Article RINFSUP1, MML version 4.99.1005

:: RINFSUP1:th 1
theorem
for b1, b2, b3 being real set holds
   b1 - b2 < b3 & b3 < b1 + b2
iff
   abs (b3 - b1) < b2;

:: RINFSUP1:funcnot 1 => RINFSUP1:func 1
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func sup A1 -> Element of REAL equals
    sup rng a1;
end;

:: RINFSUP1:def 1
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   sup b1 = sup rng b1;

:: RINFSUP1:funcnot 2 => RINFSUP1:func 2
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func inf A1 -> Element of REAL equals
    inf rng a1;
end;

:: RINFSUP1:def 2
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   inf b1 = inf rng b1;

:: RINFSUP1:th 2
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
(b1 + b2) - b2 = b1;

:: RINFSUP1:th 3
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      b1 in rng b2
   iff
      - b1 in rng - b2;

:: RINFSUP1:th 4
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   rng - b1 = - rng b1;

:: RINFSUP1:th 5
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is bounded_above
   iff
      rng b1 is bounded_above;

:: RINFSUP1:th 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is bounded_below
   iff
      rng b1 is bounded_below;

:: RINFSUP1:th 7
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above
   holds    b1 = sup b2
   iff
      (for b3 being Element of NAT holds
          b2 . b3 <= b1) &
       (for b3 being real set
             st 0 < b3
          holds ex b4 being Element of NAT st
             b1 - b3 < b2 . b4);

:: RINFSUP1:th 8
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below
   holds    b1 = inf b2
   iff
      (for b3 being Element of NAT holds
          b1 <= b2 . b3) &
       (for b3 being real set
             st 0 < b3
          holds ex b4 being Element of NAT st
             b2 . b4 < b1 + b3);

:: RINFSUP1:th 9
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      for b3 being Element of NAT holds
         b2 . b3 <= b1
   iff
      b2 is bounded_above & sup b2 <= b1;

:: RINFSUP1:th 10
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL holds
      for b3 being Element of NAT holds
         b1 <= b2 . b3
   iff
      b2 is bounded_below & b1 <= inf b2;

:: RINFSUP1:th 11
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is bounded_above
   iff
      - b1 is bounded_below;

:: RINFSUP1:th 12
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is bounded_below
   iff
      - b1 is bounded_above;

:: RINFSUP1:th 13
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above
   holds sup b1 = - inf - b1;

:: RINFSUP1:th 14
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below
   holds inf b1 = - sup - b1;

:: RINFSUP1:th 15
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b2 is bounded_below
   holds (inf b1) + inf b2 <= inf (b1 + b2);

:: RINFSUP1:th 16
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b2 is bounded_above
   holds sup (b1 + b2) <= (sup b1) + sup b2;

:: RINFSUP1:attrnot 1 => PARTFUN3:attr 4
notation
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  synonym nonnegative for nonnegative-yielding;
end;

:: RINFSUP1:attrnot 2 => PARTFUN3:attr 4
definition
  let a1 be Relation-like set;
  attr a1 is nonnegative means
    for b1 being Element of NAT holds
       0 <= a1 . b1;
end;

:: RINFSUP1:dfs 3
definiens
  let a1 be Function-like quasi_total Relation of NAT,REAL;
To prove
     a1 is nonnegative-yielding
it is sufficient to prove
  thus for b1 being Element of NAT holds
       0 <= a1 . b1;

:: RINFSUP1:def 3
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is nonnegative-yielding
   iff
      for b2 being Element of NAT holds
         0 <= b1 . b2;

:: RINFSUP1:th 17
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is nonnegative-yielding
   holds b2 ^\ b1 is nonnegative-yielding;

:: RINFSUP1:th 18
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b1 is nonnegative-yielding
   holds 0 <= inf b1;

:: RINFSUP1:th 19
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b1 is nonnegative-yielding
   holds 0 <= sup b1;

:: RINFSUP1:th 20
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below & b1 is nonnegative-yielding & b2 is bounded_below & b2 is nonnegative-yielding
   holds b1 (#) b2 is bounded_below &
    (inf b1) * inf b2 <= inf (b1 (#) b2);

