Article NCFCONT2, MML version 4.99.1005

:: NCFCONT2:prednot 1 => NCFCONT2:pred 1
definition
  let a1 be set;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
  pred A4 is_uniformly_continuous_on A1 means
    a1 c= dom a4 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));
end;

:: NCFCONT2:dfs 1
definiens
  let a1 be set;
  let a2, a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
To prove
     a4 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a4 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));

:: NCFCONT2:def 1
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
      b4 is_uniformly_continuous_on b1
   iff
      b1 c= dom b4 &
       (for b5 being Element of REAL
             st 0 < b5
          holds ex b6 being Element of REAL st
             0 < b6 &
              (for b7, b8 being Element of the carrier of b2
                    st b7 in b1 & b8 in b1 & ||.b7 - b8.|| < b6
                 holds ||.(b4 /. b7) - (b4 /. b8).|| < b5));

:: NCFCONT2:prednot 2 => NCFCONT2:pred 2
definition
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a3,the carrier of a2;
  pred A4 is_uniformly_continuous_on A1 means
    a1 c= dom a4 &
     (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 a3
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));
end;

:: NCFCONT2:dfs 2
definiens
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a3,the carrier of a2;
To prove
     a4 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a4 &
     (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 a3
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));

:: NCFCONT2:def 2
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2 holds
      b4 is_uniformly_continuous_on b1
   iff
      b1 c= dom b4 &
       (for b5 being Element of REAL
             st 0 < b5
          holds ex b6 being Element of REAL st
             0 < b6 &
              (for b7, b8 being Element of the carrier of b3
                    st b7 in b1 & b8 in b1 & ||.b7 - b8.|| < b6
                 holds ||.(b4 /. b7) - (b4 /. b8).|| < b5));

:: NCFCONT2:prednot 3 => NCFCONT2:pred 3
definition
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
  pred A4 is_uniformly_continuous_on A1 means
    a1 c= dom a4 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));
end;

:: NCFCONT2:dfs 3
definiens
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a4 be Function-like Relation of the carrier of a2,the carrier of a3;
To prove
     a4 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a4 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds ||.(a4 /. b3) - (a4 /. b4).|| < b1));

:: NCFCONT2:def 3
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
      b4 is_uniformly_continuous_on b1
   iff
      b1 c= dom b4 &
       (for b5 being Element of REAL
             st 0 < b5
          holds ex b6 being Element of REAL st
             0 < b6 &
              (for b7, b8 being Element of the carrier of b2
                    st b7 in b1 & b8 in b1 & ||.b7 - b8.|| < b6
                 holds ||.(b4 /. b7) - (b4 /. b8).|| < b5));

:: NCFCONT2:prednot 4 => NCFCONT2:pred 4
definition
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like Relation of the carrier of a2,COMPLEX;
  pred A3 is_uniformly_continuous_on A1 means
    a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds |.(a3 /. b3) - (a3 /. b4).| < b1));
end;

:: NCFCONT2:dfs 4
definiens
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like Relation of the carrier of a2,COMPLEX;
To prove
     a3 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds |.(a3 /. b3) - (a3 /. b4).| < b1));

:: NCFCONT2:def 4
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,COMPLEX holds
      b3 is_uniformly_continuous_on b1
   iff
      b1 c= dom b3 &
       (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 b2
                    st b6 in b1 & b7 in b1 & ||.b6 - b7.|| < b5
                 holds |.(b3 /. b6) - (b3 /. b7).| < b4));

:: NCFCONT2:prednot 5 => NCFCONT2:pred 5
definition
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like Relation of the carrier of a2,REAL;
  pred A3 is_uniformly_continuous_on A1 means
    a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds abs ((a3 /. b3) - (a3 /. b4)) < b1));
end;

:: NCFCONT2:dfs 5
definiens
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR;
  let a3 be Function-like Relation of the carrier of a2,REAL;
To prove
     a3 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds abs ((a3 /. b3) - (a3 /. b4)) < b1));

