Article TIETZE, MML version 4.99.1005
:: TIETZE:th 1
theorem
for b1, b2, b3 being real set
st |.b1 - b2.| <= b3
holds b2 - b3 <= b1 & b1 <= b2 + b3;
:: TIETZE:th 2
theorem
for b1, b2 being real set
st b1 < b2
holds left_closed_halfline b1 misses right_closed_halfline b2;
:: TIETZE:th 3
theorem
for b1, b2 being real set
st b1 <= b2
holds halfline b1 misses right_open_halfline b2;
:: TIETZE:th 4
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set
st b1 c= b2
holds b3 - b1 c= b3 - b2;
:: TIETZE:th 5
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set
st b1 c= b2
holds b1 - b3 c= b2 - b3;
:: TIETZE:prednot 1 => TIETZE:pred 1
definition
let a1 be Relation-like Function-like real-valued set;
let a2 be real set;
let a3 be set;
pred A1,A3 is_absolutely_bounded_by A2 means
for b1 being set
st b1 in a3 /\ proj1 a1
holds abs (a1 . b1) <= a2;
end;
:: TIETZE:dfs 1
definiens
let a1 be Relation-like Function-like real-valued set;
let a2 be real set;
let a3 be set;
To prove
a1,a3 is_absolutely_bounded_by a2
it is sufficient to prove
thus for b1 being set
st b1 in a3 /\ proj1 a1
holds abs (a1 . b1) <= a2;
:: TIETZE:def 1
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being real set
for b3 being set holds
b1,b3 is_absolutely_bounded_by b2
iff
for b4 being set
st b4 in b3 /\ proj1 b1
holds abs (b1 . b4) <= b2;
:: TIETZE:exreg 1
registration
cluster Relation-like Function-like constant non empty quasi_total complex-valued ext-real-valued real-valued summable total convergent Relation of NAT,REAL;
end;
:: TIETZE:th 6
theorem
for b1 being empty TopSpace-like TopStruct
for b2 being TopSpace-like TopStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
b3 is continuous(b1, b2);
:: TIETZE:th 7
theorem
for b1, b2 being Function-like quasi_total summable Relation of NAT,REAL
st for b3 being Element of NAT holds
b1 . b3 <= b2 . b3
holds Sum b1 <= Sum b2;
:: TIETZE:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is absolutely_summable
holds abs Sum b1 <= Sum abs b1;
:: TIETZE:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being real positive set
st b3 < 1 &
(for b4 being natural set holds
|.(b1 . b4) - (b1 . (b4 + 1)).| <= b2 * (b3 to_power b4))
holds b1 is convergent &
(for b4 being natural set holds
|.(lim b1) - (b1 . b4).| <= (b2 * (b3 to_power b4)) / (1 - b3));
:: TIETZE:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being real positive set
st b3 < 1 &
(for b4 being natural set holds
|.(b1 . b4) - (b1 . (b4 + 1)).| <= b2 * (b3 to_power b4))
holds (b1 . 0) - (b2 / (1 - b3)) <= lim b1 &
lim b1 <= (b1 . 0) + (b2 / (1 - b3));
:: TIETZE:th 11
theorem
for b1, b2 being non empty set
for b3 being Functional_Sequence of b1,REAL
st b2 common_on_dom b3
for b4, b5 being real positive set
st b5 < 1 &
(for b6 being natural set holds
(b3 . b6) - (b3 . (b6 + 1)),b2 is_absolutely_bounded_by b4 * (b5 to_power b6))
holds b3 is_unif_conv_on b2 &
(for b6 being natural set holds
(lim(b3,b2)) - (b3 . b6),b2 is_absolutely_bounded_by (b4 * (b5 to_power b6)) / (1 - b5));
:: TIETZE:th 12
theorem
for b1, b2 being non empty set
for b3 being Functional_Sequence of b1,REAL
st b2 common_on_dom b3
for b4, b5 being real positive set
st b5 < 1 &
(for b6 being natural set holds
(b3 . b6) - (b3 . (b6 + 1)),b2 is_absolutely_bounded_by b4 * (b5 to_power b6))
for b6 being Element of b2 holds
((b3 . 0) . b6) - (b4 / (1 - b5)) <= (lim(b3,b2)) . b6 &
(lim(b3,b2)) . b6 <= ((b3 . 0) . b6) + (b4 / (1 - b5));
:: TIETZE:th 13
theorem
for b1, b2 being non empty set
for b3 being Functional_Sequence of b1,REAL
st b2 common_on_dom b3
for b4, b5 being real positive set
for b6 being Function-like quasi_total Relation of b2,REAL
st b5 < 1 &
(for b7 being natural set holds
(b3 . b7) - b6,b2 is_absolutely_bounded_by b4 * (b5 to_power b7))
holds b3 is_point_conv_on b2 & lim(b3,b2) = b6;
:: TIETZE:funcreg 1
registration
let a1, a2 be TopStruct;
let a3 be empty Element of bool the carrier of a1;
let a4 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
cluster a4 | a3 -> Relation-like empty;
end;
:: TIETZE:funcreg 2
registration
let a1 be TopSpace-like TopStruct;
let a2 be closed Element of bool the carrier of a1;
cluster a1 | a2 -> strict closed;
end;
:: TIETZE:th 14
