Article RFUNCT_1, MML version 4.99.1005

:: RFUNCT_1:funcnot 1 => RFUNCT_1:func 1
definition
  let a1, a2 be Relation-like Function-like real-valued set;
  func A1 / A2 -> Relation-like Function-like set means
    proj1 it = (proj1 a1) /\ ((proj1 a2) \ (a2 " {0})) &
     (for b1 being set
           st b1 in proj1 it
        holds it . b1 = (a1 . b1) * ((a2 . b1) "));
end;

:: RFUNCT_1:def 4
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being Relation-like Function-like set holds
      b3 = b1 / b2
   iff
      proj1 b3 = (proj1 b1) /\ ((proj1 b2) \ (b2 " {0})) &
       (for b4 being set
             st b4 in proj1 b3
          holds b3 . b4 = (b1 . b4) * ((b2 . b4) "));

:: RFUNCT_1:funcreg 1
registration
  let a1, a2 be Relation-like Function-like real-valued set;
  cluster a1 / a2 -> Relation-like Function-like real-valued;
end;

:: RFUNCT_1:funcnot 2 => RFUNCT_1:func 2
definition
  let a1 be set;
  let a2 be real-membered set;
  let a3, a4 be Function-like Relation of a1,a2;
  redefine func a3 / a4 -> Function-like Relation of a1,REAL;
end;

:: RFUNCT_1:funcnot 3 => RFUNCT_1:func 3
definition
  let a1 be Relation-like Function-like real-valued set;
  func A1 ^ -> Relation-like Function-like set means
    proj1 it = (proj1 a1) \ (a1 " {0}) &
     (for b1 being set
           st b1 in proj1 it
        holds it . b1 = (a1 . b1) ");
end;

:: RFUNCT_1:def 8
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being Relation-like Function-like set holds
      b2 = b1 ^
   iff
      proj1 b2 = (proj1 b1) \ (b1 " {0}) &
       (for b3 being set
             st b3 in proj1 b2
          holds b2 . b3 = (b1 . b3) ");

:: RFUNCT_1:funcreg 2
registration
  let a1 be Relation-like Function-like real-valued set;
  cluster a1 ^ -> Relation-like Function-like real-valued;
end;

:: RFUNCT_1:funcnot 4 => RFUNCT_1:func 4
definition
  let a1 be set;
  let a2 be real-membered set;
  let a3 be Function-like Relation of a1,a2;
  redefine func a3 ^ -> Function-like Relation of a1,REAL;
end;

:: RFUNCT_1:th 11
theorem
for b1 being Relation-like Function-like real-valued set holds
   proj1 (b1 ^) c= proj1 b1 &
    (proj1 b1) /\ ((proj1 b1) \ (b1 " {0})) = (proj1 b1) \ (b1 " {0});

:: RFUNCT_1:th 12
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
(proj1 (b1 (#) b2)) \ ((b1 (#) b2) " {0}) = ((proj1 b1) \ (b1 " {0})) /\ ((proj1 b2) \ (b2 " {0}));

:: RFUNCT_1:th 13
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Relation-like Function-like real-valued set
      st b2 in proj1 (b3 ^)
   holds b3 . b2 <> 0;

:: RFUNCT_1:th 14
theorem
for b1 being Relation-like Function-like real-valued set holds
   b1 ^ " {0} = {};

:: RFUNCT_1:th 15
theorem
for b1 being Relation-like Function-like real-valued set holds
   |.b1.| " {0} = b1 " {0} &
    (- b1) " {0} = b1 " {0};

:: RFUNCT_1:th 16
theorem
for b1 being Relation-like Function-like real-valued set holds
   proj1 (b1 ^ ^) = proj1 (b1 | proj1 (b1 ^));

:: RFUNCT_1:th 17
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being real set
      st b2 <> 0
   holds (b2 (#) b1) " {0} = b1 " {0};

:: RFUNCT_1:th 19
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 + b2) + b3 = b1 + (b2 + b3);

:: RFUNCT_1:th 21
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 (#) b2) (#) b3 = b1 (#) (b2 (#) b3);

:: RFUNCT_1:th 22
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 + b2) (#) b3 = (b1 (#) b3) + (b2 (#) b3);

:: RFUNCT_1:th 23
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 (#) (b2 + b3) = (b1 (#) b2) + (b1 (#) b3);

