TPTP Problem File: ITP035^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP035^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Cartan problem prob_32__6530878_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Cartan/prob_32__6530878_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 500 ( 193 unt; 80 typ; 0 def)
% Number of atoms : 1017 ( 305 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3677 ( 42 ~; 23 |; 73 &;3098 @)
% ( 0 <=>; 441 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 284 ( 284 >; 0 *; 0 +; 0 <<)
% Number of symbols : 80 ( 77 usr; 5 con; 0-6 aty)
% Number of variables : 1042 ( 37 ^; 918 !; 15 ?;1042 :)
% ( 72 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:23:02.925
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Formal__Power__Series_Ofps,type,
formal_Power_fps: $tType > $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (75)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V2090557954_space:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Otimes,type,
times:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot0__space,type,
topological_t0_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V758409581_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo2133971006pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo503727757_space:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo2135403230pology:
!>[A: $tType] : $o ).
thf(sy_c_Complex__Analysis__Basics_Oholomorphic__on,type,
comple372758642hic_on: ( complex > complex ) > ( set @ complex ) > $o ).
thf(sy_c_Derivative_Oderiv,type,
deriv:
!>[A: $tType] : ( ( A > A ) > A > A ) ).
thf(sy_c_Formal__Power__Series_Ofps__tan,type,
formal_Power_fps_tan:
!>[A: $tType] : ( A > ( formal_Power_fps @ A ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1751647064n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oconnected,type,
topolo1099314120nected:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transitive__Closure_Orelpowp,type,
transitive_relpowp:
!>[A: $tType] : ( nat > ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_S,type,
s: set @ complex ).
thf(sy_v_T,type,
t: set @ complex ).
thf(sy_v_f,type,
f: complex > complex ).
thf(sy_v_g,type,
g: complex > complex ).
thf(sy_v_w,type,
w: complex ).
% Relevant facts (253)
thf(fact_0__092_060open_062_092_060And_062z_O_A_Ideriv_A_094_094_A1_J_Aid_Az_A_061_A_Iif_A1_A_061_A0_Athen_Az_Aelse_Aif_A1_A_061_A1_Athen_A1_058_058_063_Ha_Aelse_A_I0_058_058_063_Ha_J_J_092_060close_062,axiom,
! [A: $tType] :
( ( real_V758409581_field @ A )
=> ! [Z: A] :
( ( ( ( one_one @ nat )
= ( zero_zero @ nat ) )
=> ( ( compow @ ( ( A > A ) > A > A ) @ ( one_one @ nat ) @ ( deriv @ A ) @ ( id @ A ) @ Z )
= Z ) )
& ( ( ( one_one @ nat )
!= ( zero_zero @ nat ) )
=> ( ( compow @ ( ( A > A ) > A > A ) @ ( one_one @ nat ) @ ( deriv @ A ) @ ( id @ A ) @ Z )
= ( one_one @ A ) ) ) ) ) ).
% \<open>\<And>z. (deriv ^^ 1) id z = (if 1 = 0 then z else if 1 = 1 then 1::?'a else (0::?'a))\<close>
thf(fact_1_assms_I7_J,axiom,
member @ complex @ w @ s ).
% assms(7)
thf(fact_2_deriv__id,axiom,
! [A: $tType] :
( ( real_V758409581_field @ A )
=> ( ( deriv @ A @ ( id @ A ) )
= ( ^ [Z2: A] : ( one_one @ A ) ) ) ) ).
% deriv_id
thf(fact_3_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_apply
thf(fact_4_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X: A] : X ) ) ).
% id_def
thf(fact_5_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X: A] :
( ( F @ X )
= X ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_6_higher__deriv__id,axiom,
! [A: $tType] :
( ( real_V758409581_field @ A )
=> ! [N: nat,Z: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( compow @ ( ( A > A ) > A > A ) @ N @ ( deriv @ A ) @ ( id @ A ) @ Z )
= Z ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( ( N
= ( one_one @ nat ) )
=> ( ( compow @ ( ( A > A ) > A > A ) @ N @ ( deriv @ A ) @ ( id @ A ) @ Z )
= ( one_one @ A ) ) )
& ( ( N
!= ( one_one @ nat ) )
=> ( ( compow @ ( ( A > A ) > A > A ) @ N @ ( deriv @ A ) @ ( id @ A ) @ Z )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% higher_deriv_id
thf(fact_7_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_8_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_9_calculation,axiom,
( ( times_times @ complex @ ( deriv @ complex @ f @ w ) @ ( deriv @ complex @ g @ ( f @ w ) ) )
= ( deriv @ complex @ ( id @ complex ) @ w ) ) ).
% calculation
thf(fact_10__092_060open_062deriv_A_Ig_A_092_060circ_062_Af_J_Aw_A_061_Aderiv_Aid_Aw_092_060close_062,axiom,
( ( deriv @ complex @ ( comp @ complex @ complex @ complex @ g @ f ) @ w )
= ( deriv @ complex @ ( id @ complex ) @ w ) ) ).
% \<open>deriv (g \<circ> f) w = deriv id w\<close>
thf(fact_11_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_12__092_060open_062deriv_Af_Aw_A_K_Aderiv_Ag_A_If_Aw_J_A_061_Aderiv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_092_060close_062,axiom,
( ( times_times @ complex @ ( deriv @ complex @ f @ w ) @ ( deriv @ complex @ g @ ( f @ w ) ) )
= ( times_times @ complex @ ( deriv @ complex @ g @ ( f @ w ) ) @ ( deriv @ complex @ f @ w ) ) ) ).