:: NCFCONT2:def 5
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,REAL holds
      b3 is_uniformly_continuous_on b1
   iff
      b1 c= dom b3 &
       (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 b2
                    st b6 in b1 & b7 in b1 & ||.b6 - b7.|| < b5
                 holds abs ((b3 /. b6) - (b3 /. b7)) < b4));

:: NCFCONT2:prednot 6 => NCFCONT2:pred 6
definition
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a2,COMPLEX;
  pred A3 is_uniformly_continuous_on A1 means
    a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds |.(a3 /. b3) - (a3 /. b4).| < b1));
end;

:: NCFCONT2:dfs 6
definiens
  let a1 be set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be Function-like Relation of the carrier of a2,COMPLEX;
To prove
     a3 is_uniformly_continuous_on a1
it is sufficient to prove
  thus a1 c= dom a3 &
     (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 a2
                  st b3 in a1 & b4 in a1 & ||.b3 - b4.|| < b2
               holds |.(a3 /. b3) - (a3 /. b4).| < b1));

:: NCFCONT2:def 6
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,COMPLEX holds
      b3 is_uniformly_continuous_on b1
   iff
      b1 c= dom b3 &
       (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 b2
                    st b6 in b1 & b7 in b1 & ||.b6 - b7.|| < b5
                 holds |.(b3 /. b6) - (b3 /. b7).| < b4));

:: NCFCONT2:th 1
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b2 c= b1
   holds b5 is_uniformly_continuous_on b2;

:: NCFCONT2:th 2
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b4,the carrier of b3
      st b5 is_uniformly_continuous_on b1 & b2 c= b1
   holds b5 is_uniformly_continuous_on b2;

:: NCFCONT2:th 3
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b2 c= b1
   holds b5 is_uniformly_continuous_on b2;

:: NCFCONT2:th 4
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 + b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 5
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b4,the carrier of b3
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 + b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 6
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 + b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 7
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 - b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 8
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b4,the carrier of b3
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 - b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 9
theorem
for b1, b2 being set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1 & b6 is_uniformly_continuous_on b2
   holds b5 - b6 is_uniformly_continuous_on b1 /\ b2;

:: NCFCONT2:th 10
theorem
for b1 being set
for b2 being Element of COMPLEX
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1
   holds b2 (#) b5 is_uniformly_continuous_on b1;

:: NCFCONT2:th 11
theorem
for b1 being set
for b2 being Element of REAL
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b4,the carrier of b3
      st b5 is_uniformly_continuous_on b1
   holds b2 (#) b5 is_uniformly_continuous_on b1;

:: NCFCONT2:th 12
theorem
for b1 being set
for b2 being Element of COMPLEX
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_uniformly_continuous_on b1
   holds b2 (#) b5 is_uniformly_continuous_on b1;

:: NCFCONT2:th 13
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds - b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 14
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
      st b4 is_uniformly_continuous_on b1
   holds - b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 15
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds - b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 16
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds ||.b4.|| is_uniformly_continuous_on b1;

:: NCFCONT2:th 17
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
      st b4 is_uniformly_continuous_on b1
   holds ||.b4.|| is_uniformly_continuous_on b1;

:: NCFCONT2:th 18
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds ||.b4.|| is_uniformly_continuous_on b1;

:: NCFCONT2:th 19
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds b4 is_continuous_on b1;

:: NCFCONT2:th 20
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
      st b4 is_uniformly_continuous_on b1
   holds b4 is_continuous_on b1;

:: NCFCONT2:th 21
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_uniformly_continuous_on b1
   holds b4 is_continuous_on b1;

:: NCFCONT2:th 22
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,COMPLEX
      st b3 is_uniformly_continuous_on b1
   holds b3 is_continuous_on b1;

:: NCFCONT2:th 23
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,REAL
      st b3 is_uniformly_continuous_on b1
   holds b3 is_continuous_on b1;

:: NCFCONT2:th 24
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Function-like Relation of the carrier of b2,COMPLEX
      st b3 is_uniformly_continuous_on b1
   holds b3 is_continuous_on b1;