:: RFUNCT_1:th 24
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   b3 (#) (b1 (#) b2) = (b3 (#) b1) (#) b2;

:: RFUNCT_1:th 25
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   b3 (#) (b1 (#) b2) = b1 (#) (b3 (#) b2);

:: RFUNCT_1:th 26
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 - b2) (#) b3 = (b1 (#) b3) - (b2 (#) b3);

:: RFUNCT_1:th 27
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 (#) b2) - (b1 (#) b3) = b1 (#) (b2 - b3);

:: RFUNCT_1:th 28
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   b3 (#) (b1 + b2) = (b3 (#) b1) + (b3 (#) b2);

:: RFUNCT_1:th 29
theorem
for b1 being Relation-like Function-like real-valued set
for b2, b3 being real set holds
(b2 * b3) (#) b1 = b2 (#) (b3 (#) b1);

:: RFUNCT_1:th 30
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   b3 (#) (b1 - b2) = (b3 (#) b1) - (b3 (#) b2);

:: RFUNCT_1:th 31
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
b1 - b2 = (- 1) (#) (b2 - b1);

:: RFUNCT_1:th 32
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 - (b2 + b3) = (b1 - b2) - b3;

:: RFUNCT_1:th 33
theorem
for b1 being Relation-like Function-like real-valued set holds
   1 (#) b1 = b1;

:: RFUNCT_1:th 34
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 - (b2 - b3) = (b1 - b2) + b3;

:: RFUNCT_1:th 35
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 + (b2 - b3) = (b1 + b2) - b3;

:: RFUNCT_1:th 36
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
|.b1 (#) b2.| = |.b1.| (#) |.b2.|;

:: RFUNCT_1:th 37
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being real set holds
   |.b2 (#) b1.| = (abs b2) (#) |.b1.|;

:: RFUNCT_1:th 38
theorem
for b1 being Relation-like Function-like real-valued set holds
   - b1 = (- 1) (#) b1;

:: RFUNCT_1:th 39
theorem
for b1 being Relation-like Function-like real-valued set holds
   - - b1 = b1;

:: RFUNCT_1:th 40
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
b1 - b2 = b1 + - b2;

:: RFUNCT_1:th 41
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
b1 - - b2 = b1 + b2;

:: RFUNCT_1:th 42
theorem
for b1 being Relation-like Function-like real-valued set holds
   b1 ^ ^ = b1 | proj1 (b1 ^);

:: RFUNCT_1:th 43
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
(b1 (#) b2) ^ = b1 ^ (#) (b2 ^);

:: RFUNCT_1:th 44
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being real set
      st b2 <> 0
   holds (b2 (#) b1) ^ = b2 " (#) (b1 ^);

:: RFUNCT_1:th 45
theorem
for b1 being Relation-like Function-like real-valued set holds
   (- b1) ^ = (- 1) (#) (b1 ^);

:: RFUNCT_1:th 46
theorem
for b1 being Relation-like Function-like real-valued set holds
   |.b1.| ^ = |.b1 ^.|;

:: RFUNCT_1:th 47
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
b1 / b2 = b1 (#) (b2 ^);

:: RFUNCT_1:th 48
theorem
for b1, b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   b3 (#) (b1 / b2) = (b3 (#) b1) / b2;

:: RFUNCT_1:th 49
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
(b1 / b2) (#) b2 = b1 | proj1 (b2 ^);

:: RFUNCT_1:th 50
theorem
for b1, b2, b3, b4 being Relation-like Function-like real-valued set holds
(b1 / b2) (#) (b3 / b4) = (b1 (#) b3) / (b2 (#) b4);

:: RFUNCT_1:th 51
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
(b1 / b2) ^ = (b2 | proj1 (b2 ^)) / b1;

:: RFUNCT_1:th 52
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 (#) (b2 / b3) = (b1 (#) b2) / b3;

:: RFUNCT_1:th 53
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
b1 / (b2 / b3) = (b1 (#) (b3 | proj1 (b3 ^))) / b2;

:: RFUNCT_1:th 54
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
- (b1 / b2) = (- b1) / b2 & b1 / - b2 = - (b1 / b2);