% \<open>deriv f w * deriv g (f w) = deriv g (f w) * deriv f w\<close>
thf(fact_13_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G: C > B,X: C] : ( F2 @ ( G @ X ) ) ) ) ).
% comp_apply
thf(fact_14_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_15_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_16_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_17_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_18_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_19_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_20_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_21_id__comp,axiom,
! [B: $tType,A: $tType,G2: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G2 )
= G2 ) ).
% id_comp
thf(fact_22_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_23_funpow__0,axiom,
! [A: $tType,F: A > A,X2: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X2 )
= X2 ) ).
% funpow_0
thf(fact_24_assms_I3_J,axiom,
topolo1751647064n_open @ complex @ s ).
% assms(3)
thf(fact_25__092_060open_062deriv_Ag_A_If_Aw_J_A_K_Aderiv_Af_Aw_A_061_Aderiv_A_Ig_A_092_060circ_062_Af_J_Aw_092_060close_062,axiom,
( ( times_times @ complex @ ( deriv @ complex @ g @ ( f @ w ) ) @ ( deriv @ complex @ f @ w ) )
= ( deriv @ complex @ ( comp @ complex @ complex @ complex @ g @ f ) @ w ) ) ).
% \<open>deriv g (f w) * deriv f w = deriv (g \<circ> f) w\<close>
thf(fact_26_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_27_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_28_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_29_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C2: A,B2: A] :
( ( C2
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_30_assms_I6_J,axiom,
! [Z: complex] :
( ( member @ complex @ Z @ s )
=> ( ( g @ ( f @ Z ) )
= Z ) ) ).
% assms(6)
thf(fact_31_assms_I1_J,axiom,
comple372758642hic_on @ f @ s ).
% assms(1)
thf(fact_32_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_33_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G: A > B,X: A] : ( F2 @ ( G @ X ) ) ) ) ).
% comp_def
thf(fact_34_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G2 ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_35_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% comp_funpow
thf(fact_36_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_37_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_38_funpow__swap1,axiom,
! [A: $tType,F: A > A,N: nat,X2: A] :
( ( F @ ( compow @ ( A > A ) @ N @ F @ X2 ) )
= ( compow @ ( A > A ) @ N @ F @ ( F @ X2 ) ) ) ).
% funpow_swap1
thf(fact_39_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_40_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_41_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_42_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G2 @ X3 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_46_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_47_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B3: A] : ( times_times @ A @ B3 @ A4 ) ) ) ) ).
% mult.commute
thf(fact_48_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_49_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_50_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_51_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_52_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_53_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_54_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_55_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_56_funpow__simps__right_I1_J,axiom,
! [A: $tType,F: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_57_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T )
= T ) ).
% fun.map_id
thf(fact_58_vector__space__over__itself_Oscale__one,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( times_times @ A @ ( one_one @ A ) @ X2 )
= X2 ) ) ).
% vector_space_over_itself.scale_one
thf(fact_59_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A] :
( ( ( times_times @ A @ A2 @ X2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_60_vector__space__over__itself_Oscale__zero__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_61_vector__space__over__itself_Oscale__zero__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_62_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A,Y: A] :
( ( ( times_times @ A @ A2 @ X2 )
= ( times_times @ A @ A2 @ Y ) )
= ( ( X2 = Y )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_63_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A,B2: A] :
( ( ( times_times @ A @ A2 @ X2 )
= ( times_times @ A @ B2 @ X2 ) )
= ( ( A2 = B2 )
| ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_64_mult__if__delta,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [P: $o,Q2: A] :
( ( P
=> ( ( times_times @ A @ ( if @ A @ P @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ Q2 )
= Q2 ) )
& ( ~ P
=> ( ( times_times @ A @ ( if @ A @ P @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ Q2 )
= ( zero_zero @ A ) ) ) ) ) ).
% mult_if_delta
thf(fact_65_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_66_set__times__intro,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [A2: A,C3: set @ A,B2: A,D3: set @ A] :
( ( member @ A @ A2 @ C3 )
=> ( ( member @ A @ B2 @ D3 )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ ( set @ A ) @ C3 @ D3 ) ) ) ) ) ).
% set_times_intro
thf(fact_67_funpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% funpow_code_def
thf(fact_68_assms_I4_J,axiom,
topolo1751647064n_open @ complex @ t ).
% assms(4)
thf(fact_69_assms_I2_J,axiom,
comple372758642hic_on @ g @ t ).
% assms(2)
thf(fact_70_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_71_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_72_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_73_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_74_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_75_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_76_relpowp__Suc__D2_H,axiom,
! [A: $tType,N: nat,P: A > A > $o,X4: A,Y2: A,Z3: A] :
( ( ( compow @ ( A > A > $o ) @ N @ P @ X4 @ Y2 )
& ( P @ Y2 @ Z3 ) )
=> ? [W: A] :
( ( P @ X4 @ W )
& ( compow @ ( A > A > $o ) @ N @ P @ W @ Z3 ) ) ) ).
