Article SEQM_3, MML version 4.99.1005

:: SEQM_3:attrnot 1 => SEQM_3:attr 1
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is increasing means
    for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
       holds a1 . b1 < a1 . b2;
end;

:: SEQM_3:dfs 1
definiens
  let a1 be Function-like Relation of NAT,REAL;
To prove
     a1 is increasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
       holds a1 . b1 < a1 . b2;

:: SEQM_3:def 1
theorem
for b1 being Function-like Relation of NAT,REAL holds
      b1 is increasing
   iff
      for b2, b3 being Element of NAT
            st b2 in proj1 b1 & b3 in proj1 b1 & b2 < b3
         holds b1 . b2 < b1 . b3;

:: SEQM_3:attrnot 2 => SEQM_3:attr 2
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is decreasing means
    for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
       holds a1 . b2 < a1 . b1;
end;

:: SEQM_3:dfs 2
definiens
  let a1 be Function-like Relation of NAT,REAL;
To prove
     a1 is decreasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 < b2
       holds a1 . b2 < a1 . b1;

:: SEQM_3:def 2
theorem
for b1 being Function-like Relation of NAT,REAL holds
      b1 is decreasing
   iff
      for b2, b3 being Element of NAT
            st b2 in proj1 b1 & b3 in proj1 b1 & b2 < b3
         holds b1 . b3 < b1 . b2;

:: SEQM_3:attrnot 3 => SEQM_3:attr 3
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is non-decreasing means
    for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
       holds a1 . b1 <= a1 . b2;
end;

:: SEQM_3:dfs 3
definiens
  let a1 be Function-like Relation of NAT,REAL;
To prove
     a1 is non-decreasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
       holds a1 . b1 <= a1 . b2;

:: SEQM_3:def 3
theorem
for b1 being Function-like Relation of NAT,REAL holds
      b1 is non-decreasing
   iff
      for b2, b3 being Element of NAT
            st b2 in proj1 b1 & b3 in proj1 b1 & b2 <= b3
         holds b1 . b2 <= b1 . b3;

:: SEQM_3:attrnot 4 => SEQM_3:attr 4
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is non-increasing means
    for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
       holds a1 . b2 <= a1 . b1;
end;

:: SEQM_3:dfs 4
definiens
  let a1 be Function-like Relation of NAT,REAL;
To prove
     a1 is non-increasing
it is sufficient to prove
  thus for b1, b2 being Element of NAT
          st b1 in proj1 a1 & b2 in proj1 a1 & b1 <= b2
       holds a1 . b2 <= a1 . b1;

:: SEQM_3:def 4
theorem
for b1 being Function-like Relation of NAT,REAL holds
      b1 is non-increasing
   iff
      for b2, b3 being Element of NAT
            st b2 in proj1 b1 & b3 in proj1 b1 & b2 <= b3
         holds b1 . b3 <= b1 . b2;

:: SEQM_3:attrnot 5 => FUNCT_1:attr 3
definition
  let a1 be Relation-like Function-like set;
  attr a1 is constant means
    ex b1 being real set st
       for b2 being Element of NAT holds
          a1 . b2 = b1;
end;

:: SEQM_3:dfs 5
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is constant
it is sufficient to prove
  thus ex b1 being real set st
       for b2 being Element of NAT holds
          a1 . b2 = b1;

:: SEQM_3:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is constant
   iff
      ex b2 being real set st
         for b3 being Element of NAT holds
            b1 . b3 = b2;

:: SEQM_3:attrnot 6 => SEQM_3:attr 5
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  attr a1 is monotone means
    (a1 is not non-decreasing) implies a1 is non-increasing;
end;

:: SEQM_3:dfs 6
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is monotone
it is sufficient to prove
  thus (a1 is not non-decreasing) implies a1 is non-increasing;

:: SEQM_3:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is monotone
   iff
      (b1 is non-decreasing or b1 is non-increasing);

:: SEQM_3:th 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is increasing
   iff
      for b2, b3 being Element of NAT
            st b2 < b3
         holds b1 . b2 < b1 . b3;

:: SEQM_3:th 8
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is increasing
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 < b1 . ((b2 + 1) + b3);

:: SEQM_3:th 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is decreasing
   iff
      for b2, b3 being Element of NAT holds
      b1 . ((b2 + 1) + b3) < b1 . b2;

:: SEQM_3:th 10
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is decreasing
   iff
      for b2, b3 being Element of NAT
            st b2 < b3
         holds b1 . b3 < b1 . b2;

:: SEQM_3:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-decreasing
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 <= b1 . (b2 + b3);

:: SEQM_3:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-decreasing
   iff
      for b2, b3 being Element of NAT
            st b2 <= b3
         holds b1 . b2 <= b1 . b3;

