Article VFUNCT_1, MML version 4.99.1005

:: VFUNCT_1:funcnot 1 => VFUNCT_1:func 1
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3, a4 be Function-like Relation of a1,the carrier of a2;
  func A3 + A4 -> Function-like Relation of a1,the carrier of a2 means
    dom it = (dom a3) /\ dom a4 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = (a3 /. b1) + (a4 /. b1));
end;

:: VFUNCT_1:def 1
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
   b5 = b3 + b4
iff
   dom b5 = (dom b3) /\ dom b4 &
    (for b6 being Element of b1
          st b6 in dom b5
       holds b5 /. b6 = (b3 /. b6) + (b4 /. b6));

:: VFUNCT_1:funcnot 2 => VFUNCT_1:func 2
definition
  let a1 be non empty set;
  let a2 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR;
  let a3, a4 be Function-like Relation of a1,the carrier of a2;
  func A3 - A4 -> Function-like Relation of a1,the carrier of a2 means
    dom it = (dom a3) /\ dom a4 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = (a3 /. b1) - (a4 /. b1));
end;

:: VFUNCT_1:def 2
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
   b5 = b3 - b4
iff
   dom b5 = (dom b3) /\ dom b4 &
    (for b6 being Element of b1
          st b6 in dom b5
       holds b5 /. b6 = (b3 /. b6) - (b4 /. b6));

:: VFUNCT_1:funcnot 3 => VFUNCT_1:func 3
definition
  let a1 be non empty 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 a1,REAL;
  let a4 be Function-like Relation of a1,the carrier of a2;
  func A3 (#) A4 -> Function-like Relation of a1,the carrier of a2 means
    dom it = (dom a3) /\ dom a4 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = (a3 . b1) * (a4 /. b1));
end;

:: VFUNCT_1:def 3
theorem
for b1 being non empty 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 b1,REAL
for b4, b5 being Function-like Relation of b1,the carrier of b2 holds
   b5 = b3 (#) b4
iff
   dom b5 = (dom b3) /\ dom b4 &
    (for b6 being Element of b1
          st b6 in dom b5
       holds b5 /. b6 = (b3 . b6) * (b4 /. b6));

:: VFUNCT_1:funcnot 4 => VFUNCT_1:func 4
definition
  let a1 be non empty 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 a1,the carrier of a2;
  let a4 be Element of REAL;
  func A4 (#) A3 -> Function-like Relation of a1,the carrier of a2 means
    dom it = dom a3 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = a4 * (a3 /. b1));
end;

:: VFUNCT_1:def 4
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL
for b5 being Function-like Relation of b1,the carrier of b2 holds
      b5 = b4 (#) b3
   iff
      dom b5 = dom b3 &
       (for b6 being Element of b1
             st b6 in dom b5
          holds b5 /. b6 = b4 * (b3 /. b6));

:: VFUNCT_1:funcnot 5 => VFUNCT_1:func 5
definition
  let a1 be non empty 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 a1,the carrier of a2;
  func ||.A3.|| -> Function-like Relation of a1,REAL means
    dom it = dom a3 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it . b1 = ||.a3 /. b1.||);
end;

:: VFUNCT_1:def 5
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Function-like Relation of b1,REAL holds
      b4 = ||.b3.||
   iff
      dom b4 = dom b3 &
       (for b5 being Element of b1
             st b5 in dom b4
          holds b4 . b5 = ||.b3 /. b5.||);

:: VFUNCT_1:funcnot 6 => VFUNCT_1:func 6
definition
  let a1 be non empty 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 a1,the carrier of a2;
  func - A3 -> Function-like Relation of a1,the carrier of a2 means
    dom it = dom a3 &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = - (a3 /. b1));
end;

:: VFUNCT_1:def 6
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
   b4 = - b3
iff
   dom b4 = dom b3 &
    (for b5 being Element of b1
          st b5 in dom b4
       holds b4 /. b5 = - (b3 /. b5));

:: VFUNCT_1:th 7
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Function-like Relation of b1,REAL holds
   (dom (b4 (#) b3)) \ ((b4 (#) b3) " {0. b2}) = ((dom b4) \ (b4 " {0})) /\ ((dom b3) \ (b3 " {0. b2}));