:: RINFSUP1:th 21
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above & b1 is nonnegative-yielding & b2 is bounded_above & b2 is nonnegative-yielding
   holds b1 (#) b2 is bounded_above &
    sup (b1 (#) b2) <= (sup b1) * sup b2;

:: RINFSUP1:th 22
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b1 is bounded_above
   holds b1 is bounded;

:: RINFSUP1:th 23
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing & b1 is bounded_below
   holds b1 is bounded;

:: RINFSUP1:th 24
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b1 is bounded_above
   holds lim b1 = sup b1;

:: RINFSUP1:th 25
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing & b1 is bounded_below
   holds lim b1 = inf b1;

:: RINFSUP1:th 26
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above
   holds b2 ^\ b1 is bounded_above;

:: RINFSUP1:th 27
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below
   holds b2 ^\ b1 is bounded_below;

:: RINFSUP1:th 28
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds b2 ^\ b1 is bounded;

:: RINFSUP1:th 29
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
for b2 being Element of NAT holds
   {b1 . b3 where b3 is Element of NAT: b2 <= b3} is Element of bool REAL;

:: RINFSUP1:th 30
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   rng (b2 ^\ b1) = {b2 . b3 where b3 is Element of NAT: b1 <= b3};

:: RINFSUP1:th 31
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is bounded_above
for b2 being Element of NAT
for b3 being Element of bool REAL
      st b3 = {b1 . b4 where b4 is Element of NAT: b2 <= b4}
   holds b3 is bounded_above;

:: RINFSUP1:th 32
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is bounded_below
for b2 being Element of NAT
for b3 being Element of bool REAL
      st b3 = {b1 . b4 where b4 is Element of NAT: b2 <= b4}
   holds b3 is bounded_below;

:: RINFSUP1:th 33
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is bounded
for b2 being Element of NAT
for b3 being Element of bool REAL
      st b3 = {b1 . b4 where b4 is Element of NAT: b2 <= b4}
   holds b3 is bounded;

:: RINFSUP1:th 34
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-decreasing
for b2 being Element of NAT
for b3 being Element of bool REAL
      st b3 = {b1 . b4 where b4 is Element of NAT: b2 <= b4}
   holds inf b3 = b1 . b2;

:: RINFSUP1:th 35
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
   st b1 is non-increasing
for b2 being Element of NAT
for b3 being Element of bool REAL
      st b3 = {b1 . b4 where b4 is Element of NAT: b2 <= b4}
   holds sup b3 = b1 . b2;

:: RINFSUP1:th 36
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   ex b2 being Function-like quasi_total Relation of NAT,REAL st
      for b3 being Element of NAT
      for b4 being Element of bool REAL
            st b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5}
         holds b2 . b3 = sup b4;

:: RINFSUP1:th 37
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   ex b2 being Function-like quasi_total Relation of NAT,REAL st
      for b3 being Element of NAT
      for b4 being Element of bool REAL
            st b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5}
         holds b2 . b3 = inf b4;

:: RINFSUP1:funcnot 3 => RINFSUP1:func 3
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func inferior_realsequence A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT
    for b2 being Element of bool REAL
          st b2 = {a1 . b3 where b3 is Element of NAT: b1 <= b3}
       holds it . b1 = inf b2;
end;

:: RINFSUP1:def 4
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 = inferior_realsequence b1
iff
   for b3 being Element of NAT
   for b4 being Element of bool REAL
         st b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5}
      holds b2 . b3 = inf b4;

:: RINFSUP1:funcnot 4 => RINFSUP1:func 4
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func superior_realsequence A1 -> Function-like quasi_total Relation of NAT,REAL means
    for b1 being Element of NAT
    for b2 being Element of bool REAL
          st b2 = {a1 . b3 where b3 is Element of NAT: b1 <= b3}
       holds it . b1 = sup b2;
end;

:: RINFSUP1:def 5
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL holds
   b2 = superior_realsequence b1
iff
   for b3 being Element of NAT
   for b4 being Element of bool REAL
         st b4 = {b1 . b5 where b5 is Element of NAT: b3 <= b5}
      holds b2 . b3 = sup b4;