:: SEQM_3:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-increasing
   iff
      for b2, b3 being Element of NAT holds
      b1 . (b2 + b3) <= b1 . b2;

:: SEQM_3:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-increasing
   iff
      for b2, b3 being Element of NAT
            st b2 <= b3
         holds b1 . b3 <= b1 . b2;

:: SEQM_3:th 15
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is constant
   iff
      ex b2 being real set st
         proj2 b1 = {b2};

:: SEQM_3:th 16
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is constant
   iff
      for b2 being Element of NAT holds
         b1 . b2 = b1 . (b2 + 1);

:: SEQM_3:th 17
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is constant
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 = b1 . (b2 + b3);

:: SEQM_3:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is constant
   iff
      for b2, b3 being Element of NAT holds
      b1 . b2 = b1 . b3;

:: SEQM_3:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is increasing
for b2 being Element of NAT
      st 0 < b2
   holds b1 . 0 < b1 . b2;

:: SEQM_3:th 20
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is decreasing
for b2 being Element of NAT
      st 0 < b2
   holds b1 . b2 < b1 . 0;

:: SEQM_3:th 21
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-decreasing
for b2 being Element of NAT holds
   b1 . 0 <= b1 . b2;

:: SEQM_3:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-increasing
for b2 being Element of NAT holds
   b1 . b2 <= b1 . 0;

:: SEQM_3:condreg 1
registration
  cluster Function-like increasing -> non-decreasing (Relation of NAT,REAL);
end;

:: SEQM_3:condreg 2
registration
  cluster Function-like decreasing -> non-increasing (Relation of NAT,REAL);
end;

:: SEQM_3:condreg 3
registration
  cluster Function-like constant -> non-decreasing non-increasing (Relation of NAT,REAL);
end;

:: SEQM_3:condreg 4
registration
  cluster Function-like non-decreasing non-increasing -> constant (Relation of NAT,REAL);
end;

:: SEQM_3:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is increasing
   holds b1 is non-decreasing;

:: SEQM_3:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is decreasing
   holds b1 is non-increasing;

:: SEQM_3:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant
   holds b1 is non-decreasing;

:: SEQM_3:th 26
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant
   holds b1 is non-increasing;

:: SEQM_3:th 27
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b1 is non-increasing
   holds b1 is constant;

:: SEQM_3:attrnot 7 => VALUED_0:attr 6
notation
  let a1 be Relation-like set;
  synonym natural-yielding for natural-valued;
end;

:: SEQM_3:exreg 1
registration
  cluster Relation-like Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued increasing Relation of NAT,REAL;
end;

:: SEQM_3:modenot 1
definition
  mode Seq_of_Nat is Function-like quasi_total natural-valued Relation of NAT,REAL;
end;

:: SEQM_3:funcnot 1 => SEQM_3:func 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  let a2 be Element of NAT;
  func A1 ^\ A2 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT holds
       it . b1 = a1 . (b1 + a2);
end;

:: SEQM_3:def 9
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
      b3 = b1 ^\ b2
   iff
      for b4 being Element of NAT holds
         b3 . b4 = b1 . (b4 + b2);

:: SEQM_3:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is Function-like quasi_total natural-valued increasing Relation of NAT,REAL
   iff
      b1 is increasing &
       (for b2 being Element of NAT holds
          b1 . b2 is Element of NAT);

:: SEQM_3:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b3 being Element of NAT holds
   (b2 * b1) . b3 = b1 . (b2 . b3);

:: SEQM_3:funcnot 2 => SEQM_3:func 2
definition
  let a1 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
  let a2 be Function-like quasi_total Relation of NAT,REAL;
  redefine func a2 * a1 -> Function-like quasi_total Relation of NAT,REAL;
end;

:: SEQM_3:funcnot 3 => SEQM_3:func 3
definition
  let a1, a2 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
  redefine func a2 * a1 -> Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
end;

:: SEQM_3:funcreg 1
registration
  let a1 be Function-like quasi_total natural-valued increasing Relation of NAT,REAL;
  let a2 be Element of NAT;
  cluster a1 ^\ a2 -> Function-like quasi_total natural-valued increasing;
end;

:: SEQM_3:modenot 2 => SEQM_3:mode 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  mode subsequence of A1 -> Function-like quasi_total Relation of NAT,REAL means
    ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       it = a1 * b1;
end;

:: SEQM_3:dfs 8
definiens
  let a1, a2 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a2 is subsequence of a1
it is sufficient to prove
  thus ex b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
       a2 = a1 * b1;

:: SEQM_3:def 10
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 is subsequence of b1
iff
   ex b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL st
      b2 = b1 * b3;

:: SEQM_3:attrnot 8 => SEQM_3:attr 1
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is increasing means
    for b1 being Element of NAT holds
       a1 . b1 < a1 . (b1 + 1);
end;