% relpowp_Suc_D2'
thf(fact_77_higher__deriv__transform__within__open,axiom,
! [F: complex > complex,S: set @ complex,G2: complex > complex,Z: complex,I: nat] :
( ( comple372758642hic_on @ F @ S )
=> ( ( comple372758642hic_on @ G2 @ S )
=> ( ( topolo1751647064n_open @ complex @ S )
=> ( ( member @ complex @ Z @ S )
=> ( ! [W: complex] :
( ( member @ complex @ W @ S )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( compow @ ( ( complex > complex ) > complex > complex ) @ I @ ( deriv @ complex ) @ F @ Z )
= ( compow @ ( ( complex > complex ) > complex > complex ) @ I @ ( deriv @ complex ) @ G2 @ Z ) ) ) ) ) ) ) ).
% higher_deriv_transform_within_open
thf(fact_78_holomorphic__higher__deriv,axiom,
! [F: complex > complex,S: set @ complex,N: nat] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo1751647064n_open @ complex @ S )
=> ( comple372758642hic_on @ ( compow @ ( ( complex > complex ) > complex > complex ) @ N @ ( deriv @ complex ) @ F ) @ S ) ) ) ).
% holomorphic_higher_deriv
thf(fact_79_holomorphic__deriv,axiom,
! [F: complex > complex,S: set @ complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo1751647064n_open @ complex @ S )
=> ( comple372758642hic_on @ ( deriv @ complex @ F ) @ S ) ) ) ).
% holomorphic_deriv
thf(fact_80_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_81_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_82_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_83_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_84_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_85_vector__space__over__itself_Oscale__left__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
= ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ X2 ) ) ) ) ).
% vector_space_over_itself.scale_left_commute
thf(fact_86_vector__space__over__itself_Oscale__scale,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ X2 ) )
= ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% vector_space_over_itself.scale_scale
thf(fact_87_set__times__elim,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [X2: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ A3 @ B4 ) )
=> ~ ! [A5: A,B5: A] :
( ( X2
= ( times_times @ A @ A5 @ B5 ) )
=> ( ( member @ A @ A5 @ A3 )
=> ~ ( member @ A @ B5 @ B4 ) ) ) ) ) ).
% set_times_elim
thf(fact_88_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X2: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X2 @ X2 ) ).
% relpowp_0_I
thf(fact_89_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X2: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X2 @ Y )
=> ( X2 = Y ) ) ).
% relpowp_0_E
thf(fact_90_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R )
= ( ^ [Y3: A,Z4: A] : Y3 = Z4 ) ) ).
% relpowp.simps(1)
thf(fact_91_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_92_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ X2 )
= ( times_times @ A @ B2 @ X2 ) )
=> ( A2 = B2 ) ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_93_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,X2: A,Y: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ X2 )
= ( times_times @ A @ A2 @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_94_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_95_complex__derivative__transform__within__open,axiom,
! [F: complex > complex,S2: set @ complex,G2: complex > complex,Z: complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ S2 )
=> ( ( topolo1751647064n_open @ complex @ S2 )
=> ( ( member @ complex @ Z @ S2 )
=> ( ! [W: complex] :
( ( member @ complex @ W @ S2 )
=> ( ( F @ W )
= ( G2 @ W ) ) )
=> ( ( deriv @ complex @ F @ Z )
= ( deriv @ complex @ G2 @ Z ) ) ) ) ) ) ) ).
% complex_derivative_transform_within_open
thf(fact_96_holomorphic__on__id,axiom,
! [S2: set @ complex] : ( comple372758642hic_on @ ( id @ complex ) @ S2 ) ).
% holomorphic_on_id
thf(fact_97_relpowp__code__def,axiom,
! [A: $tType] :
( ( transitive_relpowp @ A )
= ( compow @ ( A > A > $o ) ) ) ).
% relpowp_code_def
thf(fact_98_holomorphic__on__linear,axiom,
! [C2: complex,S2: set @ complex] : ( comple372758642hic_on @ ( times_times @ complex @ C2 ) @ S2 ) ).
% holomorphic_on_linear
thf(fact_99_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G2: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G2 )
= ( id @ A ) )
=> ( ( F @ ( G2 @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_100_isomorphism__expand,axiom,
! [A: $tType,B: $tType,F: B > A,G2: A > B] :
( ( ( ( comp @ B @ A @ A @ F @ G2 )
= ( id @ A ) )
& ( ( comp @ A @ B @ B @ G2 @ F )
= ( id @ B ) ) )
= ( ! [X: A] :
( ( F @ ( G2 @ X ) )
= X )
& ! [X: B] :
( ( G2 @ ( F @ X ) )
= X ) ) ) ).
% isomorphism_expand
thf(fact_101_left__right__inverse__eq,axiom,
! [A: $tType,B: $tType,F: B > A,G2: A > B,H: B > A] :
( ( ( comp @ B @ A @ A @ F @ G2 )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G2 @ H )
= ( id @ B ) )
=> ( F = H ) ) ) ).
% left_right_inverse_eq
thf(fact_102_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G2: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G2 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_103_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R2: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= R2 )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G2 ) @ H )
= ( comp @ B @ D @ A @ F @ R2 ) ) ) ).
