_real @ ( times_2157731159493324635d_real @ z @ x ) @ ( times_2157731159493324635d_real @ z @ y ) ).
%------------------------------------------------------------------------------