:: NCFCONT2:th 25
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_Lipschitzian_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 26
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
      st b4 is_Lipschitzian_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 27
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_Lipschitzian_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 28
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 is compact(b1) & b3 is_continuous_on b4
   holds b3 is_uniformly_continuous_on b4;

:: NCFCONT2:th 29
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 is compact(b2) & b3 is_continuous_on b4
   holds b3 is_uniformly_continuous_on b4;

:: NCFCONT2:th 30
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 is compact(b1) & b3 is_continuous_on b4
   holds b3 is_uniformly_continuous_on b4;

:: NCFCONT2:th 31
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 c= dom b3 & b4 is compact(b1) & b3 is_uniformly_continuous_on b4
   holds b3 .: b4 is compact(b2);

:: NCFCONT2:th 32
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b2,the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 c= dom b3 & b4 is compact(b2) & b3 is_uniformly_continuous_on b4
   holds b3 .: b4 is compact(b1);

:: NCFCONT2:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b3 being Function-like Relation of the carrier of b1,the carrier of b2
for b4 being Element of bool the carrier of b1
      st b4 c= dom b3 & b4 is compact(b1) & b3 is_uniformly_continuous_on b4
   holds b3 .: b4 is compact(b2);

:: NCFCONT2:th 34
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of bool the carrier of b1
      st b3 <> {} & b3 c= dom b2 & b3 is compact(b1) & b2 is_uniformly_continuous_on b3
   holds ex b4, b5 being Element of the carrier of b1 st
      b4 in b3 & b5 in b3 & b2 /. b4 = upper_bound (b2 .: b3) & b2 /. b5 = lower_bound (b2 .: b3);

:: NCFCONT2:th 35
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b1 c= dom b4 & b4 is_constant_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 36
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
      st b1 c= dom b4 & b4 is_constant_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:th 37
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b1 c= dom b4 & b4 is_constant_on b1
   holds b4 is_uniformly_continuous_on b1;

:: NCFCONT2:modenot 1 => NCFCONT2:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
  mode contraction of A1 -> Function-like quasi_total Relation of the carrier of a1,the carrier of a1 means
    ex b1 being Element of REAL st
       0 < b1 &
        b1 < 1 &
        (for b2, b3 being Element of the carrier of a1 holds
        ||.(it . b2) - (it . b3).|| <= b1 * ||.b2 - b3.||);
end;

:: NCFCONT2:dfs 7
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR;
  let a2 be Function-like quasi_total Relation of the carrier of a1,the carrier of a1;
To prove
     a2 is contraction of a1
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        b1 < 1 &
        (for b2, b3 being Element of the carrier of a1 holds
        ||.(a2 . b2) - (a2 . b3).|| <= b1 * ||.b2 - b3.||);

:: NCFCONT2:def 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1 holds
      b2 is contraction of b1
   iff
      ex b3 being Element of REAL st
         0 < b3 &
          b3 < 1 &
          (for b4, b5 being Element of the carrier of b1 holds
          ||.(b2 . b4) - (b2 . b5).|| <= b3 * ||.b4 - b5.||);

:: NCFCONT2:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
   0 < ||.b2 - b3.||
iff
   b2 <> b3;

:: NCFCONT2:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like CNORMSTR
for b2, b3 being Element of the carrier of b1 holds
||.b2 - b3.|| = ||.b3 - b2.||;

:: NCFCONT2:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st b2 is contraction of b1
   holds ex b3 being Element of the carrier of b1 st
      b2 . b3 = b3 &
       (for b4 being Element of the carrier of b1
             st b2 . b4 = b4
          holds b3 = b4);

:: NCFCONT2:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed ComplexLinearSpace-like ComplexNormSpace-like complete CNORMSTR
for b2 being Function-like quasi_total Relation of the carrier of b1,the carrier of b1
      st ex b3 being Element of NAT st
           iter(b2,b3) is contraction of b1
   holds ex b3 being Element of the carrier of b1 st
      b2 . b3 = b3 &
       (for b4 being Element of the carrier of b1
             st b2 . b4 = b4
          holds b3 = b4);