Article RFUNCT_2, MML version 4.99.1005

:: RFUNCT_2:th 2
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set holds
   (b1 | b3) * (b2 | (b1 .: b3)) = (b1 * b2) | b3;

:: RFUNCT_2:th 3
theorem
for b1, b2 being Relation-like Function-like set
for b3, b4 being set holds
(b1 | b3) * (b2 | b4) = (b1 * b2) | (b3 /\ (b1 " b4));

:: RFUNCT_2:th 4
theorem
for b1, b2 being Relation-like Function-like set
for b3 being set holds
      b3 c= proj1 (b1 * b2)
   iff
      b3 c= proj1 b1 & b1 .: b3 c= proj1 b2;

:: RFUNCT_2:th 5
theorem
for b1 being Relation-like Function-like set
for b2 being set holds
   (b1 | b2) .: b2 = b1 .: b2;

:: RFUNCT_2:th 6
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
   b1 = b2 - b3
iff
   for b4 being Element of NAT holds
      b1 . b4 = (b2 . b4) - (b3 . b4);

:: RFUNCT_2:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
   rng (b1 ^\ b2) c= rng b1;

:: RFUNCT_2:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b3
   holds b1 . b2 in dom b3;

:: RFUNCT_2:th 9
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b1 in rng b2
   iff
      ex b3 being Element of NAT st
         b1 = b2 . b3;

:: RFUNCT_2:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
   b1 . b2 in rng b1;

:: RFUNCT_2:th 11
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is subsequence of b2
   holds rng b1 c= rng b2;

:: RFUNCT_2:th 12
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is subsequence of b2 & b2 is non-empty
   holds b1 is non-empty;

:: RFUNCT_2:th 13
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL holds
   (b1 + b2) * b3 = (b1 * b3) + (b2 * b3) &
    (b1 - b2) * b3 = (b1 * b3) - (b2 * b3) &
    (b1 (#) b2) * b3 = (b1 * b3) (#) (b2 * b3);

:: RFUNCT_2:th 14
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL holds
   (b1 (#) b2) * b3 = b1 (#) (b2 * b3);

:: RFUNCT_2:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL holds
   (- b1) * b2 = - (b1 * b2) & (abs b1) * b2 = abs (b1 * b2);

:: RFUNCT_2:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL holds
   (b1 * b2) " = b1 " * b2;

:: RFUNCT_2:th 17
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL holds
   (b1 /" b2) * b3 = (b1 * b3) /" (b2 * b3);

:: RFUNCT_2:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent &
         (for b2 being Element of NAT holds
            b1 . b2 <= 0)
   holds lim b1 <= 0;

:: RFUNCT_2:th 19
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st for b3 being Element of NAT holds
           b2 . b3 in b1
   holds rng b2 c= b1;

:: RFUNCT_2:funcnot 1 => RFUNCT_2:func 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be Function-like quasi_total Relation of NAT,REAL;
  assume rng a2 c= dom a1;
  func A1 * A2 -> Function-like quasi_total Relation of NAT,REAL equals
    a2 * a1;
end;

:: RFUNCT_2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
      st rng b2 c= dom b1
   holds b1 * b2 = b2 * b1;

:: RFUNCT_2:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b3
   holds (b3 * b1) . b2 = b3 . (b1 . b2);

:: RFUNCT_2:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b3
   holds (b3 * b1) ^\ b2 = b3 * (b1 ^\ b2);

:: RFUNCT_2:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st rng b1 c= (dom b2) /\ dom b3
   holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
    (b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
    (b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);

:: RFUNCT_2:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like Relation of REAL,REAL
for b3 being real set
      st rng b1 c= dom b2
   holds (b3 (#) b2) * b1 = b3 (#) (b2 * b1);

:: RFUNCT_2:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b2
   holds abs (b2 * b1) = (abs b2) * b1 & - (b2 * b1) = (- b2) * b1;

:: RFUNCT_2:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like Relation of REAL,REAL
      st rng b1 c= dom (b2 ^)
   holds b2 * b1 is non-empty;

:: RFUNCT_2:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like Relation of REAL,REAL
      st rng b1 c= dom (b2 ^)
   holds b2 ^ * b1 = (b2 * b1) ";

:: RFUNCT_2:th 28
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like natural-valued quasi_total increasing Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b3
   holds (b3 * b1) * b2 = b3 * (b1 * b2);

:: RFUNCT_2:th 29
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b3 & b2 is subsequence of b1
   holds b3 * b2 is subsequence of b3 * b1;

