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;