% rewriteR_comp_comp
thf(fact_104_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R2: E > D] :
( ( ( comp @ C @ B @ A @ F @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R2 )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G2 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R2 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_105_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G2 ) @ H )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_106_holomorphic__cong,axiom,
! [S2: set @ complex,T: set @ complex,F: complex > complex,G2: complex > complex] :
( ( S2 = T )
=> ( ! [X3: complex] :
( ( member @ complex @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( comple372758642hic_on @ F @ S2 )
= ( comple372758642hic_on @ G2 @ T ) ) ) ) ).
% holomorphic_cong
thf(fact_107_holomorphic__transform,axiom,
! [F: complex > complex,S2: set @ complex,G2: complex > complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ! [X3: complex] :
( ( member @ complex @ X3 @ S2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( comple372758642hic_on @ G2 @ S2 ) ) ) ).
% holomorphic_transform
thf(fact_108_assms_I5_J,axiom,
ord_less_eq @ ( set @ complex ) @ ( image @ complex @ complex @ f @ s ) @ t ).
% assms(5)
thf(fact_109_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_110_mult__delta__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [B2: $o,X2: A,Y: A] :
( ( B2
=> ( ( times_times @ A @ ( if @ A @ B2 @ X2 @ ( zero_zero @ A ) ) @ Y )
= ( times_times @ A @ X2 @ Y ) ) )
& ( ~ B2
=> ( ( times_times @ A @ ( if @ A @ B2 @ X2 @ ( zero_zero @ A ) ) @ Y )
= ( zero_zero @ A ) ) ) ) ) ).
% mult_delta_left
thf(fact_111_mult__delta__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [B2: $o,X2: A,Y: A] :
( ( B2
=> ( ( times_times @ A @ X2 @ ( if @ A @ B2 @ Y @ ( zero_zero @ A ) ) )
= ( times_times @ A @ X2 @ Y ) ) )
& ( ~ B2
=> ( ( times_times @ A @ X2 @ ( if @ A @ B2 @ Y @ ( zero_zero @ A ) ) )
= ( zero_zero @ A ) ) ) ) ) ).
% mult_delta_right
thf(fact_112_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_113_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_114_image__id,axiom,
! [A: $tType] :
( ( image @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_115_set__times__mono2,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [C3: set @ A,D3: set @ A,E2: set @ A,F3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D3 )
=> ( ( ord_less_eq @ ( set @ A ) @ E2 @ F3 )
=> ( ord_less_eq @ ( set @ A ) @ ( times_times @ ( set @ A ) @ C3 @ E2 ) @ ( times_times @ ( set @ A ) @ D3 @ F3 ) ) ) ) ) ).
% set_times_mono2
thf(fact_116_holomorphic__on__compose__gen,axiom,
! [F: complex > complex,S2: set @ complex,G2: complex > complex,T: set @ complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ T )
=> ( ( ord_less_eq @ ( set @ complex ) @ ( image @ complex @ complex @ F @ S2 ) @ T )
=> ( comple372758642hic_on @ ( comp @ complex @ complex @ complex @ G2 @ F ) @ S2 ) ) ) ) ).
% holomorphic_on_compose_gen
thf(fact_117_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G2: C > B,R2: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G2 @ R2 ) )
= ( image @ C @ A @ ( comp @ B @ A @ C @ F @ G2 ) @ R2 ) ) ).
% image_comp
thf(fact_118_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A3: set @ B,G2: C > A,B4: set @ C,H: A > D] :
( ( ( image @ B @ A @ F @ A3 )
= ( image @ C @ A @ G2 @ B4 ) )
=> ( ( image @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A3 )
= ( image @ C @ D @ ( comp @ A @ D @ C @ H @ G2 ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_119_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_120_holomorphic__on__subset,axiom,
! [F: complex > complex,S2: set @ complex,T: set @ complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( ord_less_eq @ ( set @ complex ) @ T @ S2 )
=> ( comple372758642hic_on @ F @ T ) ) ) ).
% holomorphic_on_subset
thf(fact_121_set__times__mono2__b,axiom,
! [A: $tType] :
( ( times @ A )
=> ! [C3: set @ A,D3: set @ A,E2: set @ A,F3: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ C3 @ D3 )
=> ( ( ord_less_eq @ ( set @ A ) @ E2 @ F3 )
=> ( ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ C3 @ E2 ) )
=> ( member @ A @ X2 @ ( times_times @ ( set @ A ) @ D3 @ F3 ) ) ) ) ) ) ).
% set_times_mono2_b
thf(fact_122_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere1490568538miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_123_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_124_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_125_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_126_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_127_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_128_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_129_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_130_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_131_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_132_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_133_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_134_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_135_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_136_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_137_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_138_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_139_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_140_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_141_holomorphic__on__compose,axiom,
! [F: complex > complex,S2: set @ complex,G2: complex > complex] :
( ( comple372758642hic_on @ F @ S2 )
=> ( ( comple372758642hic_on @ G2 @ ( image @ complex @ complex @ F @ S2 ) )
=> ( comple372758642hic_on @ ( comp @ complex @ complex @ complex @ G2 @ F ) @ S2 ) ) ) ).
