Article CFUNCT_1, MML version 4.99.1005

:: CFUNCT_1:modenot 1
definition
  mode Complex is Element of COMPLEX;
end;

:: CFUNCT_1:funcnot 1 => CFUNCT_1:func 1
definition
  let a1 be non empty set;
  let a2, a3 be Function-like Relation of a1,COMPLEX;
  func A2 / A3 -> Function-like Relation of a1,COMPLEX means
    dom it = (dom a2) /\ ((dom a3) \ (a3 " {0})) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = (a2 /. b1) * ((a3 /. b1) "));
end;

:: CFUNCT_1:def 1
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
   b4 = b2 / b3
iff
   dom b4 = (dom b2) /\ ((dom b3) \ (b3 " {0})) &
    (for b5 being Element of b1
          st b5 in dom b4
       holds b4 /. b5 = (b2 /. b5) * ((b3 /. b5) "));

:: CFUNCT_1:funcnot 2 => CFUNCT_1:func 2
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,COMPLEX;
  func A2 ^ -> Function-like Relation of a1,COMPLEX means
    dom it = (dom a2) \ (a2 " {0}) &
     (for b1 being Element of a1
           st b1 in dom it
        holds it /. b1 = (a2 /. b1) ");
end;

:: CFUNCT_1:def 2
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
   b3 = b2 ^
iff
   dom b3 = (dom b2) \ (b2 " {0}) &
    (for b4 being Element of b1
          st b4 in dom b3
       holds b3 /. b4 = (b2 /. b4) ");

:: CFUNCT_1:th 3
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
dom (b2 + b3) = (dom b2) /\ dom b3 &
 (for b4 being Element of b1
       st b4 in dom (b2 + b3)
    holds (b2 + b3) /. b4 = (b2 /. b4) + (b3 /. b4));

:: CFUNCT_1:th 4
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
dom (b2 - b3) = (dom b2) /\ dom b3 &
 (for b4 being Element of b1
       st b4 in dom (b2 - b3)
    holds (b2 - b3) /. b4 = (b2 /. b4) - (b3 /. b4));

:: CFUNCT_1:th 5
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
dom (b2 (#) b3) = (dom b2) /\ dom b3 &
 (for b4 being Element of b1
       st b4 in dom (b2 (#) b3)
    holds (b2 (#) b3) /. b4 = (b2 /. b4) * (b3 /. b4));

:: CFUNCT_1:th 7
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being Element of COMPLEX holds
   dom (b3 (#) b2) = dom b2 &
    (for b4 being Element of b1
          st b4 in dom (b3 (#) b2)
       holds (b3 (#) b2) /. b4 = b3 * (b2 /. b4));

:: CFUNCT_1:th 9
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   dom - b2 = dom b2 &
    (for b3 being Element of b1
          st b3 in dom - b2
       holds (- b2) /. b3 = - (b2 /. b3));

:: CFUNCT_1:th 15
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   dom (b2 ^) c= dom b2 &
    (dom b2) /\ ((dom b2) \ (b2 " {0})) = (dom b2) \ (b2 " {0});

:: CFUNCT_1:th 16
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
(dom (b2 (#) b3)) \ ((b2 (#) b3) " {0}) = ((dom b2) \ (b2 " {0})) /\ ((dom b3) \ (b3 " {0}));

:: CFUNCT_1:th 17
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,COMPLEX
      st b2 in dom (b3 ^)
   holds b3 /. b2 <> 0;

:: CFUNCT_1:th 18
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   b2 ^ " {0} = {};

:: CFUNCT_1:th 19
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   |.b2.| " {0} = b2 " {0} &
    (- b2) " {0} = b2 " {0};

:: CFUNCT_1:th 20
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   dom (b2 ^ ^) = dom (b2 | dom (b2 ^));

:: CFUNCT_1:th 21
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being Element of COMPLEX
      st b3 <> 0
   holds (b3 (#) b2) " {0} = b2 " {0};

:: CFUNCT_1:th 22
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 + b3) + b4 = b2 + (b3 + b4);

:: CFUNCT_1:th 23
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 (#) b3) (#) b4 = b2 (#) (b3 (#) b4);

:: CFUNCT_1:th 24
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 + b3) (#) b4 = (b2 (#) b4) + (b3 (#) b4);

:: CFUNCT_1:th 25
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 (#) (b3 + b4) = (b2 (#) b3) + (b2 (#) b4);