:: RFUNCT_2:th 30
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like Relation of REAL,REAL
      st b3 is total(REAL, REAL)
   holds (b3 * b1) . b2 = b3 . (b1 . b2);

:: RFUNCT_2:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like Relation of REAL,REAL
      st b3 is total(REAL, REAL)
   holds b3 * (b1 ^\ b2) = (b3 * b1) ^\ b2;

:: RFUNCT_2:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st b2 is total(REAL, REAL) & b3 is total(REAL, REAL)
   holds (b2 + b3) * b1 = (b2 * b1) + (b3 * b1) &
    (b2 - b3) * b1 = (b2 * b1) - (b3 * b1) &
    (b2 (#) b3) * b1 = (b2 * b1) (#) (b3 * b1);

:: RFUNCT_2:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st b3 is total(REAL, REAL)
   holds (b1 (#) b3) * b2 = b1 (#) (b3 * b2);

:: RFUNCT_2:th 34
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st rng b2 c= dom (b3 | b1)
   holds (b3 | b1) * b2 = b3 * b2;

:: RFUNCT_2:th 35
theorem
for b1, b2 being set
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being Function-like Relation of REAL,REAL
      st rng b3 c= dom (b4 | b1) &
         (rng b3 c= dom (b4 | b2) or b1 c= b2)
   holds (b4 | b1) * b3 = (b4 | b2) * b3;

:: RFUNCT_2:th 36
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st rng b2 c= dom (b3 | b1)
   holds abs ((b3 | b1) * b2) = ((abs b3) | b1) * b2;

:: RFUNCT_2:th 37
theorem
for b1 being set
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
      st rng b2 c= dom (b3 | b1) & b3 " {0} = {}
   holds (b3 ^ | b1) * b2 = ((b3 | b1) * b2) ";

:: RFUNCT_2:th 38
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like Relation of REAL,REAL
      st rng b1 c= dom b2
   holds b2 .: rng b1 = rng (b2 * b1);

:: RFUNCT_2:th 39
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2, b3 being Function-like Relation of REAL,REAL
      st rng b1 c= dom (b2 * b3)
   holds b2 * (b3 * b1) = (b2 * b3) * b1;

:: RFUNCT_2:funcreg 1
registration
  let a1 be set;
  let a2 be Relation-like Function-like one-to-one set;
  cluster a2 | a1 -> Relation-like one-to-one;
end;

:: RFUNCT_2:th 40
theorem
for b1 being set
for b2 being Relation-like Function-like one-to-one set holds
   (b2 | b1) " = b2 " | (b2 .: b1);

:: RFUNCT_2:th 41
theorem
for b1 being Function-like Relation of REAL,REAL
      st rng b1 is bounded & upper_bound rng b1 = lower_bound rng b1
   holds b1 is_constant_on dom b1;

:: RFUNCT_2:th 42
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 c= dom b2 & b2 .: b1 is bounded & upper_bound (b2 .: b1) = lower_bound (b2 .: b1)
   holds b2 is_constant_on b1;

:: RFUNCT_2:prednot 1 => RFUNCT_2:pred 1
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_increasing_on A2 means
    for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b1 < a1 . b2;
end;

:: RFUNCT_2:dfs 2
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_increasing_on a2
it is sufficient to prove
  thus for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b1 < a1 . b2;

:: RFUNCT_2:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_increasing_on b2
   iff
      for b3, b4 being Element of REAL
            st b3 in b2 /\ dom b1 & b4 in b2 /\ dom b1 & b3 < b4
         holds b1 . b3 < b1 . b4;

:: RFUNCT_2:prednot 2 => RFUNCT_2:pred 2
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_decreasing_on A2 means
    for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b2 < a1 . b1;
end;

:: RFUNCT_2:dfs 3
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_decreasing_on a2
it is sufficient to prove
  thus for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b2 < a1 . b1;

:: RFUNCT_2:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_decreasing_on b2
   iff
      for b3, b4 being Element of REAL
            st b3 in b2 /\ dom b1 & b4 in b2 /\ dom b1 & b3 < b4
         holds b1 . b4 < b1 . b3;

:: RFUNCT_2:prednot 3 => RFUNCT_2:pred 3
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_non_decreasing_on A2 means
    for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b1 <= a1 . b2;
end;

:: RFUNCT_2:dfs 4
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_non_decreasing_on a2
it is sufficient to prove
  thus for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b1 <= a1 . b2;