% holomorphic_on_compose
thf(fact_142_deriv__left__inverse,axiom,
! [F: complex > complex,S: set @ complex,G2: complex > complex,T2: set @ complex,W2: complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( comple372758642hic_on @ G2 @ T2 )
=> ( ( topolo1751647064n_open @ complex @ S )
=> ( ( topolo1751647064n_open @ complex @ T2 )
=> ( ( ord_less_eq @ ( set @ complex ) @ ( image @ complex @ complex @ F @ S ) @ T2 )
=> ( ! [Z5: complex] :
( ( member @ complex @ Z5 @ S )
=> ( ( G2 @ ( F @ Z5 ) )
= Z5 ) )
=> ( ( member @ complex @ W2 @ S )
=> ( ( times_times @ complex @ ( deriv @ complex @ F @ W2 ) @ ( deriv @ complex @ G2 @ ( F @ W2 ) ) )
= ( one_one @ complex ) ) ) ) ) ) ) ) ) ).
% deriv_left_inverse
thf(fact_143_mult__left__le,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_144_mult__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_145_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Y ) @ X2 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_146_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X2 ) @ X2 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_147_mult__eq__1,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ordered_semiring @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( A2
= ( one_one @ A ) )
& ( B2
= ( one_one @ A ) ) ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_148_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V758409581_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_149_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_150_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_151_image__eqI,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,X2: B,A3: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A3 )
=> ( member @ A @ B2 @ ( image @ B @ A @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_152_subsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% subsetI
thf(fact_153_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_154_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_155_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_156_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_157_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_158_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_159_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_160_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_161_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_162_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_163_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_164_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A3: set @ A,F: A > B] :
( ( member @ A @ X2 @ A3 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A3 ) ) ) ).
% imageI
thf(fact_165_image__iff,axiom,
! [A: $tType,B: $tType,Z: A,F: B > A,A3: set @ B] :
( ( member @ A @ Z @ ( image @ B @ A @ F @ A3 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A3 )
& ( Z
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_166_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A3 ) )
& ( P @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_167_image__cong,axiom,
! [B: $tType,A: $tType,M2: set @ A,N2: set @ A,F: A > B,G2: A > B] :
( ( M2 = N2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G2 @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M2 )
= ( image @ A @ B @ G2 @ N2 ) ) ) ) ).
% image_cong
thf(fact_168_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A3 ) )
=> ( P @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( P @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_169_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A3: set @ A,B2: B,F: A > B] :
( ( member @ A @ X2 @ A3 )
=> ( ( B2
= ( F @ X2 ) )
=> ( member @ B @ B2 @ ( image @ A @ B @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_170_in__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B4 ) ) ) ).
% in_mono
thf(fact_171_subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_172_equalityE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_173_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X: A] :
( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B6 ) ) ) ) ).
% subset_eq
thf(fact_174_equalityD1,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_175_equalityD2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_176_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_177_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_178_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_179_subset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_180_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z4: set @ A] : Y3 = Z4 )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_181_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B4: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ ( image @ A @ B @ F @ B4 ) ) ) ).
% image_mono
thf(fact_183_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F: A > B,B4: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A3 )
=> ( member @ B @ ( F @ X3 ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A3 ) @ B4 ) ) ).
% image_subsetI
thf(fact_184_subset__imageE,axiom,
! [A: $tType,B: $tType,B4: set @ A,F: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A3 ) )
=> ~ ! [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A3 )
=> ( B4
!= ( image @ B @ A @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_185_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A3 ) @ B4 )
= ( ! [X: B] :
( ( member @ B @ X @ A3 )
=> ( member @ A @ ( F @ X ) @ B4 ) ) ) ) ).
% image_subset_iff
thf(fact_186_subset__image__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,F: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B4
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_187_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_188_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q: nat > $o] :
( ! [X3: nat > real] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( ! [X3: nat > real] :
( ( P @ X3 )
=> ! [I2: nat] :
( ( Q @ I2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X3 @ I2 ) )
& ( ord_less_eq @ real @ ( X3 @ I2 ) @ ( one_one @ real ) ) ) ) )
=> ? [L3: ( nat > real ) > nat > nat] :
( ! [X4: nat > real,I3: nat] : ( ord_less_eq @ nat @ ( L3 @ X4 @ I3 ) @ ( one_one @ nat ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( X4 @ I3 )
= ( zero_zero @ real ) ) )
=> ( ( L3 @ X4 @ I3 )
= ( zero_zero @ nat ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( X4 @ I3 )
= ( one_one @ real ) ) )
=> ( ( L3 @ X4 @ I3 )
= ( one_one @ nat ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( L3 @ X4 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ord_less_eq @ real @ ( X4 @ I3 ) @ ( F @ X4 @ I3 ) ) )
& ! [X4: nat > real,I3: nat] :
( ( ( P @ X4 )
& ( Q @ I3 )
& ( ( L3 @ X4 @ I3 )
= ( one_one @ nat ) ) )
=> ( ord_less_eq @ real @ ( F @ X4 @ I3 ) @ ( X4 @ I3 ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_189_Sup_OSUP__id__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_id_eq
thf(fact_190_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_191_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_192_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_193_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_194_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_195_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_196_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_197_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z4: A] : Y3 = Z4 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( ord_less_eq @ A @ A4 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_198_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_199_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_200_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_201_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_202_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_203_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_204_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_205_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z4: A] : Y3 = Z4 )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( ord_less_eq @ A @ B3 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_206_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_207_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_208_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_209_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_210_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_211_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_212_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_213_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z4: A] : Y3 = Z4 )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_214_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_215_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_216_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_217_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_218_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_219_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G2 @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_220_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funE
thf(fact_221_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% le_funD
thf(fact_222_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B4: set @ B,C3: B > A,D3: B > A,Sup: ( set @ A ) > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C3 @ A3 ) )
= ( Sup @ ( image @ B @ A @ D3 @ B4 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_223_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A3: set @ B,B4: set @ B,C3: B > A,D3: B > A,Inf: ( set @ A ) > A] :
( ( A3 = B4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B4 )
=> ( ( C3 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C3 @ A3 ) )
= ( Inf @ ( image @ B @ A @ D3 @ B4 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_224_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G2: B > A,F: C > B,A3: set @ C] :
( ( Inf @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F @ A3 ) ) )
= ( Inf @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F ) @ A3 ) ) ) ).