:: SEQM_3:dfs 9
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is increasing
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a1 . b1 < a1 . (b1 + 1);

:: SEQM_3:def 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is increasing
   iff
      for b2 being Element of NAT holds
         b1 . b2 < b1 . (b2 + 1);

:: SEQM_3:attrnot 9 => SEQM_3:attr 2
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is decreasing means
    for b1 being Element of NAT holds
       a1 . (b1 + 1) < a1 . b1;
end;

:: SEQM_3:dfs 10
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is decreasing
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a1 . (b1 + 1) < a1 . b1;

:: SEQM_3:def 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is decreasing
   iff
      for b2 being Element of NAT holds
         b1 . (b2 + 1) < b1 . b2;

:: SEQM_3:attrnot 10 => SEQM_3:attr 3
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is non-decreasing means
    for b1 being Element of NAT holds
       a1 . b1 <= a1 . (b1 + 1);
end;

:: SEQM_3:dfs 11
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is non-decreasing
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a1 . b1 <= a1 . (b1 + 1);

:: SEQM_3:def 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-decreasing
   iff
      for b2 being Element of NAT holds
         b1 . b2 <= b1 . (b2 + 1);

:: SEQM_3:attrnot 11 => SEQM_3:attr 4
definition
  let a1 be Function-like Relation of NAT,REAL;
  attr a1 is non-increasing means
    for b1 being Element of NAT holds
       a1 . (b1 + 1) <= a1 . b1;
end;

:: SEQM_3:dfs 12
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is non-increasing
it is sufficient to prove
  thus for b1 being Element of NAT holds
       a1 . (b1 + 1) <= a1 . b1;

:: SEQM_3:def 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is non-increasing
   iff
      for b2 being Element of NAT holds
         b1 . (b2 + 1) <= b1 . b2;

:: SEQM_3:th 33
theorem
for b1 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL
for b2 being Element of NAT holds
   b2 <= b1 . b2;

:: SEQM_3:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   b1 ^\ 0 = b1;

:: SEQM_3:th 35
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
   (b3 ^\ b1) ^\ b2 = (b3 ^\ b2) ^\ b1;

:: SEQM_3:th 36
theorem
for b1, b2 being Element of NAT
for b3 being Function-like quasi_total Relation of NAT,REAL holds
   (b3 ^\ b1) ^\ b2 = b3 ^\ (b1 + b2);

:: SEQM_3:th 37
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 + b3) ^\ b1 = (b2 ^\ b1) + (b3 ^\ b1);

:: SEQM_3:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (- b2) ^\ b1 = - (b2 ^\ b1);

:: SEQM_3:th 39
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 - b3) ^\ b1 = (b2 ^\ b1) - (b3 ^\ b1);

:: SEQM_3:th 40
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-empty
   holds b2 ^\ b1 is non-empty;

:: SEQM_3:th 41
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 " ^\ b1 = (b2 ^\ b1) ";

:: SEQM_3:th 42
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 (#) b3) ^\ b1 = (b2 ^\ b1) (#) (b3 ^\ b1);

:: SEQM_3:th 43
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL holds
(b2 /" b3) ^\ b1 = (b2 ^\ b1) /" (b3 ^\ b1);

:: SEQM_3:th 44
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 (#) b3) ^\ b1 = b2 (#) (b3 ^\ b1);

:: SEQM_3:th 45
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like quasi_total natural-valued increasing Relation of NAT,REAL holds
   (b2 * b3) ^\ b1 = b2 * (b3 ^\ b1);

:: SEQM_3:th 46
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   b1 is subsequence of b1;

:: SEQM_3:th 47
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 ^\ b1 is subsequence of b2;

:: SEQM_3:th 48
theorem
for b1, b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b1 is subsequence of b2 & b2 is subsequence of b3
   holds b1 is subsequence of b3;

:: SEQM_3:th 49
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is increasing & b2 is subsequence of b1
   holds b2 is increasing;

:: SEQM_3:th 50
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is decreasing & b2 is subsequence of b1
   holds b2 is decreasing;

:: SEQM_3:th 51
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b2 is subsequence of b1
   holds b2 is non-decreasing;

:: SEQM_3:th 52
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing & b2 is subsequence of b1
   holds b2 is non-increasing;

:: SEQM_3:th 53
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is monotone & b2 is subsequence of b1
   holds b2 is monotone;

:: SEQM_3:th 54
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant & b2 is subsequence of b1
   holds b2 is constant;

:: SEQM_3:th 55
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant & b2 is subsequence of b1
   holds b1 = b2;

:: SEQM_3:th 56
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b2 is subsequence of b1
   holds b2 is bounded_above;

:: SEQM_3:th 57
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b2 is subsequence of b1
   holds b2 is bounded_below;

:: SEQM_3:th 58
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded & b2 is subsequence of b1
   holds b2 is bounded;

