Article NFCONT_1, MML version 4.99.1005

:: NFCONT_1:th 1
theorem
for b1 being non empty addLoopStr
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
b2 - b3 = b2 + - b3;

:: NFCONT_1:th 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1 holds
   - b2 = (- 1) * b2;

:: NFCONT_1:funcnot 1 => NFCONT_1:func 1
definition
  let a1, 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 a1,the carrier of a2;
  func ||.A3.|| -> Function-like Relation of the carrier of a1,REAL means
    dom it = dom a3 &
     (for b1 being Element of the carrier of a1
           st b1 in dom it
        holds it . b1 = ||.a3 /. b1.||);
end;

:: NFCONT_1:def 2
theorem
for b1, 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 b1,the carrier of b2
for b4 being Function-like Relation of the carrier of b1,REAL holds
      b4 = ||.b3.||
   iff
      dom b4 = dom b3 &
       (for b5 being Element of the carrier of b1
             st b5 in dom b4
          holds b4 . b5 = ||.b3 /. b5.||);

:: NFCONT_1:modenot 1 => NFCONT_1:mode 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of the carrier of a1;
  mode Neighbourhood of A2 -> Element of bool the carrier of a1 means
    ex b1 being Element of REAL st
       0 < b1 &
        {b2 where b2 is Element of the carrier of a1: ||.b2 - a2.|| < b1} c= it;
end;

:: NFCONT_1:dfs 2
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of the carrier of a1;
  let a3 be Element of bool the carrier of a1;
To prove
     a3 is Neighbourhood of a2
it is sufficient to prove
  thus ex b1 being Element of REAL st
       0 < b1 &
        {b2 where b2 is Element of the carrier of a1: ||.b2 - a2.|| < b1} c= a3;

:: NFCONT_1:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of bool the carrier of b1 holds
      b3 is Neighbourhood of b2
   iff
      ex b4 being Element of REAL st
         0 < b4 &
          {b5 where b5 is Element of the carrier of b1: ||.b5 - b2.|| < b4} c= b3;

:: NFCONT_1:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Element of REAL
      st 0 < b3
   holds {b4 where b4 is Element of the carrier of b1: ||.b4 - b2.|| < b3} is Neighbourhood of b2;

:: NFCONT_1:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Neighbourhood of b2 holds
   b2 in b3;

:: NFCONT_1:attrnot 1 => NFCONT_1:attr 1
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
  attr a2 is compact means
    for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st rng b1 c= a2
       holds ex b2 being Function-like quasi_total Relation of NAT,the carrier of a1 st
          b2 is subsequence of b1 & b2 is convergent(a1) & lim b2 in a2;
end;

:: NFCONT_1:dfs 3
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is compact
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st rng b1 c= a2
       holds ex b2 being Function-like quasi_total Relation of NAT,the carrier of a1 st
          b2 is subsequence of b1 & b2 is convergent(a1) & lim b2 in a2;

:: NFCONT_1:def 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is compact(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
            st rng b3 c= b2
         holds ex b4 being Function-like quasi_total Relation of NAT,the carrier of b1 st
            b4 is subsequence of b3 & b4 is convergent(b1) & lim b4 in b2;

:: NFCONT_1:attrnot 2 => NFCONT_1:attr 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
  attr a2 is closed means
    for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st rng b1 c= a2 & b1 is convergent(a1)
       holds lim b1 in a2;
end;

:: NFCONT_1:dfs 4
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is closed
it is sufficient to prove
  thus for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
          st rng b1 c= a2 & b1 is convergent(a1)
       holds lim b1 in a2;

:: NFCONT_1:def 5
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is closed(b1)
   iff
      for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
            st rng b3 c= b2 & b3 is convergent(b1)
         holds lim b3 in b2;

:: NFCONT_1:attrnot 3 => NFCONT_1:attr 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
  attr a2 is open means
    a2 ` is closed(a1);
end;

:: NFCONT_1:dfs 5
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Element of bool the carrier of a1;
To prove
     a2 is open
it is sufficient to prove
  thus a2 ` is closed(a1);

:: NFCONT_1:def 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
      b2 is open(b1)
   iff
      b2 ` is closed(b1);

:: NFCONT_1:funcnot 2 => NFCONT_1:func 2
definition
  let a1, 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 a1,the carrier of a2;
  let a4 be Function-like quasi_total Relation of NAT,the carrier of a1;
  assume rng a4 c= dom a3;
  func A3 * A4 -> Function-like quasi_total Relation of NAT,the carrier of a2 equals
    a4 * a3;
end;