% Inf.INF_image
thf(fact_225_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G2: B > A,F: C > B,A3: set @ C] :
( ( Sup @ ( image @ B @ A @ G2 @ ( image @ C @ B @ F @ A3 ) ) )
= ( Sup @ ( image @ C @ A @ ( comp @ B @ A @ C @ G2 @ F ) @ A3 ) ) ) ).
% Sup.SUP_image
thf(fact_226_Inf_OINF__id__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf @ ( image @ A @ A @ ( id @ A ) @ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_id_eq
thf(fact_227_topological__space__class_OopenI,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ! [S: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ? [T4: set @ A] :
( ( topolo1751647064n_open @ A @ T4 )
& ( member @ A @ X3 @ T4 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ S ) ) )
=> ( topolo1751647064n_open @ A @ S ) ) ) ).
% topological_space_class.openI
thf(fact_228_open__subopen,axiom,
! [A: $tType] :
( ( topolo503727757_space @ A )
=> ( ( topolo1751647064n_open @ A )
= ( ^ [S3: set @ A] :
! [X: A] :
( ( member @ A @ X @ S3 )
=> ? [T5: set @ A] :
( ( topolo1751647064n_open @ A @ T5 )
& ( member @ A @ X @ T5 )
& ( ord_less_eq @ ( set @ A ) @ T5 @ S3 ) ) ) ) ) ) ).
% open_subopen
thf(fact_229_t0__space,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( ( member @ A @ X2 @ U )
!= ( member @ A @ Y @ U ) ) ) ) ) ).
% t0_space
thf(fact_230_t1__space,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ? [U: set @ A] :
( ( topolo1751647064n_open @ A @ U )
& ( member @ A @ X2 @ U )
& ~ ( member @ A @ Y @ U ) ) ) ) ).
% t1_space
thf(fact_231_separation__t0,axiom,
! [A: $tType] :
( ( topological_t0_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( ( member @ A @ X2 @ U2 )
!= ( member @ A @ Y @ U2 ) ) ) ) ) ) ).
% separation_t0
thf(fact_232_separation__t1,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ? [U2: set @ A] :
( ( topolo1751647064n_open @ A @ U2 )
& ( member @ A @ X2 @ U2 )
& ~ ( member @ A @ Y @ U2 ) ) ) ) ) ).
% separation_t1
thf(fact_233_open__discrete,axiom,
! [A: $tType] :
( ( topolo2133971006pology @ A )
=> ! [A3: set @ A] : ( topolo1751647064n_open @ A @ A3 ) ) ).
% open_discrete
thf(fact_234_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo2135403230pology @ A )
=> ! [X2: A] :
? [A7: nat > ( set @ A )] :
( ! [I3: nat] :
( ( member @ A @ X2 @ ( A7 @ I3 ) )
& ( topolo1751647064n_open @ A @ ( A7 @ I3 ) ) )
& ! [S4: set @ A] :
( ( ( topolo1751647064n_open @ A @ S4 )
& ( member @ A @ X2 @ S4 ) )
=> ? [I2: nat] : ( ord_less_eq @ ( set @ A ) @ ( A7 @ I2 ) @ S4 ) ) ) ) ).
% first_countable_basis
thf(fact_235_landau__omega_OR__mult__right__mono,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ real @ ( times_times @ real @ B2 @ C2 ) @ ( times_times @ real @ A2 @ C2 ) ) ) ) ).
% landau_omega.R_mult_right_mono
thf(fact_236_landau__omega_OR__mult__left__mono,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ real @ ( times_times @ real @ C2 @ B2 ) @ ( times_times @ real @ C2 @ A2 ) ) ) ) ).
% landau_omega.R_mult_left_mono
thf(fact_237_landau__omega_OR__linear,axiom,
! [Y: real,X2: real] :
( ~ ( ord_less_eq @ real @ Y @ X2 )
=> ( ord_less_eq @ real @ X2 @ Y ) ) ).
% landau_omega.R_linear
thf(fact_238_landau__omega_OR__trans,axiom,
! [B2: real,A2: real,C2: real] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ord_less_eq @ real @ C2 @ A2 ) ) ) ).
% landau_omega.R_trans
thf(fact_239_landau__omega_OR__refl,axiom,
! [X2: real] : ( ord_less_eq @ real @ X2 @ X2 ) ).
% landau_omega.R_refl
thf(fact_240_landau__o_OR__linear,axiom,
! [X2: real,Y: real] :
( ~ ( ord_less_eq @ real @ X2 @ Y )
=> ( ord_less_eq @ real @ Y @ X2 ) ) ).
% landau_o.R_linear
thf(fact_241_landau__o_OR__trans,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ B2 @ C2 )
=> ( ord_less_eq @ real @ A2 @ C2 ) ) ) ).
