Article UNIFORM1, MML version 4.99.1005

:: UNIFORM1:th 2
theorem
for b1 being Element of REAL
      st 0 < b1
   holds ex b2 being Element of NAT st
      0 < b2 & 1 / b2 < b1;

:: UNIFORM1:attrnot 1 => UNIFORM1:attr 1
definition
  let a1, a2 be non empty MetrStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
  attr a3 is uniformly_continuous means
    for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of REAL st
          0 < b2 &
           (for b3, b4 being Element of the carrier of a1
                 st dist(b3,b4) < b2
              holds dist(a3 /. b3,a3 /. b4) < b1);
end;

:: UNIFORM1:dfs 1
definiens
  let a1, a2 be non empty MetrStruct;
  let a3 be Function-like quasi_total Relation of the carrier of a1,the carrier of a2;
To prove
     a3 is uniformly_continuous
it is sufficient to prove
  thus for b1 being Element of REAL
          st 0 < b1
       holds ex b2 being Element of REAL st
          0 < b2 &
           (for b3, b4 being Element of the carrier of a1
                 st dist(b3,b4) < b2
              holds dist(a3 /. b3,a3 /. b4) < b1);

:: UNIFORM1:def 1
theorem
for b1, b2 being non empty MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2 holds
      b3 is uniformly_continuous(b1, b2)
   iff
      for b4 being Element of REAL
            st 0 < b4
         holds ex b5 being Element of REAL st
            0 < b5 &
             (for b6, b7 being Element of the carrier of b1
                   st dist(b6,b7) < b5
                holds dist(b3 /. b6,b3 /. b7) < b4);

:: UNIFORM1:th 3
theorem
for b1 being non empty TopSpace-like TopStruct
for b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of TopSpaceMetr b2
   st b3 is continuous(b1, TopSpaceMetr b2)
for b4 being Element of REAL
for b5 being Element of the carrier of b2
for b6 being Element of bool the carrier of TopSpaceMetr b2
      st b6 = Ball(b5,b4)
   holds b3 " b6 is open(b1);

:: UNIFORM1:th 4
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
      st for b4 being real set
        for b5 being Element of the carrier of b1
        for b6 being Element of the carrier of b2
              st 0 < b4 & b6 = b3 . b5
           holds ex b7 being real set st
              0 < b7 &
               (for b8 being Element of the carrier of b1
               for b9 being Element of the carrier of b2
                     st b9 = b3 . b8 & dist(b5,b8) < b7
                  holds dist(b6,b9) < b4)
   holds b3 is continuous(TopSpaceMetr b1, TopSpaceMetr b2);

:: UNIFORM1:th 5
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
   st b3 is continuous(TopSpaceMetr b1, TopSpaceMetr b2)
for b4 being Element of REAL
for b5 being Element of the carrier of b1
for b6 being Element of the carrier of b2
      st 0 < b4 & b6 = b3 . b5
   holds ex b7 being Element of REAL st
      0 < b7 &
       (for b8 being Element of the carrier of b1
       for b9 being Element of the carrier of b2
             st b9 = b3 . b8 & dist(b5,b8) < b7
          holds dist(b6,b9) < b4);

:: UNIFORM1:th 6
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
      st b3 = b4 & b3 is uniformly_continuous(b1, b2)
   holds b4 is continuous(TopSpaceMetr b1, TopSpaceMetr b2);

:: UNIFORM1:th 7
theorem
for b1 being non empty Reflexive discerning symmetric triangle MetrStruct
for b2 being Element of bool bool the carrier of TopSpaceMetr b1
      st b2 is_a_cover_of TopSpaceMetr b1 & b2 is open(TopSpaceMetr b1) & TopSpaceMetr b1 is compact
   holds ex b3 being Element of REAL st
      0 < b3 &
       (for b4, b5 being Element of the carrier of b1
             st dist(b4,b5) < b3
          holds ex b6 being Element of bool the carrier of TopSpaceMetr b1 st
             b4 in b6 & b5 in b6 & b6 in b2);

:: UNIFORM1:th 8
theorem
for b1, b2 being non empty Reflexive discerning symmetric triangle MetrStruct
for b3 being Function-like quasi_total Relation of the carrier of b1,the carrier of b2
for b4 being Function-like quasi_total Relation of the carrier of TopSpaceMetr b1,the carrier of TopSpaceMetr b2
      st b4 = b3 & TopSpaceMetr b1 is compact & b4 is continuous(TopSpaceMetr b1, TopSpaceMetr b2)
   holds b3 is uniformly_continuous(b1, b2);