:: NFCONT_1:def 7
theorem
for b1, 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 b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b4 c= dom b3
   holds b3 * b4 = b4 * b3;

:: NFCONT_1:funcnot 3 => NFCONT_1:func 3
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like Relation of the carrier of a1,REAL;
  let a3 be Function-like quasi_total Relation of NAT,the carrier of a1;
  assume rng a3 c= dom a2;
  func A2 * A3 -> Function-like quasi_total Relation of NAT,REAL equals
    a3 * a2;
end;

:: NFCONT_1:def 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b3 c= dom b2
   holds b2 * b3 = b3 * b2;

:: NFCONT_1:prednot 1 => NFCONT_1:pred 1
definition
  let a1, 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 a1,the carrier of a2;
  let a4 be Element of the carrier of a1;
  pred A3 is_continuous_in A4 means
    a4 in dom a3 &
     (for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
           st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
        holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));
end;

:: NFCONT_1:dfs 8
definiens
  let a1, 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 a1,the carrier of a2;
  let a4 be Element of the carrier of a1;
To prove
     a3 is_continuous_in a4
it is sufficient to prove
  thus a4 in dom a3 &
     (for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
           st rng b1 c= dom a3 & b1 is convergent(a1) & lim b1 = a4
        holds a3 * b1 is convergent(a2) & a3 /. a4 = lim (a3 * b1));

:: NFCONT_1:def 9
theorem
for b1, 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 b1,the carrier of b2
for b4 being Element of the carrier of b1 holds
      b3 is_continuous_in b4
   iff
      b4 in dom b3 &
       (for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
             st rng b5 c= dom b3 & b5 is convergent(b1) & lim b5 = b4
          holds b3 * b5 is convergent(b2) & b3 /. b4 = lim (b3 * b5));

:: NFCONT_1:prednot 2 => NFCONT_1:pred 2
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like Relation of the carrier of a1,REAL;
  let a3 be Element of the carrier of a1;
  pred A2 is_continuous_in A3 means
    a3 in dom a2 &
     (for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
           st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
        holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));
end;

:: NFCONT_1:dfs 9
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like Relation of the carrier of a1,REAL;
  let a3 be Element of the carrier of a1;
To prove
     a2 is_continuous_in a3
it is sufficient to prove
  thus a3 in dom a2 &
     (for b1 being Function-like quasi_total Relation of NAT,the carrier of a1
           st rng b1 c= dom a2 & b1 is convergent(a1) & lim b1 = a3
        holds a2 * b1 is convergent & a2 /. a3 = lim (a2 * b1));

:: NFCONT_1:def 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Element of the carrier of b1 holds
      b2 is_continuous_in b3
   iff
      b3 in dom b2 &
       (for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
             st rng b4 c= dom b2 & b4 is convergent(b1) & lim b4 = b3
          holds b2 * b4 is convergent & b2 /. b3 = lim (b2 * b4));

:: NFCONT_1:th 5
theorem
for b1 being Element of NAT
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
for b5 being Function-like Relation of the carrier of b2,the carrier of b3
      st rng b4 c= dom b5
   holds b4 . b1 in dom b5;

:: NFCONT_1:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like quasi_total Relation of NAT,the carrier of b1
for b3 being set holds
      b3 in rng b2
   iff
      ex b4 being Element of NAT st
         b3 = b2 . b4;

:: NFCONT_1:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2, b3 being Function-like quasi_total Relation of NAT,the carrier of b1
      st b3 is subsequence of b2
   holds rng b3 c= rng b2;

:: NFCONT_1:th 8
theorem
for b1, 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,the carrier of b1
for b4 being Function-like quasi_total Relation of NAT,the carrier of b2
   st rng b4 c= dom b3
for b5 being Element of NAT holds
   (b3 * b4) . b5 = b3 /. (b4 . b5);

:: NFCONT_1:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
   st rng b3 c= dom b2
for b4 being Element of NAT holds
   (b2 * b3) . b4 = b2 /. (b3 . b4);

:: NFCONT_1:th 10
theorem
for b1, 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 b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL
      st rng b4 c= dom b3
   holds (b3 * b4) * b5 = b3 * (b4 * b5);