% landau_o.R_trans
thf(fact_242_landau__o_OR__mult__left__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ real @ ( times_times @ real @ C2 @ A2 ) @ ( times_times @ real @ C2 @ B2 ) ) ) ) ).
% landau_o.R_mult_left_mono
thf(fact_243_landau__o_OR__mult__right__mono,axiom,
! [A2: real,B2: real,C2: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ real @ ( times_times @ real @ A2 @ C2 ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).
% landau_o.R_mult_right_mono
thf(fact_244_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A3: set @ B,P: ( set @ A ) > $o] :
( ( ! [B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image @ B @ A @ F @ A3 ) )
=> ( P @ B6 ) ) )
= ( ! [B6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A3 )
=> ( P @ ( image @ B @ A @ F @ B6 ) ) ) ) ) ).
% all_subset_image
thf(fact_245_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_246_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_247_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_248_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_249_ln__ge__zero,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X2 ) ) ) ).
% ln_ge_zero
thf(fact_250_Ln__1,axiom,
( ( ln_ln @ complex @ ( one_one @ complex ) )
= ( zero_zero @ complex ) ) ).
% Ln_1
thf(fact_251_fps__tan__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( formal_Power_fps_tan @ A @ ( zero_zero @ A ) )
= ( zero_zero @ ( formal_Power_fps @ A ) ) ) ) ).
% fps_tan_0
thf(fact_252_holomorphic__fun__eq__0__on__connected,axiom,
! [F: complex > complex,S: set @ complex,Z: complex,W2: complex] :
( ( comple372758642hic_on @ F @ S )
=> ( ( topolo1751647064n_open @ complex @ S )
=> ( ( topolo1099314120nected @ complex @ S )
=> ( ! [N3: nat] :
( ( compow @ ( ( complex > complex ) > complex > complex ) @ N3 @ ( deriv @ complex ) @ F @ Z )
= ( zero_zero @ complex ) )
=> ( ( member @ complex @ Z @ S )
=> ( ( member @ complex @ W2 @ S )
=> ( ( F @ W2 )
= ( zero_zero @ complex ) ) ) ) ) ) ) ) ).
% holomorphic_fun_eq_0_on_connected
% Type constructors (163)
thf(tcon_Formal__Power__Series_Ofps___Real__Vector__Spaces_Ometric__space,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( real_V2090557954_space @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Ogroup__add,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( group_add @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Ocancel__semigroup__add,axiom,
! [A8: $tType] :
( ( cancel_semigroup_add @ A8 )
=> ( cancel_semigroup_add @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Ocomm__semiring__0,axiom,
! [A8: $tType] :
( ( comm_semiring_0 @ A8 )
=> ( comm_semiring_0 @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Osemiring__0,axiom,
! [A8: $tType] :
( ( semiring_0 @ A8 )
=> ( semiring_0 @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Osemiring__1,axiom,
! [A8: $tType] :
( ( semiring_1 @ A8 )
=> ( semiring_1 @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Ocomm__monoid__add,axiom,
! [A8: $tType] :
( ( comm_monoid_add @ A8 )
=> ( comm_monoid_add @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Oring__1,axiom,
! [A8: $tType] :
( ( ring_1 @ A8 )
=> ( ring_1 @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_1,axiom,
real_V2090557954_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_2,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_3,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_4,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_5,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_6,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_7,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_8,axiom,
ring_1 @ complex ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space_9,axiom,
real_V2090557954_space @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_10,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_11,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_12,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_13,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_14,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_15,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_16,axiom,
ring_1 @ real ).
thf(tcon_HOL_Obool___Countable_Ocountable,axiom,
countable @ $o ).
thf(tcon_Set_Oset___Countable_Ocountable_17,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( countable @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__add_18,axiom,
! [A8: $tType] :
( ( comm_monoid_add @ A8 )
=> ( comm_monoid_add @ ( set @ A8 ) ) ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_19,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_20,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_21,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_22,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_23,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_24,axiom,
comm_monoid_add @ nat ).
thf(tcon_fun___Countable_Ocountable_25,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( countable @ A9 ) )
=> ( countable @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Real__Vector__Spaces_Ometric__space_26,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( real_V2090557954_space @ A9 ) )
=> ( real_V2090557954_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 )
& ( finite_finite @ A9 ) )
=> ( finite_finite @ ( A8 > A9 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_27,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 )
=> ( finite_finite @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_28,axiom,
finite_finite @ $o ).