:: RINFSUP1:th 38
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (inferior_realsequence b2) . b1 = inf (b2 ^\ b1);

:: RINFSUP1:th 39
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL holds
   (superior_realsequence b2) . b1 = sup (b2 ^\ b1);

:: RINFSUP1:th 40
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below
   holds (inferior_realsequence b1) . 0 = inf b1;

:: RINFSUP1:th 41
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above
   holds (superior_realsequence b1) . 0 = sup b1;

:: RINFSUP1:th 42
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is bounded_below
   holds    b2 = (inferior_realsequence b3) . b1
   iff
      (for b4 being Element of NAT holds
          b2 <= b3 . (b1 + b4)) &
       (for b4 being real set
             st 0 < b4
          holds ex b5 being Element of NAT st
             b3 . (b1 + b5) < b2 + b4);

:: RINFSUP1:th 43
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is bounded_above
   holds    b2 = (superior_realsequence b3) . b1
   iff
      (for b4 being Element of NAT holds
          b3 . (b1 + b4) <= b2) &
       (for b4 being real set
             st 0 < b4
          holds ex b5 being Element of NAT st
             b2 - b4 < b3 . (b1 + b5));

:: RINFSUP1: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
      st b3 is bounded_below
   holds    for b4 being Element of NAT holds
         b2 <= b3 . (b1 + b4)
   iff
      b2 <= (inferior_realsequence b3) . b1;

:: RINFSUP1:th 45
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is bounded_below
   holds    for b4 being Element of NAT
            st b1 <= b4
         holds b2 <= b3 . b4
   iff
      b2 <= (inferior_realsequence b3) . b1;

:: RINFSUP1:th 46
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is bounded_above
   holds    for b4 being Element of NAT holds
         b3 . (b1 + b4) <= b2
   iff
      (superior_realsequence b3) . b1 <= b2;

:: RINFSUP1:th 47
theorem
for b1 being Element of NAT
for b2 being real set
for b3 being Function-like quasi_total Relation of NAT,REAL
      st b3 is bounded_above
   holds    for b4 being Element of NAT
            st b1 <= b4
         holds b3 . b4 <= b2
   iff
      (superior_realsequence b3) . b1 <= b2;

:: RINFSUP1:th 48
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below
   holds (inferior_realsequence b2) . b1 = min((inferior_realsequence b2) . (b1 + 1),b2 . b1);

:: RINFSUP1:th 49
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above
   holds (superior_realsequence b2) . b1 = max((superior_realsequence b2) . (b1 + 1),b2 . b1);

:: RINFSUP1:th 50
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below
   holds (inferior_realsequence b2) . b1 <= (inferior_realsequence b2) . (b1 + 1);

:: RINFSUP1:th 51
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above
   holds (superior_realsequence b2) . (b1 + 1) <= (superior_realsequence b2) . b1;

:: RINFSUP1:th 52
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below
   holds inferior_realsequence b1 is non-decreasing;

:: RINFSUP1:th 53
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above
   holds superior_realsequence b1 is non-increasing;

:: RINFSUP1:th 54
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds (inferior_realsequence b2) . b1 <= (superior_realsequence b2) . b1;

:: RINFSUP1:th 55
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds (inferior_realsequence b2) . b1 <= inf superior_realsequence b2;

:: RINFSUP1:th 56
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds sup inferior_realsequence b2 <= (superior_realsequence b2) . b1;

:: RINFSUP1:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds sup inferior_realsequence b1 <= inf superior_realsequence b1;

:: RINFSUP1:th 58
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds superior_realsequence b1 is bounded & inferior_realsequence b1 is bounded;

:: RINFSUP1:th 59
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds inferior_realsequence b1 is convergent & lim inferior_realsequence b1 = sup inferior_realsequence b1;

:: RINFSUP1:th 60
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds superior_realsequence b1 is convergent & lim superior_realsequence b1 = inf superior_realsequence b1;

:: RINFSUP1:th 61
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below
   holds (inferior_realsequence b2) . b1 = - ((superior_realsequence - b2) . b1);

:: RINFSUP1:th 62
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above
   holds (superior_realsequence b2) . b1 = - ((inferior_realsequence - b2) . b1);