:: RFUNCT_2:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_non_decreasing_on b2
   iff
      for b3, b4 being Element of REAL
            st b3 in b2 /\ dom b1 & b4 in b2 /\ dom b1 & b3 < b4
         holds b1 . b3 <= b1 . b4;

:: RFUNCT_2:prednot 4 => RFUNCT_2:pred 4
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_non_increasing_on A2 means
    for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b2 <= a1 . b1;
end;

:: RFUNCT_2:dfs 5
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_non_increasing_on a2
it is sufficient to prove
  thus for b1, b2 being Element of REAL
          st b1 in a2 /\ dom a1 & b2 in a2 /\ dom a1 & b1 < b2
       holds a1 . b2 <= a1 . b1;

:: RFUNCT_2:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_non_increasing_on b2
   iff
      for b3, b4 being Element of REAL
            st b3 in b2 /\ dom b1 & b4 in b2 /\ dom b1 & b3 < b4
         holds b1 . b4 <= b1 . b3;

:: RFUNCT_2:prednot 5 => RFUNCT_2:pred 5
definition
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
  pred A1 is_monotone_on A2 means
    (not a1 is_non_decreasing_on a2) implies a1 is_non_increasing_on a2;
end;

:: RFUNCT_2:dfs 6
definiens
  let a1 be Function-like Relation of REAL,REAL;
  let a2 be set;
To prove
     a1 is_monotone_on a2
it is sufficient to prove
  thus (not a1 is_non_decreasing_on a2) implies a1 is_non_increasing_on a2;

:: RFUNCT_2:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being set holds
      b1 is_monotone_on b2
   iff
      (b1 is_non_decreasing_on b2 or b1 is_non_increasing_on b2);

:: RFUNCT_2:th 48
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_non_decreasing_on b1
   iff
      for b3, b4 being Element of REAL
            st b3 in b1 /\ dom b2 & b4 in b1 /\ dom b2 & b3 <= b4
         holds b2 . b3 <= b2 . b4;

:: RFUNCT_2:th 49
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_non_increasing_on b1
   iff
      for b3, b4 being Element of REAL
            st b3 in b1 /\ dom b2 & b4 in b1 /\ dom b2 & b3 <= b4
         holds b2 . b4 <= b2 . b3;

:: RFUNCT_2:th 50
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_increasing_on b1
   iff
      b2 | b1 is_increasing_on b1;

:: RFUNCT_2:th 51
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_decreasing_on b1
   iff
      b2 | b1 is_decreasing_on b1;

:: RFUNCT_2:th 52
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_non_decreasing_on b1
   iff
      b2 | b1 is_non_decreasing_on b1;

:: RFUNCT_2:th 53
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
      b2 is_non_increasing_on b1
   iff
      b2 | b1 is_non_increasing_on b1;

:: RFUNCT_2:th 54
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b1 misses dom b2
   holds b2 is_increasing_on b1 & b2 is_decreasing_on b1 & b2 is_non_decreasing_on b1 & b2 is_non_increasing_on b1 & b2 is_monotone_on b1;

:: RFUNCT_2:th 55
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_increasing_on b1
   holds b2 is_non_decreasing_on b1;

:: RFUNCT_2:th 56
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_decreasing_on b1
   holds b2 is_non_increasing_on b1;

:: RFUNCT_2:th 57
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_constant_on b1
   holds b2 is_non_decreasing_on b1;

:: RFUNCT_2:th 58
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_constant_on b1
   holds b2 is_non_increasing_on b1;

:: RFUNCT_2:th 59
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b3 is_non_decreasing_on b1 & b3 is_non_increasing_on b2
   holds b3 is_constant_on b1 /\ b2;

:: RFUNCT_2:th 60
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b1 c= b2 & b3 is_increasing_on b2
   holds b3 is_increasing_on b1;

:: RFUNCT_2:th 61
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b1 c= b2 & b3 is_decreasing_on b2
   holds b3 is_decreasing_on b1;

:: RFUNCT_2:th 62
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b1 c= b2 & b3 is_non_decreasing_on b2
   holds b3 is_non_decreasing_on b1;

:: RFUNCT_2:th 63
theorem
for b1, b2 being set
for b3 being Function-like Relation of REAL,REAL
      st b1 c= b2 & b3 is_non_increasing_on b2
   holds b3 is_non_increasing_on b1;

:: RFUNCT_2:th 64
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_increasing_on b1 & 0 < b2 implies b2 (#) b3 is_increasing_on b1) &
    (b2 = 0 implies b2 (#) b3 is_constant_on b1) &
    (b3 is_increasing_on b1 & b2 < 0 implies b2 (#) b3 is_decreasing_on b1);