:: UNIFORM1:th 9
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1
for b3 being Function-like quasi_total Relation of the carrier of Closed-Interval-MSpace(0,1),the carrier of Euclid b1
      st b2 is continuous(I[01], TOP-REAL b1) & b3 = b2
   holds b3 is uniformly_continuous(Closed-Interval-MSpace(0,1), Euclid b1);

:: UNIFORM1:th 10
theorem
for b1 being Element of NAT
for b2 being Element of bool the carrier of TOP-REAL b1
for b3 being non empty Element of bool the carrier of Euclid b1
for b4 being Function-like quasi_total Relation of the carrier of I[01],the carrier of (TOP-REAL b1) | b2
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-MSpace(0,1),the carrier of (Euclid b1) | b3
      st b2 = b3 & b4 is continuous(I[01], (TOP-REAL b1) | b2) & b5 = b4
   holds b5 is uniformly_continuous(Closed-Interval-MSpace(0,1), (Euclid b1) | b3);

:: UNIFORM1:th 11
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1 holds
   ex b3 being Function-like quasi_total Relation of the carrier of Closed-Interval-MSpace(0,1),the carrier of Euclid b1 st
      b3 = b2;

:: UNIFORM1:th 13
theorem
for b1, b2 being real set holds
abs (b1 - b2) = abs (b2 - b1);

:: UNIFORM1:th 14
theorem
for b1, b2, b3, b4 being Element of REAL
      st b1 in [.b3,b4.] & b2 in [.b3,b4.]
   holds abs (b1 - b2) <= b4 - b3;

:: UNIFORM1:attrnot 2 => UNIFORM1:attr 2
definition
  let a1 be FinSequence of REAL;
  attr a1 is decreasing means
    for b1, b2 being Element of NAT
          st b1 in dom a1 & b2 in dom a1 & b1 < b2
       holds a1 . b2 < a1 . b1;
end;

:: UNIFORM1:dfs 2
definiens
  let a1 be FinSequence of REAL;
To prove
     a1 is decreasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 in dom a1 & b2 in dom a1 & b1 < b2
       holds a1 . b2 < a1 . b1;

:: UNIFORM1:def 2
theorem
for b1 being FinSequence of REAL holds
      b1 is decreasing
   iff
      for b2, b3 being Element of NAT
            st b2 in dom b1 & b3 in dom b1 & b2 < b3
         holds b1 . b3 < b1 . b2;

:: UNIFORM1:th 15
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of TOP-REAL b1
      st 0 < b2 & b3 is continuous(I[01], TOP-REAL b1) & b3 is one-to-one & b3 . 0 = b4 & b3 . 1 = b5
   holds ex b6 being FinSequence of REAL st
      b6 . 1 = 0 &
       b6 . len b6 = 1 &
       5 <= len b6 &
       rng b6 c= the carrier of I[01] &
       b6 is increasing &
       (for b7 being Element of NAT
       for b8 being Element of bool the carrier of I[01]
       for b9 being Element of bool the carrier of Euclid b1
             st 1 <= b7 &
                b7 < len b6 &
                b8 = [.b6 /. b7,b6 /. (b7 + 1).] &
                b9 = b3 .: b8
          holds diameter b9 < b2);

:: UNIFORM1:th 16
theorem
for b1 being Element of NAT
for b2 being Element of REAL
for b3 being Function-like quasi_total Relation of the carrier of I[01],the carrier of TOP-REAL b1
for b4, b5 being Element of the carrier of TOP-REAL b1
      st 0 < b2 & b3 is continuous(I[01], TOP-REAL b1) & b3 is one-to-one & b3 . 0 = b4 & b3 . 1 = b5
   holds ex b6 being FinSequence of REAL st
      b6 . 1 = 1 &
       b6 . len b6 = 0 &
       5 <= len b6 &
       rng b6 c= the carrier of I[01] &
       b6 is decreasing &
       (for b7 being Element of NAT
       for b8 being Element of bool the carrier of I[01]
       for b9 being Element of bool the carrier of Euclid b1
             st 1 <= b7 &
                b7 < len b6 &
                b8 = [.b6 /. (b7 + 1),b6 /. b7.] &
                b9 = b3 .: b8
          holds diameter b9 < b2);