:: NFCONT_1:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being Function-like quasi_total Relation of NAT,the carrier of b1
for b4 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL
      st rng b3 c= dom b2
   holds (b2 * b3) * b4 = b2 * (b3 * b4);

:: NFCONT_1:th 12
theorem
for b1, 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 b1,the carrier of b2
for b4, b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b4 c= dom b3 & b5 is subsequence of b4
   holds b3 * b5 is subsequence of b3 * b4;

:: NFCONT_1:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3, b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b3 c= dom b2 & b4 is subsequence of b3
   holds b2 * b4 is subsequence of b2 * b3;

:: NFCONT_1:th 14
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2 holds
      b3 is_continuous_in b4
   iff
      b4 in dom b3 &
       (for b5 being Element of REAL
             st 0 < b5
          holds ex b6 being Element of REAL st
             0 < b6 &
              (for b7 being Element of the carrier of b2
                    st b7 in dom b3 & ||.b7 - b4.|| < b6
                 holds ||.(b3 /. b7) - (b3 /. b4).|| < b5));

:: NFCONT_1:th 15
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of the carrier of b1
for b3 being Function-like Relation of the carrier of b1,REAL holds
      b3 is_continuous_in b2
   iff
      b2 in dom b3 &
       (for b4 being Element of REAL
             st 0 < b4
          holds ex b5 being Element of REAL st
             0 < b5 &
              (for b6 being Element of the carrier of b1
                    st b6 in dom b3 & ||.b6 - b2.|| < b5
                 holds abs ((b3 /. b6) - (b3 /. b2)) < b4));

:: NFCONT_1:th 16
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2 holds
      b3 is_continuous_in b4
   iff
      b4 in dom b3 &
       (for b5 being Neighbourhood of b3 /. b4 holds
          ex b6 being Neighbourhood of b4 st
             for b7 being Element of the carrier of b2
                   st b7 in dom b3 & b7 in b6
                holds b3 /. b7 in b5);

:: NFCONT_1:th 17
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2 holds
      b3 is_continuous_in b4
   iff
      b4 in dom b3 &
       (for b5 being Neighbourhood of b3 /. b4 holds
          ex b6 being Neighbourhood of b4 st
             b3 .: b6 c= b5);

:: NFCONT_1:th 18
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2
      st b4 in dom b3 &
         (ex b5 being Neighbourhood of b4 st
            (dom b3) /\ b5 = {b4})
   holds b3 is_continuous_in b4;

:: NFCONT_1:th 19
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b1,the carrier of b2
for b5 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b5 c= (dom b3) /\ dom b4
   holds (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) &
    (b3 - b4) * b5 = (b3 * b5) - (b4 * b5);

:: NFCONT_1:th 20
theorem
for b1, 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 b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
for b5 being Element of REAL
      st rng b4 c= dom b3
   holds (b5 (#) b3) * b4 = b5 * (b3 * b4);

:: NFCONT_1:th 21
theorem
for b1, 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 b1,the carrier of b2
for b4 being Function-like quasi_total Relation of NAT,the carrier of b1
      st rng b4 c= dom b3
   holds ||.b3 * b4.|| = ||.b3.|| * b4 &
    - (b3 * b4) = (- b3) * b4;

:: NFCONT_1:th 22
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of the carrier of b2,the carrier of b1
for b5 being Element of the carrier of b2
      st b3 is_continuous_in b5 & b4 is_continuous_in b5
   holds b3 + b4 is_continuous_in b5 & b3 - b4 is_continuous_in b5;

:: NFCONT_1:th 23
theorem
for b1 being Element of REAL
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2
for b5 being Element of the carrier of b3
      st b4 is_continuous_in b5
   holds b1 (#) b4 is_continuous_in b5;

:: NFCONT_1:th 24
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2
      st b3 is_continuous_in b4
   holds ||.b3.|| is_continuous_in b4 & - b3 is_continuous_in b4;

:: NFCONT_1:prednot 3 => NFCONT_1:pred 3
definition
  let a1, 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 a1,the carrier of a2;
  let a4 be set;
  pred A3 is_continuous_on A4 means
    a4 c= dom a3 &
     (for b1 being Element of the carrier of a1
           st b1 in a4
        holds a3 | a4 is_continuous_in b1);
end;

:: NFCONT_1:dfs 10
definiens
  let a1, 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 a1,the carrier of a2;
  let a4 be set;
To prove
     a3 is_continuous_on a4
it is sufficient to prove
  thus a4 c= dom a3 &
     (for b1 being Element of the carrier of a1
           st b1 in a4
        holds a3 | a4 is_continuous_in b1);