:: RFUNCT_1:th 55
theorem
for b1, b2, b3 being Relation-like Function-like real-valued set holds
(b1 / b2) + (b3 / b2) = (b1 + b3) / b2 &
 (b1 / b2) - (b3 / b2) = (b1 - b3) / b2;

:: RFUNCT_1:th 56
theorem
for b1, b2, b3, b4 being Relation-like Function-like real-valued set holds
(b1 / b2) + (b3 / b4) = ((b1 (#) b4) + (b3 (#) b2)) / (b2 (#) b4);

:: RFUNCT_1:th 57
theorem
for b1, b2, b3, b4 being Relation-like Function-like real-valued set holds
(b1 / b2) / (b3 / b4) = (b1 (#) (b4 | proj1 (b4 ^))) / (b2 (#) b3);

:: RFUNCT_1:th 58
theorem
for b1, b2, b3, b4 being Relation-like Function-like real-valued set holds
(b1 / b2) - (b3 / b4) = ((b1 (#) b4) - (b3 (#) b2)) / (b2 (#) b4);

:: RFUNCT_1:th 59
theorem
for b1, b2 being Relation-like Function-like real-valued set holds
|.b1 / b2.| = |.b1.| / |.b2.|;

:: RFUNCT_1:th 60
theorem
for b1 being set
for b2, b3 being Relation-like Function-like real-valued set holds
(b2 + b3) | b1 = (b2 | b1) + (b3 | b1) &
 (b2 + b3) | b1 = (b2 | b1) + b3 &
 (b2 + b3) | b1 = b2 + (b3 | b1);

:: RFUNCT_1:th 61
theorem
for b1 being set
for b2, b3 being Relation-like Function-like real-valued set holds
(b2 (#) b3) | b1 = (b2 | b1) (#) (b3 | b1) &
 (b2 (#) b3) | b1 = (b2 | b1) (#) b3 &
 (b2 (#) b3) | b1 = b2 (#) (b3 | b1);

:: RFUNCT_1:th 62
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set holds
   (- b2) | b1 = - (b2 | b1) &
    b2 ^ | b1 = (b2 | b1) ^ &
    |.b2.| | b1 = |.b2 | b1.|;

:: RFUNCT_1:th 63
theorem
for b1 being set
for b2, b3 being Relation-like Function-like real-valued set holds
(b2 - b3) | b1 = (b2 | b1) - (b3 | b1) &
 (b2 - b3) | b1 = (b2 | b1) - b3 &
 (b2 - b3) | b1 = b2 - (b3 | b1);

:: RFUNCT_1:th 64
theorem
for b1 being set
for b2, b3 being Relation-like Function-like real-valued set holds
(b2 / b3) | b1 = (b2 | b1) / (b3 | b1) &
 (b2 / b3) | b1 = (b2 | b1) / b3 &
 (b2 / b3) | b1 = b2 / (b3 | b1);

:: RFUNCT_1:th 65
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set
for b3 being real set holds
   (b3 (#) b2) | b1 = b3 (#) (b2 | b1);

:: RFUNCT_1:th 66
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
(b2 is total(b1, REAL) & b3 is total(b1, REAL) implies b2 + b3 is total(b1, REAL)) &
 (b2 + b3 is total(b1, REAL) implies b2 is total(b1, REAL) & b3 is total(b1, REAL)) &
 (b2 is total(b1, REAL) & b3 is total(b1, REAL) implies b2 - b3 is total(b1, REAL)) &
 (b2 - b3 is total(b1, REAL) implies b2 is total(b1, REAL) & b3 is total(b1, REAL)) &
 (b2 is total(b1, REAL) & b3 is total(b1, REAL) implies b2 (#) b3 is total(b1, REAL)) &
 (b2 (#) b3 is total(b1, REAL) implies b2 is total(b1, REAL) & b3 is total(b1, REAL));

:: RFUNCT_1:th 67
theorem
for b1 being non empty set
for b2 being real set
for b3 being Function-like Relation of b1,REAL holds
      b3 is total(b1, REAL)
   iff
      b2 (#) b3 is total(b1, REAL);

:: RFUNCT_1:th 68
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
      b2 is total(b1, REAL)
   iff
      - b2 is total(b1, REAL);

:: RFUNCT_1:th 69
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
      b2 is total(b1, REAL)
   iff
      abs b2 is total(b1, REAL);