:: SEQM_3:th 59
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is increasing & 0 < b1 implies b1 (#) b2 is increasing) &
    (0 = b1 implies b1 (#) b2 is constant) &
    (b2 is increasing & b1 < 0 implies b1 (#) b2 is decreasing);

:: SEQM_3:th 60
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is decreasing & 0 < b1 implies b1 (#) b2 is decreasing) &
    (b2 is decreasing & b1 < 0 implies b1 (#) b2 is increasing);

:: SEQM_3:th 61
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is non-decreasing & 0 <= b1 implies b1 (#) b2 is non-decreasing) &
    (b2 is non-decreasing & b1 <= 0 implies b1 (#) b2 is non-increasing);

:: SEQM_3:th 62
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is non-increasing & 0 <= b1 implies b1 (#) b2 is non-increasing) &
    (b2 is non-increasing & b1 <= 0 implies b1 (#) b2 is non-decreasing);

:: SEQM_3:th 63
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is increasing & b2 is increasing implies b1 + b2 is increasing) &
 (b1 is decreasing & b2 is decreasing implies b1 + b2 is decreasing) &
 (b1 is non-decreasing & b2 is non-decreasing implies b1 + b2 is non-decreasing) &
 (b1 is non-increasing & b2 is non-increasing implies b1 + b2 is non-increasing);

:: SEQM_3:th 64
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is increasing & b2 is constant implies b1 + b2 is increasing) &
 (b1 is decreasing & b2 is constant implies b1 + b2 is decreasing) &
 (b1 is non-decreasing & b2 is constant implies b1 + b2 is non-decreasing) &
 (b1 is non-increasing & b2 is constant implies b1 + b2 is non-increasing);

:: SEQM_3:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant
   holds (for b2 being real set holds
       b2 (#) b1 is constant) &
    - b1 is constant &
    abs b1 is constant;

:: SEQM_3:th 66
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant & b2 is constant
   holds b1 (#) b2 is constant & b1 + b2 is constant;

:: SEQM_3:th 67
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant & b2 is constant
   holds b1 - b2 is constant;

:: SEQM_3:th 68
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is bounded_above & 0 < b1 implies b1 (#) b2 is bounded_above) &
    (0 = b1 implies b1 (#) b2 is bounded) &
    (b2 is bounded_above & b1 < 0 implies b1 (#) b2 is bounded_below);

:: SEQM_3:th 69
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (b2 is bounded_below & 0 < b1 implies b1 (#) b2 is bounded_below) &
    (b2 is bounded_below & b1 < 0 implies b1 (#) b2 is bounded_above);

:: SEQM_3:th 70
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   (b1 is bounded implies for b2 being real set holds
       b2 (#) b1 is bounded) &
    (b1 is bounded implies - b1 is bounded) &
    (b1 is bounded implies abs b1 is bounded) &
    (abs b1 is bounded implies b1 is bounded);

:: SEQM_3:th 71
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is bounded_above implies b1 + b2 is bounded_above) &
 (b1 is bounded_below & b2 is bounded_below implies b1 + b2 is bounded_below) &
 (b1 is bounded & b2 is bounded implies b1 + b2 is bounded);

:: SEQM_3:th 72
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded & b2 is bounded
   holds b1 (#) b2 is bounded & b1 - b2 is bounded;

:: SEQM_3:th 73
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant
   holds b1 is bounded;

:: SEQM_3:th 74
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is constant
   holds (for b2 being real set holds
       b2 (#) b1 is bounded) &
    - b1 is bounded &
    abs b1 is bounded;

:: SEQM_3:th 75
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is constant implies b1 + b2 is bounded_above) &
 (b1 is bounded_below & b2 is constant implies b1 + b2 is bounded_below) &
 (b1 is bounded & b2 is constant implies b1 + b2 is bounded);

:: SEQM_3:th 76
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 is bounded_above & b2 is constant implies b1 - b2 is bounded_above) &
 (b1 is bounded_below & b2 is constant implies b1 - b2 is bounded_below) &
 (b1 is bounded & b2 is constant implies b1 - b2 is bounded & b2 - b1 is bounded & b1 (#) b2 is bounded);

:: SEQM_3:th 77
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b2 is non-increasing
   holds b1 + b2 is bounded_above;

:: SEQM_3:th 78
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b2 is non-decreasing
   holds b1 + b2 is bounded_below;

:: SEQM_3:th 79
theorem
for b1, b2 being set holds
b1 --> b2 is constant;

:: SEQM_3:funcreg 2
registration
  let a1, a2 be set;
  cluster a1 --> a2 -> constant;
end;

:: SEQM_3:th 80
theorem
incl NAT is increasing & incl NAT is natural-valued;

:: SEQM_3:condreg 5
registration
  cluster -> natural-valued (FinSequence of NAT);
end;