:: NFCONT_1:def 11
theorem
for b1, 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 b1,the carrier of b2
for b4 being set holds
      b3 is_continuous_on b4
   iff
      b4 c= dom b3 &
       (for b5 being Element of the carrier of b1
             st b5 in b4
          holds b3 | b4 is_continuous_in b5);

:: NFCONT_1:prednot 4 => NFCONT_1:pred 4
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like Relation of the carrier of a1,REAL;
  let a3 be set;
  pred A2 is_continuous_on A3 means
    a3 c= dom a2 &
     (for b1 being Element of the carrier of a1
           st b1 in a3
        holds a2 | a3 is_continuous_in b1);
end;

:: NFCONT_1:dfs 11
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be Function-like Relation of the carrier of a1,REAL;
  let a3 be set;
To prove
     a2 is_continuous_on a3
it is sufficient to prove
  thus a3 c= dom a2 &
     (for b1 being Element of the carrier of a1
           st b1 in a3
        holds a2 | a3 is_continuous_in b1);

:: NFCONT_1:def 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
for b3 being set holds
      b2 is_continuous_on b3
   iff
      b3 c= dom b2 &
       (for b4 being Element of the carrier of b1
             st b4 in b3
          holds b2 | b3 is_continuous_in b4);

:: NFCONT_1:th 25
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b2,the carrier of b1 holds
      b4 is_continuous_on b3
   iff
      b3 c= dom b4 &
       (for b5 being Function-like quasi_total Relation of NAT,the carrier of b2
             st rng b5 c= b3 & b5 is convergent(b2) & lim b5 in b3
          holds b4 * b5 is convergent(b1) & b4 /. lim b5 = lim (b4 * b5));

:: NFCONT_1:th 26
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b3,the carrier of b2 holds
      b4 is_continuous_on b1
   iff
      b1 c= dom b4 &
       (for b5 being Element of the carrier of b3
       for b6 being Element of REAL
             st b5 in b1 & 0 < b6
          holds ex b7 being Element of REAL st
             0 < b7 &
              (for b8 being Element of the carrier of b3
                    st b8 in b1 & ||.b8 - b5.|| < b7
                 holds ||.(b4 /. b8) - (b4 /. b5).|| < b6));

:: NFCONT_1: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 Function-like Relation of the carrier of b2,REAL holds
      b3 is_continuous_on b1
   iff
      b1 c= dom b3 &
       (for b4 being Element of the carrier of b2
       for b5 being Element of REAL
             st b4 in b1 & 0 < b5
          holds ex b6 being Element of REAL st
             0 < b6 &
              (for b7 being Element of the carrier of b2
                    st b7 in b1 & ||.b7 - b4.|| < b6
                 holds abs ((b3 /. b7) - (b3 /. b4)) < b5));

:: NFCONT_1:th 28
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
      b4 is_continuous_on b1
   iff
      b4 | b1 is_continuous_on b1;

:: NFCONT_1:th 29
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,REAL holds
      b3 is_continuous_on b1
   iff
      b3 | b1 is_continuous_on b1;

:: NFCONT_1:th 30
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_continuous_on b1 & b2 c= b1
   holds b5 is_continuous_on b2;

:: NFCONT_1:th 31
theorem
for b1, 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,the carrier of b1
for b4 being Element of the carrier of b2
      st b4 in dom b3
   holds b3 is_continuous_on {b4};

:: NFCONT_1:th 32
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4, b5 being Function-like Relation of the carrier of b2,the carrier of b1
      st b4 is_continuous_on b3 & b5 is_continuous_on b3
   holds b4 + b5 is_continuous_on b3 & b4 - b5 is_continuous_on b3;

:: NFCONT_1:th 33
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being set
for b5, b6 being Function-like Relation of the carrier of b2,the carrier of b1
      st b5 is_continuous_on b3 & b6 is_continuous_on b4
   holds b5 + b6 is_continuous_on b3 /\ b4 & b5 - b6 is_continuous_on b3 /\ b4;

:: NFCONT_1:th 34
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being Element of REAL
for b4 being set
for b5 being Function-like Relation of the carrier of b2,the carrier of b1
      st b5 is_continuous_on b4
   holds b3 (#) b5 is_continuous_on b4;

