Article SUPINF_2, MML version 4.99.1005

:: SUPINF_2:funcnot 1 => XBOOLE_0:func 1
notation
  synonym 0. for {};
end;

:: SUPINF_2:funcnot 2 => SUPINF_2:func 1
definition
  redefine func 0. -> Element of ExtREAL;
end;

:: SUPINF_2:funcnot 3 => SUPINF_2:func 2
definition
  let a1, a2 be Element of ExtREAL;
  func A1 + A2 -> Element of ExtREAL means
    ex b1, b2 being Element of REAL st
       a1 = b1 & a2 = b2 & it = b1 + b2
    if a1 in REAL & a2 in REAL,
it = +infty
    if (a1 = +infty & a2 <> -infty or a2 = +infty & a1 <> -infty),
it = -infty
    if (a1 = -infty & a2 <> +infty or a2 = -infty & a1 <> +infty)
    otherwise it = 0.;
  commutativity;
::  for a1, a2 being Element of ExtREAL holds
::  a1 + a2 = a2 + a1;
end;

:: SUPINF_2:def 2
theorem
for b1, b2, b3 being Element of ExtREAL holds
(b1 in REAL & b2 in REAL implies    (b3 = b1 + b2
 iff
    ex b4, b5 being Element of REAL st
       b1 = b4 & b2 = b5 & b3 = b4 + b5)) &
 ((b1 = +infty implies b2 = -infty) & (b2 = +infty implies b1 = -infty) or    (b3 = b1 + b2
 iff
    b3 = +infty)) &
 ((b1 = -infty implies b2 = +infty) & (b2 = -infty implies b1 = +infty) or    (b3 = b1 + b2
 iff
    b3 = -infty)) &
 ((b1 in REAL implies not b2 in REAL) & (b1 = +infty implies b2 = -infty) & (b2 = +infty implies b1 = -infty) & (b1 = -infty implies b2 = +infty) & (b2 = -infty implies b1 = +infty) implies    (b3 = b1 + b2
 iff
    b3 = 0.));

:: SUPINF_2:th 1
theorem
for b1, b2 being Element of ExtREAL
for b3, b4 being Element of REAL
      st b1 = b3 & b2 = b4
   holds b1 + b2 = b3 + b4;

:: SUPINF_2:th 2
theorem
for b1 being Element of ExtREAL
      st not b1 in REAL & b1 <> +infty
   holds b1 = -infty;

:: SUPINF_2:funcnot 4 => SUPINF_2:func 3
definition
  let a1 be Element of ExtREAL;
  func - A1 -> Element of ExtREAL means
    ex b1 being Element of REAL st
       a1 = b1 & it = - b1
    if a1 in REAL,
it = -infty
    if a1 = +infty
    otherwise it = +infty;
  involutiveness;
::  for a1 being Element of ExtREAL holds
::     - - a1 = a1;
end;

:: SUPINF_2:def 3
theorem
for b1, b2 being Element of ExtREAL holds
(b1 in REAL implies    (b2 = - b1
 iff
    ex b3 being Element of REAL st
       b1 = b3 & b2 = - b3)) &
 (b1 = +infty implies    (b2 = - b1
 iff
    b2 = -infty)) &
 (not b1 in REAL & b1 <> +infty implies    (b2 = - b1
 iff
    b2 = +infty));

:: SUPINF_2:funcnot 5 => SUPINF_2:func 4
definition
  let a1, a2 be Element of ExtREAL;
  func A1 - A2 -> Element of ExtREAL equals
    a1 + - a2;
end;

:: SUPINF_2:def 4
theorem
for b1, b2 being Element of ExtREAL holds
b1 - b2 = b1 + - b2;

:: SUPINF_2:th 3
theorem
for b1 being Element of ExtREAL
for b2 being Element of REAL
      st b1 = b2
   holds - b1 = - b2;

:: SUPINF_2:th 4
theorem
- -infty = +infty;

:: SUPINF_2:th 5
theorem
for b1, b2 being Element of ExtREAL
for b3, b4 being Element of REAL
      st b1 = b3 & b2 = b4
   holds b1 - b2 = b3 - b4;

:: SUPINF_2:th 6
theorem
for b1 being Element of ExtREAL
      st b1 <> +infty
   holds +infty - b1 = +infty & b1 - +infty = -infty;

:: SUPINF_2:th 7
theorem
for b1 being Element of ExtREAL
      st b1 <> -infty
   holds -infty - b1 = -infty & b1 - -infty = +infty;

:: SUPINF_2:th 8
theorem
for b1, b2 being Element of ExtREAL
      st b1 + b2 = +infty & b1 <> +infty
   holds b2 = +infty;

:: SUPINF_2:th 9
theorem
for b1, b2 being Element of ExtREAL
      st b1 + b2 = -infty & b1 <> -infty
   holds b2 = -infty;

:: SUPINF_2:th 10
theorem
for b1, b2 being Element of ExtREAL
      st b1 - b2 = +infty & b1 <> +infty
   holds b2 = -infty;

:: SUPINF_2:th 11
theorem
for b1, b2 being Element of ExtREAL
      st b1 - b2 = -infty & b1 <> -infty
   holds b2 = +infty;

:: SUPINF_2:th 12
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & b1 + b2 in REAL
   holds b1 in REAL & b2 in REAL;