:: CFUNCT_1:th 26
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX holds
   b4 (#) (b2 (#) b3) = (b4 (#) b2) (#) b3;

:: CFUNCT_1:th 27
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX holds
   b4 (#) (b2 (#) b3) = b2 (#) (b4 (#) b3);

:: CFUNCT_1:th 28
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 - b3) (#) b4 = (b2 (#) b4) - (b3 (#) b4);

:: CFUNCT_1:th 29
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 (#) b3) - (b2 (#) b4) = b2 (#) (b3 - b4);

:: CFUNCT_1:th 30
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX holds
   b4 (#) (b2 + b3) = (b4 (#) b2) + (b4 (#) b3);

:: CFUNCT_1:th 31
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3, b4 being Element of COMPLEX holds
(b3 * b4) (#) b2 = b3 (#) (b4 (#) b2);

:: CFUNCT_1:th 32
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX holds
   b4 (#) (b2 - b3) = (b4 (#) b2) - (b4 (#) b3);

:: CFUNCT_1:th 33
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
b2 - b3 = (- 1r) (#) (b3 - b2);

:: CFUNCT_1:th 34
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 - (b3 + b4) = (b2 - b3) - b4;

:: CFUNCT_1:th 35
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   1r (#) b2 = b2;

:: CFUNCT_1:th 36
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 - (b3 - b4) = (b2 - b3) + b4;

:: CFUNCT_1:th 37
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 + (b3 - b4) = (b2 + b3) - b4;

:: CFUNCT_1:th 38
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
|.b2 (#) b3.| = |.b2.| (#) |.b3.|;

:: CFUNCT_1:th 39
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being Element of COMPLEX holds
   |.b3 (#) b2.| = |.b3.| (#) |.b2.|;

:: CFUNCT_1:th 40
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   - b2 = (- 1r) (#) b2;

:: CFUNCT_1:th 41
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   - - b2 = b2;

:: CFUNCT_1:th 43
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
b2 - - b3 = b2 + b3;

:: CFUNCT_1:th 44
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   b2 ^ ^ = b2 | dom (b2 ^);

:: CFUNCT_1:th 45
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
(b2 (#) b3) ^ = b2 ^ (#) (b3 ^);

:: CFUNCT_1:th 46
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being Element of COMPLEX
      st b3 <> 0
   holds (b3 (#) b2) ^ = b3 " (#) (b2 ^);

:: CFUNCT_1:th 49
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   (- b2) ^ = (- 1r) (#) (b2 ^);

:: CFUNCT_1:th 50
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
   |.b2.| ^ = |.b2 ^.|;

:: CFUNCT_1:th 51
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
b2 / b3 = b2 (#) (b3 ^);

:: CFUNCT_1:th 52
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX holds
   b4 (#) (b2 / b3) = (b4 (#) b2) / b3;

:: CFUNCT_1:th 53
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) (#) b3 = b2 | dom (b3 ^);

:: CFUNCT_1:th 54
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) (#) (b4 / b5) = (b2 (#) b4) / (b3 (#) b5);

:: CFUNCT_1:th 55
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) ^ = (b3 | dom (b3 ^)) / b2;

:: CFUNCT_1:th 56
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 (#) (b3 / b4) = (b2 (#) b3) / b4;

:: CFUNCT_1:th 57
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
b2 / (b3 / b4) = (b2 (#) (b4 | dom (b4 ^))) / b3;

:: CFUNCT_1:th 58
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
- (b2 / b3) = (- b2) / b3 & b2 / - b3 = - (b2 / b3);

:: CFUNCT_1:th 59
theorem
for b1 being non empty set
for b2, b3, b4 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) + (b4 / b3) = (b2 + b4) / b3 &
 (b2 / b3) - (b4 / b3) = (b2 - b4) / b3;

:: CFUNCT_1:th 60
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) + (b4 / b5) = ((b2 (#) b5) + (b4 (#) b3)) / (b3 (#) b5);

:: CFUNCT_1:th 61
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) / (b4 / b5) = (b2 (#) (b5 | dom (b5 ^))) / (b3 (#) b4);

:: CFUNCT_1:th 62
theorem
for b1 being non empty set
for b2, b3, b4, b5 being Function-like Relation of b1,COMPLEX holds
(b2 / b3) - (b4 / b5) = ((b2 (#) b5) - (b4 (#) b3)) / (b3 (#) b5);

:: CFUNCT_1:th 63
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
|.b2 / b3.| = |.b2.| / |.b3.|;