:: RFUNCT_2:th 65
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_decreasing_on b1 & 0 < b2 implies b2 (#) b3 is_decreasing_on b1) &
    (b3 is_decreasing_on b1 & b2 < 0 implies b2 (#) b3 is_increasing_on b1);

:: RFUNCT_2:th 66
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_non_decreasing_on b1 & 0 <= b2 implies b2 (#) b3 is_non_decreasing_on b1) &
    (b3 is_non_decreasing_on b1 & b2 <= 0 implies b2 (#) b3 is_non_increasing_on b1);

:: RFUNCT_2:th 67
theorem
for b1 being set
for b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
   (b3 is_non_increasing_on b1 & 0 <= b2 implies b2 (#) b3 is_non_increasing_on b1) &
    (b3 is_non_increasing_on b1 & b2 <= 0 implies b2 (#) b3 is_non_decreasing_on b1);

:: RFUNCT_2:th 68
theorem
for b1, b2 being set
for b3 being Element of REAL
for b4, b5 being Function-like Relation of REAL,REAL
      st b3 in (b1 /\ b2) /\ dom (b4 + b5)
   holds b3 in b1 /\ dom b4 & b3 in b2 /\ dom b5;

:: RFUNCT_2:th 69
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL holds
(b3 is_increasing_on b1 & b4 is_increasing_on b2 implies b3 + b4 is_increasing_on b1 /\ b2) &
 (b3 is_decreasing_on b1 & b4 is_decreasing_on b2 implies b3 + b4 is_decreasing_on b1 /\ b2) &
 (b3 is_non_decreasing_on b1 & b4 is_non_decreasing_on b2 implies b3 + b4 is_non_decreasing_on b1 /\ b2) &
 (b3 is_non_increasing_on b1 & b4 is_non_increasing_on b2 implies b3 + b4 is_non_increasing_on b1 /\ b2);

:: RFUNCT_2:th 70
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL holds
(b3 is_increasing_on b1 & b4 is_constant_on b2 implies b3 + b4 is_increasing_on b1 /\ b2) &
 (b3 is_decreasing_on b1 & b4 is_constant_on b2 implies b3 + b4 is_decreasing_on b1 /\ b2);

:: RFUNCT_2:th 71
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_increasing_on b1 & b4 is_non_decreasing_on b2
   holds b3 + b4 is_increasing_on b1 /\ b2;

:: RFUNCT_2:th 72
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_non_increasing_on b1 & b4 is_constant_on b2
   holds b3 + b4 is_non_increasing_on b1 /\ b2;

:: RFUNCT_2:th 73
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_decreasing_on b1 & b4 is_non_increasing_on b2
   holds b3 + b4 is_decreasing_on b1 /\ b2;

:: RFUNCT_2:th 74
theorem
for b1, b2 being set
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_non_decreasing_on b1 & b4 is_constant_on b2
   holds b3 + b4 is_non_decreasing_on b1 /\ b2;

:: RFUNCT_2:th 75
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
   b2 is_increasing_on {b1};

:: RFUNCT_2:th 76
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
   b2 is_decreasing_on {b1};

:: RFUNCT_2:th 77
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
   b2 is_non_decreasing_on {b1};

:: RFUNCT_2:th 78
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL holds
   b2 is_non_increasing_on {b1};

:: RFUNCT_2:th 79
theorem
for b1 being Element of bool REAL holds
   id b1 is_increasing_on b1;

:: RFUNCT_2:th 80
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_increasing_on b1
   holds - b2 is_decreasing_on b1;

:: RFUNCT_2:th 81
theorem
for b1 being set
for b2 being Function-like Relation of REAL,REAL
      st b2 is_non_decreasing_on b1
   holds - b2 is_non_increasing_on b1;

:: RFUNCT_2:th 82
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
      st (b3 is_increasing_on [.b1,b2.] or b3 is_decreasing_on [.b1,b2.])
   holds b3 | [.b1,b2.] is one-to-one;

:: RFUNCT_2:th 83
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
      st b3 is_increasing_on [.b1,b2.]
   holds (b3 | [.b1,b2.]) " is_increasing_on b3 .: [.b1,b2.];

:: RFUNCT_2:th 84
theorem
for b1, b2 being Element of REAL
for b3 being Function-like one-to-one Relation of REAL,REAL
      st b3 is_decreasing_on [.b1,b2.]
   holds (b3 | [.b1,b2.]) " is_decreasing_on b3 .: [.b1,b2.];