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st (b3 misses b4 or b5 | (b3 meet b4) = b6 | (b3 meet b4))
for b7 being Element of the carrier of b1 holds
(b7 in the carrier of b3 implies (b5 union b6) . b7 = b5 . b7) &
(b7 in the carrier of b4 implies (b5 union b6) . b7 = b6 . b7);
:: TIETZE:th 15
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st (b3 misses b4 or b5 | (b3 meet b4) = b6 | (b3 meet b4))
holds rng (b5 union b6) c= (rng b5) \/ rng b6;
:: TIETZE:th 16
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being non empty SubSpace of b1
for b5 being Function-like quasi_total Relation of the carrier of b3,the carrier of b2
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
st (b3 misses b4 or b5 | (b3 meet b4) = b6 | (b3 meet b4))
holds (for b7 being Element of bool the carrier of b3 holds
(b5 union b6) .: b7 = b5 .: b7) &
(for b7 being Element of bool the carrier of b4 holds
(b5 union b6) .: b7 = b6 .: b7);
:: TIETZE:th 17
theorem
for b1 being real set
for b2 being set
for b3, b4 being Relation-like Function-like real-valued set
st b3 c= b4 & b4,b2 is_absolutely_bounded_by b1
holds b3,b2 is_absolutely_bounded_by b1;
:: TIETZE:th 18
theorem
for b1 being real set
for b2 being set
for b3, b4 being Relation-like Function-like real-valued set
st (b2 c= proj1 b3 or proj1 b4 c= proj1 b3) & b3 | b2 = b4 | b2 & b3,b2 is_absolutely_bounded_by b1
holds b4,b2 is_absolutely_bounded_by b1;
:: TIETZE:th 19
theorem
for b1 being real set
for b2 being non empty TopSpace-like TopStruct
for b3 being closed Element of bool the carrier of b2
st 0 < b1 & b2 is being_T4
for b4 being Function-like quasi_total continuous Relation of the carrier of b2 | b3,the carrier of R^1
st b4,b3 is_absolutely_bounded_by b1
holds ex b5 being Function-like quasi_total continuous Relation of the carrier of b2,the carrier of R^1 st
b5,dom b5 is_absolutely_bounded_by b1 / 3 &
b4 - b5,b3 is_absolutely_bounded_by (2 * b1) / 3;
:: TIETZE:th 20
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2, b3 being non empty closed Element of bool the carrier of b1
st b2 misses b3
holds ex b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of R^1 st
b4 .: b2 = {0} & b4 .: b3 = {1}
holds b1 is being_T4;
:: TIETZE:th 21
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being Element of the carrier of b1 holds
b2 is_continuous_at b3
iff
for b4 being real set
st 0 < b4
holds ex b5 being Element of bool the carrier of b1 st
b5 is open(b1) &
b3 in b5 &
(for b6 being Element of the carrier of b1
st b6 in b5
holds abs ((b2 . b6) - (b2 . b3)) < b4);
:: TIETZE:th 22
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Functional_Sequence of the carrier of b1,REAL
st b2 is_unif_conv_on the carrier of b1 &
(for b3 being Element of NAT holds
b2 . b3 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of R^1)
holds lim(b2,the carrier of b1) is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of R^1;
:: TIETZE:th 23
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of R^1
for b3 being real positive set holds
b2,the carrier of b1 is_absolutely_bounded_by b3
iff
b2 is Function-like quasi_total Relation of the carrier of b1,the carrier of Closed-Interval-TSpace(- b3,b3);
:: TIETZE:th 24
theorem
for b1 being real set
for b2 being set
for b3, b4 being Relation-like Function-like real-valued set
st b3 - b4,b2 is_absolutely_bounded_by b1
holds b4 - b3,b2 is_absolutely_bounded_by b1;
:: TIETZE:th 25
theorem
for b1 being non empty TopSpace-like TopStruct
st b1 is being_T4
for b2 being closed Element of bool the carrier of b1
for b3 being Function-like quasi_total Relation of the carrier of b1 | b2,the carrier of Closed-Interval-TSpace(- 1,1)
st b3 is continuous(b1 | b2, Closed-Interval-TSpace(- 1,1))
holds ex b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of Closed-Interval-TSpace(- 1,1) st
b4 | b2 = b3;
:: TIETZE:th 26
theorem
for b1 being non empty TopSpace-like TopStruct
st for b2 being non empty closed Element of bool the carrier of b1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1 | b2,the carrier of Closed-Interval-TSpace(- 1,1) holds
ex b4 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of Closed-Interval-TSpace(- 1,1) st
b4 | b2 = b3
holds b1 is being_T4;