:: CFUNCT_1:th 64
theorem
for b1 being set
for b2 being non empty set
for b3, b4 being Function-like Relation of b2,COMPLEX holds
(b3 + b4) | b1 = (b3 | b1) + (b4 | b1) &
 (b3 + b4) | b1 = (b3 | b1) + b4 &
 (b3 + b4) | b1 = b3 + (b4 | b1);

:: CFUNCT_1:th 65
theorem
for b1 being set
for b2 being non empty set
for b3, b4 being Function-like Relation of b2,COMPLEX holds
(b3 (#) b4) | b1 = (b3 | b1) (#) (b4 | b1) &
 (b3 (#) b4) | b1 = (b3 | b1) (#) b4 &
 (b3 (#) b4) | b1 = b3 (#) (b4 | b1);

:: CFUNCT_1:th 66
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX holds
   (- b3) | b1 = - (b3 | b1) &
    b3 ^ | b1 = (b3 | b1) ^ &
    |.b3.| | b1 = |.b3 | b1.|;

:: CFUNCT_1:th 67
theorem
for b1 being set
for b2 being non empty set
for b3, b4 being Function-like Relation of b2,COMPLEX holds
(b3 - b4) | b1 = (b3 | b1) - (b4 | b1) &
 (b3 - b4) | b1 = (b3 | b1) - b4 &
 (b3 - b4) | b1 = b3 - (b4 | b1);

:: CFUNCT_1:th 68
theorem
for b1 being set
for b2 being non empty set
for b3, b4 being Function-like Relation of b2,COMPLEX holds
(b3 / b4) | b1 = (b3 | b1) / (b4 | b1) &
 (b3 / b4) | b1 = (b3 | b1) / b4 &
 (b3 / b4) | b1 = b3 / (b4 | b1);

:: CFUNCT_1:th 69
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
for b4 being Element of COMPLEX holds
   (b4 (#) b3) | b1 = b4 (#) (b3 | b1);

:: CFUNCT_1:th 70
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
(b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX) implies b2 + b3 is total(b1, COMPLEX)) &
 (b2 + b3 is total(b1, COMPLEX) implies b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX)) &
 (b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX) implies b2 - b3 is total(b1, COMPLEX)) &
 (b2 - b3 is total(b1, COMPLEX) implies b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX)) &
 (b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX) implies b2 (#) b3 is total(b1, COMPLEX)) &
 (b2 (#) b3 is total(b1, COMPLEX) implies b2 is total(b1, COMPLEX) & b3 is total(b1, COMPLEX));

:: CFUNCT_1:th 71
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being Element of COMPLEX holds
      b2 is total(b1, COMPLEX)
   iff
      b3 (#) b2 is total(b1, COMPLEX);

:: CFUNCT_1:th 72
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
      b2 is total(b1, COMPLEX)
   iff
      - b2 is total(b1, COMPLEX);

:: CFUNCT_1:th 73
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
      b2 is total(b1, COMPLEX)
   iff
      |.b2.| is total(b1, REAL);

:: CFUNCT_1:th 74
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX holds
      b2 ^ is total(b1, COMPLEX)
   iff
      b2 " {0} = {} & b2 is total(b1, COMPLEX);

:: CFUNCT_1:th 75
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,COMPLEX holds
   b2 is total(b1, COMPLEX) & b3 " {0} = {} & b3 is total(b1, COMPLEX)
iff
   b2 / b3 is total(b1, COMPLEX);

:: CFUNCT_1:th 76
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like Relation of b1,COMPLEX
      st b3 is total(b1, COMPLEX) & b4 is total(b1, COMPLEX)
   holds (b3 + b4) /. b2 = (b3 /. b2) + (b4 /. b2) &
    (b3 - b4) /. b2 = (b3 /. b2) - (b4 /. b2) &
    (b3 (#) b4) /. b2 = (b3 /. b2) * (b4 /. b2);

:: CFUNCT_1:th 77
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,COMPLEX
for b4 being Element of COMPLEX
      st b3 is total(b1, COMPLEX)
   holds (b4 (#) b3) /. b2 = b4 * (b3 /. b2);

:: CFUNCT_1:th 78
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,COMPLEX
      st b3 is total(b1, COMPLEX)
   holds (- b3) /. b2 = - (b3 /. b2) &
    |.b3.| . b2 = |.b3 /. b2.|;