:: VFUNCT_1:th 8
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
   ||.b3.|| " {0} = b3 " {0. b2} &
    (- b3) " {0. b2} = b3 " {0. b2};

:: VFUNCT_1:th 9
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL
      st b4 <> 0
   holds (b4 (#) b3) " {0. b2} = b3 " {0. b2};

:: VFUNCT_1:th 10
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
b3 + b4 = b4 + b3;

:: VFUNCT_1:th 11
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
(b3 + b4) + b5 = b3 + (b4 + b5);

:: VFUNCT_1:th 12
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,REAL
for b5 being Function-like Relation of b1,the carrier of b2 holds
   (b3 (#) b4) (#) b5 = b3 (#) (b4 (#) b5);

:: VFUNCT_1:th 13
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4, b5 being Function-like Relation of b1,REAL holds
(b4 + b5) (#) b3 = (b4 (#) b3) + (b5 (#) b3);

:: VFUNCT_1:th 14
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2
for b5 being Function-like Relation of b1,REAL holds
   b5 (#) (b3 + b4) = (b5 (#) b3) + (b5 (#) b4);

:: VFUNCT_1:th 15
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL
for b5 being Function-like Relation of b1,REAL holds
   b4 (#) (b5 (#) b3) = (b4 (#) b5) (#) b3;

:: VFUNCT_1:th 16
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL
for b5 being Function-like Relation of b1,REAL holds
   b4 (#) (b5 (#) b3) = b5 (#) (b4 (#) b3);

:: VFUNCT_1:th 17
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4, b5 being Function-like Relation of b1,REAL holds
(b4 - b5) (#) b3 = (b4 (#) b3) - (b5 (#) b3);

:: VFUNCT_1:th 18
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2
for b5 being Function-like Relation of b1,REAL holds
   (b5 (#) b3) - (b5 (#) b4) = b5 (#) (b3 - b4);

:: VFUNCT_1:th 19
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2
for b5 being Element of REAL holds
   b5 (#) (b3 + b4) = (b5 (#) b3) + (b5 (#) b4);

:: VFUNCT_1:th 20
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4, b5 being Element of REAL holds
(b4 * b5) (#) b3 = b4 (#) (b5 (#) b3);

:: VFUNCT_1:th 21
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2
for b5 being Element of REAL holds
   b5 (#) (b3 - b4) = (b5 (#) b3) - (b5 (#) b4);

:: VFUNCT_1:th 22
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
b3 - b4 = (- 1) (#) (b4 - b3);

:: VFUNCT_1:th 23
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
b3 - (b4 + b5) = (b3 - b4) - b5;

:: VFUNCT_1:th 24
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
   1 (#) b3 = b3;

:: VFUNCT_1:th 25
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
b3 - (b4 - b5) = (b3 - b4) + b5;

:: VFUNCT_1:th 26
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4, b5 being Function-like Relation of b1,the carrier of b2 holds
b3 + (b4 - b5) = (b3 + b4) - b5;

:: VFUNCT_1:th 27
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Function-like Relation of b1,REAL holds
   ||.b4 (#) b3.|| = (abs b4) (#) ||.b3.||;

:: VFUNCT_1:th 28
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL holds
   ||.b4 (#) b3.|| = (abs b4) (#) ||.b3.||;

:: VFUNCT_1:th 29
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
   - b3 = (- 1) (#) b3;

:: VFUNCT_1:th 30
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
   - - b3 = b3;

:: VFUNCT_1:th 31
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
b3 - b4 = b3 + - b4;

:: VFUNCT_1:th 32
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
b3 - - b4 = b3 + b4;

:: VFUNCT_1:th 33
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4, b5 being Function-like Relation of b2,the carrier of b3 holds
(b4 + b5) | b1 = (b4 | b1) + (b5 | b1) &
 (b4 + b5) | b1 = (b4 | b1) + b5 &
 (b4 + b5) | b1 = b4 + (b5 | b1);