:: RINFSUP1:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_below
   holds inferior_realsequence b1 = - superior_realsequence - b1;

:: RINFSUP1:th 64
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded_above
   holds superior_realsequence b1 = - inferior_realsequence - b1;

:: RINFSUP1:th 65
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-decreasing
   holds b2 . b1 <= (inferior_realsequence b2) . (b1 + 1);

:: RINFSUP1:th 66
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing
   holds inferior_realsequence b1 = b1;

:: RINFSUP1:th 67
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-decreasing & b2 is bounded_above
   holds b2 . b1 <= (superior_realsequence b2) . (b1 + 1);

:: RINFSUP1:th 68
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-decreasing & b2 is bounded_above
   holds (superior_realsequence b2) . b1 = (superior_realsequence b2) . (b1 + 1);

:: RINFSUP1:th 69
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-decreasing & b2 is bounded_above
   holds (superior_realsequence b2) . b1 = sup b2 & superior_realsequence b2 is constant;

:: RINFSUP1:th 70
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-decreasing & b1 is bounded_above
   holds inf superior_realsequence b1 = sup b1;

:: RINFSUP1:th 71
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-increasing
   holds (superior_realsequence b2) . (b1 + 1) <= b2 . b1;

:: RINFSUP1:th 72
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing
   holds superior_realsequence b1 = b1;

:: RINFSUP1:th 73
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-increasing & b2 is bounded_below
   holds (inferior_realsequence b2) . (b1 + 1) <= b2 . b1;

:: RINFSUP1:th 74
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-increasing & b2 is bounded_below
   holds (inferior_realsequence b2) . b1 = (inferior_realsequence b2) . (b1 + 1);

:: RINFSUP1:th 75
theorem
for b1 being Element of NAT
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is non-increasing & b2 is bounded_below
   holds (inferior_realsequence b2) . b1 = inf b2 & inferior_realsequence b2 is constant;

:: RINFSUP1:th 76
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is non-increasing & b1 is bounded_below
   holds sup inferior_realsequence b1 = inf b1;

:: RINFSUP1:th 77
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded &
         b2 is bounded &
         (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3)
   holds (for b3 being Element of NAT holds
       (superior_realsequence b1) . b3 <= (superior_realsequence b2) . b3) &
    (for b3 being Element of NAT holds
       (inferior_realsequence b1) . b3 <= (inferior_realsequence b2) . b3);

:: RINFSUP1:th 78
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below & b3 is bounded_below
   holds ((inferior_realsequence b2) . b1) + ((inferior_realsequence b3) . b1) <= (inferior_realsequence (b2 + b3)) . b1;

:: RINFSUP1:th 79
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above & b3 is bounded_above
   holds (superior_realsequence (b2 + b3)) . b1 <= ((superior_realsequence b2) . b1) + ((superior_realsequence b3) . b1);

:: RINFSUP1:th 80
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below & b2 is nonnegative-yielding & b3 is bounded_below & b3 is nonnegative-yielding
   holds ((inferior_realsequence b2) . b1) * ((inferior_realsequence b3) . b1) <= (inferior_realsequence (b2 (#) b3)) . b1;

:: RINFSUP1:th 81
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_below & b2 is nonnegative-yielding & b3 is bounded_below & b3 is nonnegative-yielding
   holds ((inferior_realsequence b2) . b1) * ((inferior_realsequence b3) . b1) <= (inferior_realsequence (b2 (#) b3)) . b1;

:: RINFSUP1:th 82
theorem
for b1 being Element of NAT
for b2, b3 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded_above & b2 is nonnegative-yielding & b3 is bounded_above & b3 is nonnegative-yielding
   holds (superior_realsequence (b2 (#) b3)) . b1 <= ((superior_realsequence b2) . b1) * ((superior_realsequence b3) . b1);

:: RINFSUP1:funcnot 5 => RINFSUP1:func 5
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func lim_sup A1 -> Element of REAL equals
    inf superior_realsequence a1;
end;

:: RINFSUP1:def 6
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   lim_sup b1 = inf superior_realsequence b1;