:: CFUNCT_1:th 79
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,COMPLEX
      st b3 ^ is total(b1, COMPLEX)
   holds b3 ^ /. b2 = (b3 /. b2) ";

:: CFUNCT_1:th 80
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like Relation of b1,COMPLEX
      st b3 is total(b1, COMPLEX) & b4 ^ is total(b1, COMPLEX)
   holds (b3 / b4) /. b2 = (b3 /. b2) * ((b4 /. b2) ");

:: CFUNCT_1:prednot 1 => CFUNCT_1:pred 1
definition
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,COMPLEX;
  let a3 be set;
  pred A2 is_bounded_on A3 means
    |.a2.| is_bounded_on a3;
end;

:: CFUNCT_1:dfs 3
definiens
  let a1 be non empty set;
  let a2 be Function-like Relation of a1,COMPLEX;
  let a3 be set;
To prove
     a2 is_bounded_on a3
it is sufficient to prove
  thus |.a2.| is_bounded_on a3;

:: CFUNCT_1:def 3
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,COMPLEX
for b3 being set holds
      b2 is_bounded_on b3
   iff
      |.b2.| is_bounded_on b3;

:: CFUNCT_1:th 81
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX holds
      b3 is_bounded_on b1
   iff
      ex b4 being real set st
         for b5 being Element of b2
               st b5 in b1 /\ dom b3
            holds |.b3 /. b5.| <= b4;

:: CFUNCT_1:th 82
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being Function-like Relation of b3,COMPLEX
      st b1 c= b2 & b4 is_bounded_on b2
   holds b4 is_bounded_on b1;

:: CFUNCT_1:th 83
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
      st b1 misses dom b3
   holds b3 is_bounded_on b1;

:: CFUNCT_1:th 84
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
for b4 being Element of COMPLEX
      st b3 is_bounded_on b1
   holds b4 (#) b3 is_bounded_on b1;

:: CFUNCT_1:th 85
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX holds
   |.b3.| is_bounded_below_on b1;

:: CFUNCT_1:th 86
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
      st b3 is_bounded_on b1
   holds |.b3.| is_bounded_on b1 & - b3 is_bounded_on b1;

:: CFUNCT_1:th 87
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,COMPLEX
      st b4 is_bounded_on b1 & b5 is_bounded_on b2
   holds b4 + b5 is_bounded_on b1 /\ b2;

:: CFUNCT_1:th 88
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,COMPLEX
      st b4 is_bounded_on b1 & b5 is_bounded_on b2
   holds b4 (#) b5 is_bounded_on b1 /\ b2 & b4 - b5 is_bounded_on b1 /\ b2;

:: CFUNCT_1:th 89
theorem
for b1, b2 being set
for b3 being non empty set
for b4 being Function-like Relation of b3,COMPLEX
      st b4 is_bounded_on b1 & b4 is_bounded_on b2
   holds b4 is_bounded_on b1 \/ b2;

:: CFUNCT_1:th 90
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,COMPLEX
      st b4 is_constant_on b1 & b5 is_constant_on b2
   holds b4 + b5 is_constant_on b1 /\ b2 & b4 - b5 is_constant_on b1 /\ b2 & b4 (#) b5 is_constant_on b1 /\ b2;

:: CFUNCT_1:th 91
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
for b4 being Element of COMPLEX
      st b3 is_constant_on b1
   holds b4 (#) b3 is_constant_on b1;

:: CFUNCT_1:th 92
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
      st b3 is_constant_on b1
   holds |.b3.| is_constant_on b1 & - b3 is_constant_on b1;

:: CFUNCT_1:th 93
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
      st b3 is_constant_on b1
   holds b3 is_bounded_on b1;

:: CFUNCT_1:th 94
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,COMPLEX
      st b3 is_constant_on b1
   holds (for b4 being Element of COMPLEX holds
       b4 (#) b3 is_bounded_on b1) &
    - b3 is_bounded_on b1 &
    |.b3.| is_bounded_on b1;

:: CFUNCT_1:th 95
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,COMPLEX
      st b4 is_bounded_on b1 & b5 is_constant_on b2
   holds b4 + b5 is_bounded_on b1 /\ b2;

:: CFUNCT_1:th 96
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,COMPLEX
      st b4 is_bounded_on b1 & b5 is_constant_on b2
   holds b4 - b5 is_bounded_on b1 /\ b2 & b5 - b4 is_bounded_on b1 /\ b2 & b4 (#) b5 is_bounded_on b1 /\ b2;