:: NFCONT_1:th 35
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_continuous_on b1
   holds ||.b4.|| is_continuous_on b1 & - b4 is_continuous_on b1;

:: NFCONT_1:th 36
theorem
for b1, 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,the carrier of b1
      st b3 is total(the carrier of b2, the carrier of b1) &
         (for b4, b5 being Element of the carrier of b2 holds
         b3 /. (b4 + b5) = (b3 /. b4) + (b3 /. b5)) &
         (ex b4 being Element of the carrier of b2 st
            b3 is_continuous_in b4)
   holds b3 is_continuous_on the carrier of b2;

:: NFCONT_1:th 37
theorem
for b1, 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,the carrier of b1
      st dom b3 is compact(b2) & b3 is_continuous_on dom b3
   holds rng b3 is compact(b1);

:: NFCONT_1:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
      st dom b2 is compact(b1) & b2 is_continuous_on dom b2
   holds rng b2 is compact;

:: NFCONT_1:th 39
theorem
for b1, 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,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_continuous_on b4
   holds b3 .: b4 is compact(b1);

:: NFCONT_1:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
      st dom b2 <> {} & dom b2 is compact(b1) & b2 is_continuous_on dom b2
   holds ex b3, b4 being Element of the carrier of b1 st
      b3 in dom b2 & b4 in dom b2 & b2 /. b3 = upper_bound rng b2 & b2 /. b4 = lower_bound rng b2;

:: NFCONT_1:th 41
theorem
for b1, 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,the carrier of b1
      st dom b3 <> {} & dom b3 is compact(b2) & b3 is_continuous_on dom b3
   holds ex b4, b5 being Element of the carrier of b2 st
      b4 in dom b3 &
       b5 in dom b3 &
       ||.b3.|| /. b4 = upper_bound rng ||.b3.|| &
       ||.b3.|| /. b5 = lower_bound rng ||.b3.||;

:: NFCONT_1:th 42
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3 holds
   ||.b4.|| | b1 = ||.b4 | b1.||;

:: NFCONT_1:th 43
theorem
for b1, 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,the carrier of b1
for b4 being Element of bool the carrier of b2
      st b4 <> {} & b4 c= dom b3 & b4 is compact(b2) & b3 is_continuous_on b4
   holds ex b5, b6 being Element of the carrier of b2 st
      b5 in b4 &
       b6 in b4 &
       ||.b3.|| /. b5 = upper_bound (||.b3.|| .: b4) &
       ||.b3.|| /. b6 = lower_bound (||.b3.|| .: b4);

:: NFCONT_1:th 44
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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_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);

:: NFCONT_1:prednot 5 => NFCONT_1:pred 5
definition
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be set;
  let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
  pred A4 is_Lipschitzian_on A3 means
    a3 c= dom a4 &
     (ex b1 being Element of REAL st
        0 < b1 &
         (for b2, b3 being Element of the carrier of a1
               st b2 in a3 & b3 in a3
            holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));
end;

:: NFCONT_1:dfs 12
definiens
  let a1, a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3 be set;
  let a4 be Function-like Relation of the carrier of a1,the carrier of a2;
To prove
     a4 is_Lipschitzian_on a3
it is sufficient to prove
  thus a3 c= dom a4 &
     (ex b1 being Element of REAL st
        0 < b1 &
         (for b2, b3 being Element of the carrier of a1
               st b2 in a3 & b3 in a3
            holds ||.(a4 /. b2) - (a4 /. b3).|| <= b1 * ||.b2 - b3.||));

:: NFCONT_1:def 13
theorem
for b1, b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3 being set
for b4 being Function-like Relation of the carrier of b1,the carrier of b2 holds
      b4 is_Lipschitzian_on b3
   iff
      b3 c= dom b4 &
       (ex b5 being Element of REAL st
          0 < b5 &
           (for b6, b7 being Element of the carrier of b1
                 st b6 in b3 & b7 in b3
              holds ||.(b4 /. b6) - (b4 /. b7).|| <= b5 * ||.b6 - b7.||));

:: NFCONT_1:prednot 6 => NFCONT_1:pred 6
definition
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be set;
  let a3 be Function-like Relation of the carrier of a1,REAL;
  pred A3 is_Lipschitzian_on A2 means
    a2 c= dom a3 &
     (ex b1 being Element of REAL st
        0 < b1 &
         (for b2, b3 being Element of the carrier of a1
               st b2 in a2 & b3 in a2
            holds abs ((a3 /. b2) - (a3 /. b3)) <= b1 * ||.b2 - b3.||));
end;