:: RINFSUP1:funcnot 6 => RINFSUP1:func 6
definition
  let a1 be Function-like quasi_total Relation of NAT,REAL;
  func lim_inf A1 -> Element of REAL equals
    sup inferior_realsequence a1;
end;

:: RINFSUP1:def 7
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
   lim_inf b1 = sup inferior_realsequence b1;

:: RINFSUP1:th 83
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    lim_inf b2 <= b1
   iff
      for b3 being real set
         st 0 < b3
      for b4 being Element of NAT holds
         ex b5 being Element of NAT st
            b2 . (b4 + b5) < b1 + b3;

:: RINFSUP1:th 84
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    b1 <= lim_inf b2
   iff
      for b3 being real set
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5 being Element of NAT holds
               b1 - b3 < b2 . (b4 + b5);

:: RINFSUP1:th 85
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    b1 = lim_inf b2
   iff
      for b3 being real set
            st 0 < b3
         holds (for b4 being Element of NAT holds
             ex b5 being Element of NAT st
                b2 . (b4 + b5) < b1 + b3) &
          (ex b4 being Element of NAT st
             for b5 being Element of NAT holds
                b1 - b3 < b2 . (b4 + b5));

:: RINFSUP1:th 86
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    b1 <= lim_sup b2
   iff
      for b3 being real set
         st 0 < b3
      for b4 being Element of NAT holds
         ex b5 being Element of NAT st
            b1 - b3 < b2 . (b4 + b5);

:: RINFSUP1:th 87
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    lim_sup b2 <= b1
   iff
      for b3 being real set
            st 0 < b3
         holds ex b4 being Element of NAT st
            for b5 being Element of NAT holds
               b2 . (b4 + b5) < b1 + b3;

:: RINFSUP1:th 88
theorem
for b1 being real set
for b2 being Function-like quasi_total Relation of NAT,REAL
      st b2 is bounded
   holds    b1 = lim_sup b2
   iff
      for b3 being real set
            st 0 < b3
         holds (for b4 being Element of NAT holds
             ex b5 being Element of NAT st
                b1 - b3 < b2 . (b4 + b5)) &
          (ex b4 being Element of NAT st
             for b5 being Element of NAT holds
                b2 . (b4 + b5) < b1 + b3);

:: RINFSUP1:th 89
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds lim_inf b1 <= lim_sup b1;

:: RINFSUP1:th 90
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL holds
      b1 is bounded & lim_sup b1 = lim_inf b1
   iff
      b1 is convergent;

:: RINFSUP1:th 91
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is convergent
   holds lim b1 = lim_sup b1 & lim b1 = lim_inf b1;

:: RINFSUP1:th 92
theorem
for b1 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded
   holds lim_sup - b1 = - lim_inf b1 & lim_inf - b1 = - lim_sup b1;

:: RINFSUP1:th 93
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded &
         b2 is bounded &
         (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3)
   holds lim_sup b1 <= lim_sup b2 & lim_inf b1 <= lim_inf b2;

:: RINFSUP1:th 94
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded & b2 is bounded
   holds (lim_inf b1) + lim_inf b2 <= lim_inf (b1 + b2) &
    lim_inf (b1 + b2) <= (lim_inf b1) + lim_sup b2 &
    lim_inf (b1 + b2) <= (lim_sup b1) + lim_inf b2 &
    (lim_inf b1) + lim_sup b2 <= lim_sup (b1 + b2) &
    (lim_sup b1) + lim_inf b2 <= lim_sup (b1 + b2) &
    lim_sup (b1 + b2) <= (lim_sup b1) + lim_sup b2 &
    (b1 is not convergent & b2 is not convergent or lim_inf (b1 + b2) = (lim_inf b1) + lim_inf b2 &
     lim_sup (b1 + b2) = (lim_sup b1) + lim_sup b2);

:: RINFSUP1:th 95
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,REAL
      st b1 is bounded & b1 is nonnegative-yielding & b2 is bounded & b2 is nonnegative-yielding
   holds (lim_inf b1) * lim_inf b2 <= lim_inf (b1 (#) b2) &
    lim_sup (b1 (#) b2) <= (lim_sup b1) * lim_sup b2;