:: RFUNCT_1:th 70
theorem
for b1 being non empty set
for b2 being Function-like Relation of b1,REAL holds
      b2 ^ is total(b1, REAL)
   iff
      b2 " {0} = {} & b2 is total(b1, REAL);

:: RFUNCT_1:th 71
theorem
for b1 being non empty set
for b2, b3 being Function-like Relation of b1,REAL holds
   b2 is total(b1, REAL) & b3 " {0} = {} & b3 is total(b1, REAL)
iff
   b2 / b3 is total(b1, REAL);

:: RFUNCT_1:th 72
theorem
for b1 being non empty set
for b2 being Element of b1
for b3, b4 being Function-like Relation of b1,REAL
      st b3 is total(b1, REAL) & b4 is total(b1, REAL)
   holds (b3 + b4) . b2 = (b3 . b2) + (b4 . b2) &
    (b3 - b4) . b2 = (b3 . b2) - (b4 . b2) &
    (b3 (#) b4) . b2 = (b3 . b2) * (b4 . b2);

:: RFUNCT_1:th 73
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being real set
for b4 being Function-like Relation of b1,REAL
      st b4 is total(b1, REAL)
   holds (b3 (#) b4) . b2 = b3 * (b4 . b2);

:: RFUNCT_1:th 74
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,REAL
      st b3 is total(b1, REAL)
   holds (- b3) . b2 = - (b3 . b2) & (abs b3) . b2 = abs (b3 . b2);

:: RFUNCT_1:th 75
theorem
for b1 being non empty set
for b2 being Element of b1
for b3 being Function-like Relation of b1,REAL
      st b3 ^ is total(b1, REAL)
   holds b3 ^ . b2 = (b3 . b2) ";

:: RFUNCT_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,REAL
      st b3 is total(b1, REAL) & b4 ^ is total(b1, REAL)
   holds (b3 / b4) . b2 = (b3 . b2) * ((b4 . b2) ");

:: RFUNCT_1:funcnot 5 => RFUNCT_1:func 5
definition
  let a1, a2 be set;
  redefine func chi(a1,a2) -> Function-like Relation of a2,REAL;
end;

:: RFUNCT_1:th 77
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,REAL holds
      b3 = chi(b1,b2)
   iff
      dom b3 = b2 &
       (for b4 being Element of b2 holds
          (b4 in b1 implies b3 . b4 = 1) & (b4 in b1 or b3 . b4 = 0));

:: RFUNCT_1:th 78
theorem
for b1 being set
for b2 being non empty set holds
   chi(b1,b2) is total(b2, REAL);

:: RFUNCT_1:th 79
theorem
for b1 being set
for b2 being non empty set
for b3 being Element of b2 holds
      b3 in b1
   iff
      (chi(b1,b2)) . b3 = 1;

:: RFUNCT_1:th 80
theorem
for b1 being set
for b2 being non empty set
for b3 being Element of b2 holds
      not b3 in b1
   iff
      (chi(b1,b2)) . b3 = 0;

:: RFUNCT_1:th 81
theorem
for b1 being set
for b2 being non empty set
for b3 being Element of b2 holds
      b3 in b2 \ b1
   iff
      (chi(b1,b2)) . b3 = 0;

:: RFUNCT_1:th 83
theorem
for b1 being non empty set
for b2 being Element of b1 holds
   (chi(b1,b1)) . b2 = 1;

:: RFUNCT_1:th 84
theorem
for b1 being set
for b2 being non empty set
for b3 being Element of b2 holds
      (chi(b1,b2)) . b3 <> 1
   iff
      (chi(b1,b2)) . b3 = 0;

:: RFUNCT_1:th 85
theorem
for b1, b2 being set
for b3 being non empty set
      st b1 misses b2
   holds (chi(b1,b3)) + chi(b2,b3) = chi(b1 \/ b2,b3);

:: RFUNCT_1:th 86
theorem
for b1, b2 being set
for b3 being non empty set holds
   (chi(b1,b3)) (#) chi(b2,b3) = chi(b1 /\ b2,b3);

:: RFUNCT_1:prednot 1 => RFUNCT_1:pred 1
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
  pred A1 is_bounded_above_on A2 means
    ex b1 being real set st
       for b2 being set
             st b2 in a2 /\ proj1 a1
          holds a1 . b2 <= b1;
end;