:: VFUNCT_1:th 34
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
for b5 being Function-like Relation of b2,REAL holds
   (b5 (#) b4) | b1 = (b5 | b1) (#) (b4 | b1) &
    (b5 (#) b4) | b1 = (b5 | b1) (#) b4 &
    (b5 (#) b4) | b1 = b5 (#) (b4 | b1);

:: VFUNCT_1:th 35
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3 holds
   (- b4) | b1 = - (b4 | b1) &
    ||.b4.|| | b1 = ||.b4 | b1.||;

:: VFUNCT_1:th 36
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4, b5 being Function-like Relation of b2,the carrier of b3 holds
(b4 - b5) | b1 = (b4 | b1) - (b5 | b1) &
 (b4 - b5) | b1 = (b4 | b1) - b5 &
 (b4 - b5) | b1 = b4 - (b5 | b1);

:: VFUNCT_1:th 37
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
for b5 being Element of REAL holds
   (b5 (#) b4) | b1 = b5 (#) (b4 | b1);

:: VFUNCT_1:th 38
theorem
for b1 being non empty set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b3, b4 being Function-like Relation of b1,the carrier of b2 holds
(b3 is total(b1, the carrier of b2) & b4 is total(b1, the carrier of b2) implies b3 + b4 is total(b1, the carrier of b2)) &
 (b3 + b4 is total(b1, the carrier of b2) implies b3 is total(b1, the carrier of b2) & b4 is total(b1, the carrier of b2)) &
 (b3 is total(b1, the carrier of b2) & b4 is total(b1, the carrier of b2) implies b3 - b4 is total(b1, the carrier of b2)) &
 (b3 - b4 is total(b1, the carrier of b2) implies b3 is total(b1, the carrier of b2) & b4 is total(b1, the carrier of b2));

:: VFUNCT_1:th 39
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Function-like Relation of b1,REAL holds
      b4 is total(b1, REAL) & b3 is total(b1, the carrier of b2)
   iff
      b4 (#) b3 is total(b1, the carrier of b2);

:: VFUNCT_1:th 40
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being Element of REAL holds
      b3 is total(b1, the carrier of b2)
   iff
      b4 (#) b3 is total(b1, the carrier of b2);

:: VFUNCT_1:th 41
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
      b3 is total(b1, the carrier of b2)
   iff
      - b3 is total(b1, the carrier of b2);

:: VFUNCT_1:th 42
theorem
for b1 being non empty 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 b1,the carrier of b2 holds
      b3 is total(b1, the carrier of b2)
   iff
      ||.b3.|| is total(b1, REAL);

:: VFUNCT_1:th 43
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4, b5 being Function-like Relation of b1,the carrier of b3
      st b4 is total(b1, the carrier of b3) & b5 is total(b1, the carrier of b3)
   holds (b4 + b5) /. b2 = (b4 /. b2) + (b5 /. b2) &
    (b4 - b5) /. b2 = (b4 /. b2) - (b5 /. b2);

:: VFUNCT_1:th 44
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b1,the carrier of b3
for b5 being Function-like Relation of b1,REAL
      st b5 is total(b1, REAL) & b4 is total(b1, the carrier of b3)
   holds (b5 (#) b4) /. b2 = (b5 . b2) * (b4 /. b2);

:: VFUNCT_1:th 45
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b1,the carrier of b3
for b5 being Element of REAL
      st b4 is total(b1, the carrier of b3)
   holds (b5 (#) b4) /. b2 = b5 * (b4 /. b2);

:: VFUNCT_1:th 46
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b1,the carrier of b3
      st b4 is total(b1, the carrier of b3)
   holds (- b4) /. b2 = - (b4 /. b2) &
    ||.b4.|| . b2 = ||.b4 /. b2.||;

:: VFUNCT_1:prednot 1 => VFUNCT_1:pred 1
definition
  let a1 be non empty 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 a1,the carrier of a2;
  let a4 be set;
  pred A3 is_bounded_on A4 means
    ex b1 being Element of REAL st
       for b2 being Element of a1
             st b2 in a4 /\ dom a3
          holds ||.a3 /. b2.|| <= b1;
end;