:: NFCONT_1:dfs 13
definiens
  let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a2 be set;
  let a3 be Function-like Relation of the carrier of a1,REAL;
To prove
     a3 is_Lipschitzian_on a2
it is sufficient to prove
  thus a2 c= dom a3 &
     (ex b1 being Element of REAL st
        0 < b1 &
         (for b2, b3 being Element of the carrier of a1
               st b2 in a2 & b3 in a2
            holds abs ((a3 /. b2) - (a3 /. b3)) <= b1 * ||.b2 - b3.||));

:: NFCONT_1:def 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being set
for b3 being Function-like Relation of the carrier of b1,REAL holds
      b3 is_Lipschitzian_on b2
   iff
      b2 c= dom b3 &
       (ex b4 being Element of REAL st
          0 < b4 &
           (for b5, b6 being Element of the carrier of b1
                 st b5 in b2 & b6 in b2
              holds abs ((b3 /. b5) - (b3 /. b6)) <= b4 * ||.b5 - b6.||));

:: NFCONT_1:th 45
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_Lipschitzian_on b1 & b2 c= b1
   holds b5 is_Lipschitzian_on b2;

:: NFCONT_1:th 46
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_Lipschitzian_on b1 & b6 is_Lipschitzian_on b2
   holds b5 + b6 is_Lipschitzian_on b1 /\ b2;

:: NFCONT_1:th 47
theorem
for b1, b2 being set
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_Lipschitzian_on b1 & b6 is_Lipschitzian_on b2
   holds b5 - b6 is_Lipschitzian_on b1 /\ b2;

:: NFCONT_1:th 48
theorem
for b1 being set
for b2 being Element of REAL
for b3, b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of the carrier of b3,the carrier of b4
      st b5 is_Lipschitzian_on b1
   holds b2 (#) b5 is_Lipschitzian_on b1;

:: NFCONT_1:th 49
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_Lipschitzian_on b1
   holds - b4 is_Lipschitzian_on b1 & ||.b4.|| is_Lipschitzian_on b1;

:: NFCONT_1:th 50
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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_Lipschitzian_on b1;

:: NFCONT_1:th 51
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1 holds
   id b2 is_Lipschitzian_on b2;

:: NFCONT_1:th 52
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of the carrier of b2,the carrier of b3
      st b4 is_Lipschitzian_on b1
   holds b4 is_continuous_on b1;

:: NFCONT_1:th 53
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,REAL
      st b3 is_Lipschitzian_on b1
   holds b3 is_continuous_on b1;

:: NFCONT_1:th 54
theorem
for b1, 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,the carrier of b1
      st ex b4 being Element of the carrier of b1 st
           rng b3 = {b4}
   holds b3 is_continuous_on dom b3;

:: NFCONT_1:th 55
theorem
for b1 being set
for b2, b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
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_continuous_on b1;

:: NFCONT_1:th 56
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
      st for b3 being Element of the carrier of b1
              st b3 in dom b2
           holds b2 /. b3 = b3
   holds b2 is_continuous_on dom b2;

:: NFCONT_1:th 57
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,the carrier of b1
      st b2 = id dom b2
   holds b2 is_continuous_on dom b2;

:: NFCONT_1:th 58
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Element of bool the carrier of b1
for b3 being Function-like Relation of the carrier of b1,the carrier of b1
      st b2 c= dom b3 & b3 | b2 = id b2
   holds b3 is_continuous_on b2;

:: NFCONT_1:th 59
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,the carrier of b2
for b4 being Element of REAL
for b5 being Element of the carrier of b2
      st b1 c= dom b3 &
         (for b6 being Element of the carrier of b2
               st b6 in b1
            holds b3 /. b6 = (b4 * b6) + b5)
   holds b3 is_continuous_on b1;

:: NFCONT_1:th 60
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b2 being Function-like Relation of the carrier of b1,REAL
      st for b3 being Element of the carrier of b1
              st b3 in dom b2
           holds b2 /. b3 = ||.b3.||
   holds b2 is_continuous_on dom b2;

:: NFCONT_1:th 61
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,REAL
      st b1 c= dom b3 &
         (for b4 being Element of the carrier of b2
               st b4 in b1
            holds b3 /. b4 = ||.b4.||)
   holds b3 is_continuous_on b1;