:: RFUNCT_1:dfs 3
definiens
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
To prove
     a1 is_bounded_above_on a2
it is sufficient to prove
  thus ex b1 being real set st
       for b2 being set
             st b2 in a2 /\ proj1 a1
          holds a1 . b2 <= b1;

:: RFUNCT_1:def 9
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being set holds
      b1 is_bounded_above_on b2
   iff
      ex b3 being real set st
         for b4 being set
               st b4 in b2 /\ proj1 b1
            holds b1 . b4 <= b3;

:: RFUNCT_1:prednot 2 => RFUNCT_1:pred 2
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
  pred A1 is_bounded_below_on A2 means
    ex b1 being real set st
       for b2 being set
             st b2 in a2 /\ proj1 a1
          holds b1 <= a1 . b2;
end;

:: RFUNCT_1:dfs 4
definiens
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
To prove
     a1 is_bounded_below_on a2
it is sufficient to prove
  thus ex b1 being real set st
       for b2 being set
             st b2 in a2 /\ proj1 a1
          holds b1 <= a1 . b2;

:: RFUNCT_1:def 10
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being set holds
      b1 is_bounded_below_on b2
   iff
      ex b3 being real set st
         for b4 being set
               st b4 in b2 /\ proj1 b1
            holds b3 <= b1 . b4;

:: RFUNCT_1:prednot 3 => RFUNCT_1:pred 3
definition
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
  pred A1 is_bounded_on A2 means
    a1 is_bounded_above_on a2 & a1 is_bounded_below_on a2;
end;

:: RFUNCT_1:dfs 5
definiens
  let a1 be Relation-like Function-like real-valued set;
  let a2 be set;
To prove
     a1 is_bounded_on a2
it is sufficient to prove
  thus a1 is_bounded_above_on a2 & a1 is_bounded_below_on a2;

:: RFUNCT_1:def 11
theorem
for b1 being Relation-like Function-like real-valued set
for b2 being set holds
      b1 is_bounded_on b2
   iff
      b1 is_bounded_above_on b2 & b1 is_bounded_below_on b2;

:: RFUNCT_1:th 90
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set holds
      b2 is_bounded_on b1
   iff
      ex b3 being real set st
         for b4 being set
               st b4 in b1 /\ proj1 b2
            holds abs (b2 . b4) <= b3;

:: RFUNCT_1:th 91
theorem
for b1, b2 being set
for b3 being Relation-like Function-like real-valued set holds
   (b1 c= b2 & b3 is_bounded_above_on b2 implies b3 is_bounded_above_on b1) & (b1 c= b2 & b3 is_bounded_below_on b2 implies b3 is_bounded_below_on b1) & (b1 c= b2 & b3 is_bounded_on b2 implies b3 is_bounded_on b1);

:: RFUNCT_1:th 92
theorem
for b1, b2 being set
for b3 being Relation-like Function-like real-valued set
      st b3 is_bounded_above_on b1 & b3 is_bounded_below_on b2
   holds b3 is_bounded_on b1 /\ b2;

:: RFUNCT_1:th 93
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set
      st b1 misses proj1 b2
   holds b2 is_bounded_on b1;

:: RFUNCT_1:th 94
theorem
for b1 being set
for b2 being real set
for b3 being Relation-like Function-like real-valued set
      st 0 = b2
   holds b2 (#) b3 is_bounded_on b1;

:: RFUNCT_1:th 95
theorem
for b1 being set
for b2 being real set
for b3 being Relation-like Function-like real-valued set holds
   (b3 is_bounded_above_on b1 & 0 <= b2 implies b2 (#) b3 is_bounded_above_on b1) &
    (b3 is_bounded_above_on b1 & b2 <= 0 implies b2 (#) b3 is_bounded_below_on b1);

:: RFUNCT_1:th 96
theorem
for b1 being set
for b2 being real set
for b3 being Relation-like Function-like real-valued set holds
   (b3 is_bounded_below_on b1 & 0 <= b2 implies b2 (#) b3 is_bounded_below_on b1) &
    (b3 is_bounded_below_on b1 & b2 <= 0 implies b2 (#) b3 is_bounded_above_on b1);