thf(tcon_fun___Topological__Spaces_Ofirst__countable__topology,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( topolo2135403230pology @ A9 ) )
=> ( topolo2135403230pology @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Otopological__space,axiom,
! [A8: $tType,A9: $tType] :
( ( topolo503727757_space @ A9 )
=> ( topolo503727757_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Ot1__space,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( real_V2090557954_space @ A9 ) )
=> ( topological_t1_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Topological__Spaces_Ot0__space,axiom,
! [A8: $tType,A9: $tType] :
( ( ( countable @ A8 )
& ( real_V2090557954_space @ A9 ) )
=> ( topological_t0_space @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Topological__Spaces_Ofirst__countable__topology_29,axiom,
topolo2135403230pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_30,axiom,
topolo503727757_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology,axiom,
topolo2133971006pology @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_31,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot0__space_32,axiom,
topological_t0_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_33,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_34,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_35,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Set_Oset___Groups_Oab__semigroup__mult_36,axiom,
! [A8: $tType] :
( ( ab_semigroup_mult @ A8 )
=> ( ab_semigroup_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ocomm__monoid__mult_37,axiom,
! [A8: $tType] :
( ( comm_monoid_mult @ A8 )
=> ( comm_monoid_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Osemigroup__mult_38,axiom,
! [A8: $tType] :
( ( semigroup_mult @ A8 )
=> ( semigroup_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_39,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Omonoid__mult_40,axiom,
! [A8: $tType] :
( ( monoid_mult @ A8 )
=> ( monoid_mult @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_41,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_42,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Otimes_43,axiom,
! [A8: $tType] :
( ( times @ A8 )
=> ( times @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ozero_44,axiom,
! [A8: $tType] :
( ( zero @ A8 )
=> ( zero @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Groups_Oone_45,axiom,
! [A8: $tType] :
( ( one @ A8 )
=> ( one @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_46,axiom,
topolo503727757_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_47,axiom,
topolo2133971006pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_48,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot0__space_49,axiom,
topological_t0_space @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_50,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_51,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_52,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_53,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology_54,axiom,
topolo2135403230pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_55,axiom,
semiri1923998003cancel @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_56,axiom,
topolo503727757_space @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V758409581_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_57,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_58,axiom,
semiri1193490041visors @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_59,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot0__space_60,axiom,
topological_t0_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_61,axiom,
ordere1490568538miring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_62,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_63,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_64,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_65,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_66,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_67,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_68,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_69,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_70,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_71,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_72,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_73,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_74,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Otimes_75,axiom,
times @ real ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_76,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oone_77,axiom,
one @ real ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_78,axiom,
topolo2135403230pology @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_79,axiom,
semiri1923998003cancel @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_80,axiom,
topolo503727757_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_81,axiom,
real_V758409581_field @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_82,axiom,
semiri1193490041visors @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_83,axiom,
ring_11004092258visors @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_84,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot0__space_85,axiom,
topological_t0_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_86,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_87,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_88,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_89,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_90,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_91,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_92,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Transcendental_Oln_93,axiom,
ln @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_94,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_95,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Otimes_96,axiom,
times @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_97,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_98,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_99,axiom,
one @ complex ).
thf(tcon_Formal__Power__Series_Ofps___Topological__Spaces_Ofirst__countable__topology_100,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( topolo2135403230pology @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Osemiring__no__zero__divisors__cancel_101,axiom,
! [A8: $tType] :
( ( ( cancel_semigroup_add @ A8 )
& ( semiri1923998003cancel @ A8 ) )
=> ( semiri1923998003cancel @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Topological__Spaces_Otopological__space_102,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( topolo503727757_space @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Osemiring__no__zero__divisors_103,axiom,
! [A8: $tType] :
( ( semiri1193490041visors @ A8 )
=> ( semiri1193490041visors @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Oring__1__no__zero__divisors_104,axiom,
! [A8: $tType] :
( ( ring_11004092258visors @ A8 )
=> ( ring_11004092258visors @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Topological__Spaces_Ot1__space_105,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( topological_t1_space @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Topological__Spaces_Ot0__space_106,axiom,
! [A8: $tType] :
( ( group_add @ A8 )
=> ( topological_t0_space @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Oab__semigroup__mult_107,axiom,
! [A8: $tType] :
( ( comm_semiring_0 @ A8 )
=> ( ab_semigroup_mult @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Ocomm__monoid__mult_108,axiom,
! [A8: $tType] :
( ( comm_semiring_1 @ A8 )
=> ( comm_monoid_mult @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Ocomm__semiring__1_109,axiom,
! [A8: $tType] :
( ( comm_semiring_1 @ A8 )
=> ( comm_semiring_1 @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Osemigroup__mult_110,axiom,
! [A8: $tType] :
( ( semiring_0 @ A8 )
=> ( semigroup_mult @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Ozero__neq__one_111,axiom,
! [A8: $tType] :
( ( zero_neq_one @ A8 )
=> ( zero_neq_one @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Omonoid__mult_112,axiom,
! [A8: $tType] :
( ( semiring_1 @ A8 )
=> ( monoid_mult @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Rings_Omult__zero_113,axiom,
! [A8: $tType] :
( ( ( comm_monoid_add @ A8 )
& ( mult_zero @ A8 ) )
=> ( mult_zero @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Num_Oneg__numeral_114,axiom,
! [A8: $tType] :
( ( ring_1 @ A8 )
=> ( neg_numeral @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Otimes_115,axiom,
! [A8: $tType] :
( ( ( comm_monoid_add @ A8 )
& ( times @ A8 ) )
=> ( times @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Ozero_116,axiom,
! [A8: $tType] :
( ( zero @ A8 )
=> ( zero @ ( formal_Power_fps @ A8 ) ) ) ).
thf(tcon_Formal__Power__Series_Ofps___Groups_Oone_117,axiom,
! [A8: $tType] :
( ( ( one @ A8 )
& ( zero @ A8 ) )
=> ( one @ ( formal_Power_fps @ A8 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( deriv @ complex @ ( id @ complex ) @ w )
= ( one_one @ complex ) ) ).
%------------------------------------------------------------------------------