:: VFUNCT_1:dfs 7
definiens
  let a1 be non empty 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 a1,the carrier of a2;
  let a4 be set;
To prove
     a3 is_bounded_on a4
it is sufficient to prove
  thus ex b1 being Element of REAL st
       for b2 being Element of a1
             st b2 in a4 /\ dom a3
          holds ||.a3 /. b2.|| <= b1;

:: VFUNCT_1:def 7
theorem
for b1 being non empty 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 b1,the carrier of b2
for b4 being set holds
      b3 is_bounded_on b4
   iff
      ex b5 being Element of REAL st
         for b6 being Element of b1
               st b6 in b4 /\ dom b3
            holds ||.b3 /. b6.|| <= b5;

:: VFUNCT_1:th 48
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of b3,the carrier of b4
      st b1 c= b2 & b5 is_bounded_on b2
   holds b5 is_bounded_on b1;

:: VFUNCT_1:th 49
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
      st b1 misses dom b4
   holds b4 is_bounded_on b1;

:: VFUNCT_1:th 50
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3 holds
   0 (#) b4 is_bounded_on b1;

:: VFUNCT_1:th 51
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
for b5 being Element of REAL
      st b4 is_bounded_on b1
   holds b5 (#) b4 is_bounded_on b1;

:: VFUNCT_1:th 52
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
      st b4 is_bounded_on b1
   holds ||.b4.|| is_bounded_on b1 & - b4 is_bounded_on b1;

:: VFUNCT_1:th 53
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of b3,the carrier of b4
      st b5 is_bounded_on b1 & b6 is_bounded_on b2
   holds b5 + b6 is_bounded_on b1 /\ b2;

:: VFUNCT_1:th 54
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of b3,the carrier of b4
for b6 being Function-like Relation of b3,REAL
      st b6 is_bounded_on b1 & b5 is_bounded_on b2
   holds b6 (#) b5 is_bounded_on b1 /\ b2;

:: VFUNCT_1:th 55
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of b3,the carrier of b4
      st b5 is_bounded_on b1 & b6 is_bounded_on b2
   holds b5 - b6 is_bounded_on b1 /\ b2;

:: VFUNCT_1:th 56
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of b3,the carrier of b4
      st b5 is_bounded_on b1 & b5 is_bounded_on b2
   holds b5 is_bounded_on b1 \/ b2;

:: VFUNCT_1:th 57
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of b3,the carrier of b4
      st b5 is_constant_on b1 & b6 is_constant_on b2
   holds b5 + b6 is_constant_on b1 /\ b2 & b5 - b6 is_constant_on b1 /\ b2;

:: VFUNCT_1:th 58
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5 being Function-like Relation of b3,the carrier of b4
for b6 being Function-like Relation of b3,REAL
      st b6 is_constant_on b1 & b5 is_constant_on b2
   holds b6 (#) b5 is_constant_on b1 /\ b2;

:: VFUNCT_1:th 59
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
for b5 being Element of REAL
      st b4 is_constant_on b1
   holds b5 (#) b4 is_constant_on b1;

:: VFUNCT_1:th 60
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
      st b4 is_constant_on b1
   holds ||.b4.|| is_constant_on b1 & - b4 is_constant_on b1;

:: VFUNCT_1:th 61
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
      st b4 is_constant_on b1
   holds b4 is_bounded_on b1;

:: VFUNCT_1:th 62
theorem
for b1 being set
for b2 being non empty set
for b3 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b4 being Function-like Relation of b2,the carrier of b3
      st b4 is_constant_on b1
   holds (for b5 being Element of REAL holds
       b5 (#) b4 is_bounded_on b1) &
    - b4 is_bounded_on b1 &
    ||.b4.|| is_bounded_on b1;

:: VFUNCT_1:th 63
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of b3,the carrier of b4
      st b5 is_bounded_on b1 & b6 is_constant_on b2
   holds b5 + b6 is_bounded_on b1 /\ b2;

:: VFUNCT_1:th 64
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RealNormSpace-like NORMSTR
for b5, b6 being Function-like Relation of b3,the carrier of b4
      st b5 is_bounded_on b1 & b6 is_constant_on b2
   holds b5 - b6 is_bounded_on b1 /\ b2 & b6 - b5 is_bounded_on b1 /\ b2;