:: RFUNCT_1:th 97
theorem
for b1 being set
for b2 being real set
for b3 being Relation-like Function-like real-valued set
      st b3 is_bounded_on b1
   holds b2 (#) b3 is_bounded_on b1;

:: RFUNCT_1:th 98
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set holds
   |.b2.| is_bounded_below_on b1;

:: RFUNCT_1:th 99
theorem
for b1 being set
for b2 being Relation-like Function-like real-valued set
      st b2 is_bounded_on b1
   holds |.b2.| is_bounded_on b1 & - b2 is_bounded_on b1;

:: RFUNCT_1:th 100
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like real-valued set holds
(b3 is_bounded_above_on b1 & b4 is_bounded_above_on b2 implies b3 + b4 is_bounded_above_on b1 /\ b2) &
 (b3 is_bounded_below_on b1 & b4 is_bounded_below_on b2 implies b3 + b4 is_bounded_below_on b1 /\ b2) &
 (b3 is_bounded_on b1 & b4 is_bounded_on b2 implies b3 + b4 is_bounded_on b1 /\ b2);

:: RFUNCT_1:th 101
theorem
for b1, b2 being set
for b3, b4 being Relation-like Function-like real-valued set
      st b3 is_bounded_on b1 & b4 is_bounded_on b2
   holds b3 (#) b4 is_bounded_on b1 /\ b2 & b3 - b4 is_bounded_on b1 /\ b2;

:: RFUNCT_1:th 102
theorem
for b1, b2 being set
for b3 being Relation-like Function-like real-valued set
      st b3 is_bounded_above_on b1 & b3 is_bounded_above_on b2
   holds b3 is_bounded_above_on b1 \/ b2;

:: RFUNCT_1:th 103
theorem
for b1, b2 being set
for b3 being Relation-like Function-like real-valued set
      st b3 is_bounded_below_on b1 & b3 is_bounded_below_on b2
   holds b3 is_bounded_below_on b1 \/ b2;

:: RFUNCT_1:th 104
theorem
for b1, b2 being set
for b3 being Relation-like Function-like real-valued set
      st b3 is_bounded_on b1 & b3 is_bounded_on b2
   holds b3 is_bounded_on b1 \/ b2;

:: RFUNCT_1:th 105
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,REAL
      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;

:: RFUNCT_1:th 106
theorem
for b1 being set
for b2 being non empty set
for b3 being real set
for b4 being Function-like Relation of b2,REAL
      st b4 is_constant_on b1
   holds b3 (#) b4 is_constant_on b1;

:: RFUNCT_1:th 107
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,REAL
      st b3 is_constant_on b1
   holds abs b3 is_constant_on b1 & - b3 is_constant_on b1;

:: RFUNCT_1:th 108
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,REAL
      st b3 is_constant_on b1
   holds b3 is_bounded_on b1;

:: RFUNCT_1:th 109
theorem
for b1 being set
for b2 being non empty set
for b3 being Function-like Relation of b2,REAL
      st b3 is_constant_on b1
   holds (for b4 being real set holds
       b4 (#) b3 is_bounded_on b1) &
    - b3 is_bounded_on b1 &
    abs b3 is_bounded_on b1;

:: RFUNCT_1:th 110
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,REAL holds
(b4 is_bounded_above_on b1 & b5 is_constant_on b2 implies b4 + b5 is_bounded_above_on b1 /\ b2) &
 (b4 is_bounded_below_on b1 & b5 is_constant_on b2 implies b4 + b5 is_bounded_below_on b1 /\ b2) &
 (b4 is_bounded_on b1 & b5 is_constant_on b2 implies b4 + b5 is_bounded_on b1 /\ b2);

:: RFUNCT_1:th 111
theorem
for b1, b2 being set
for b3 being non empty set
for b4, b5 being Function-like Relation of b3,REAL holds
(b4 is_bounded_above_on b1 & b5 is_constant_on b2 implies b4 - b5 is_bounded_above_on b1 /\ b2) &
 (b4 is_bounded_below_on b1 & b5 is_constant_on b2 implies b4 - b5 is_bounded_below_on b1 /\ b2) &
 (b4 is_bounded_on b1 & b5 is_constant_on b2 implies b4 - b5 is_bounded_on b1 /\ b2 & b5 - b4 is_bounded_on b1 /\ b2 & b4 (#) b5 is_bounded_on b1 /\ b2);