:: SUPINF_2:th 13
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & b1 - b2 in REAL
   holds b1 in REAL & b2 in REAL;

:: SUPINF_2:th 14
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st (b1 = +infty implies b3 <> -infty) & (b1 = -infty implies b3 <> +infty) & (b2 = +infty implies b4 <> -infty) & (b2 = -infty implies b4 <> +infty) & b1 <= b2 & b3 <= b4
   holds b1 + b3 <= b2 + b4;

:: SUPINF_2:th 15
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st (b1 = +infty implies b4 <> +infty) & (b1 = -infty implies b4 <> -infty) & (b2 = +infty implies b3 <> +infty) & (b2 = -infty implies b3 <> -infty) & b1 <= b2 & b3 <= b4
   holds b1 - b4 <= b2 - b3;

:: SUPINF_2:th 16
theorem
for b1, b2 being Element of ExtREAL holds
   b1 <= b2
iff
   - b2 <= - b1;

:: SUPINF_2:th 17
theorem
for b1, b2 being Element of ExtREAL holds
   b1 < b2
iff
   - b2 < - b1;

:: SUPINF_2:th 18
theorem
for b1 being Element of ExtREAL holds
   b1 + 0. = b1 & 0. + b1 = b1;

:: SUPINF_2:th 19
theorem
-infty < 0. & 0. < +infty;

:: SUPINF_2:th 20
theorem
for b1, b2, b3 being Element of ExtREAL
      st 0. <= b3 & 0. <= b1 & b2 = b1 + b3
   holds b1 <= b2;

:: SUPINF_2:th 21
theorem
for b1 being Element of ExtREAL
      st b1 in NAT
   holds 0. <= b1;

:: SUPINF_2:funcnot 6 => SUPINF_2:func 5
definition
  let a1, a2 be non empty Element of bool ExtREAL;
  func A1 + A2 -> Element of bool ExtREAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          ex b2, b3 being Element of ExtREAL st
             b2 in a1 & b3 in a2 & b1 = b2 + b3;
end;

:: SUPINF_2:def 5
theorem
for b1, b2 being non empty Element of bool ExtREAL
for b3 being Element of bool ExtREAL holds
      b3 = b1 + b2
   iff
      for b4 being Element of ExtREAL holds
            b4 in b3
         iff
            ex b5, b6 being Element of ExtREAL st
               b5 in b1 & b6 in b2 & b4 = b5 + b6;

:: SUPINF_2:funcreg 1
registration
  let a1, a2 be non empty Element of bool ExtREAL;
  cluster a1 + a2 -> non empty;
end;

:: SUPINF_2:funcnot 7 => SUPINF_2:func 6
definition
  let a1 be non empty Element of bool ExtREAL;
  func - A1 -> Element of bool ExtREAL means
    for b1 being Element of ExtREAL holds
          b1 in it
       iff
          ex b2 being Element of ExtREAL st
             b2 in a1 & b1 = - b2;
end;

:: SUPINF_2:def 6
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being Element of bool ExtREAL holds
      b2 = - b1
   iff
      for b3 being Element of ExtREAL holds
            b3 in b2
         iff
            ex b4 being Element of ExtREAL st
               b4 in b1 & b3 = - b4;

:: SUPINF_2:funcreg 2
registration
  let a1 be non empty Element of bool ExtREAL;
  cluster - a1 -> non empty;
end;

:: SUPINF_2:th 22
theorem
for b1 being non empty Element of bool ExtREAL holds
   - - b1 = b1;

:: SUPINF_2:th 23
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being Element of ExtREAL holds
      b2 in b1
   iff
      - b2 in - b1;

:: SUPINF_2:th 24
theorem
for b1, b2 being non empty Element of bool ExtREAL holds
   b1 c= b2
iff
   - b1 c= - b2;

:: SUPINF_2:th 25
theorem
for b1 being Element of ExtREAL holds
      b1 in REAL
   iff
      - b1 in REAL;

:: SUPINF_2:funcnot 8 => SUPINF_2:func 7
definition
  let a1 be non empty Element of bool ExtREAL;
  redefine func inf a1 -> Element of ExtREAL;
end;

:: SUPINF_2:funcnot 9 => SUPINF_2:func 8
definition
  let a1 be non empty Element of bool ExtREAL;
  redefine func sup a1 -> Element of ExtREAL;
end;

:: SUPINF_2:th 26
theorem
for b1, b2 being non empty Element of bool ExtREAL
      st (-infty in b1 implies not +infty in b2) &
         (+infty in b1 implies not -infty in b2) &
         (sup b1 = +infty implies sup b2 <> -infty) &
         (sup b1 = -infty implies sup b2 <> +infty)
   holds sup (b1 + b2) <= (sup b1) + sup b2;

:: SUPINF_2:th 27
theorem
for b1, b2 being non empty Element of bool ExtREAL
      st (-infty in b1 implies not +infty in b2) &
         (+infty in b1 implies not -infty in b2) &
         (inf b1 = +infty implies inf b2 <> -infty) &
         (inf b1 = -infty implies inf b2 <> +infty)
   holds (inf b1) + inf b2 <= inf (b1 + b2);

:: SUPINF_2:th 28
theorem
for b1, b2 being non empty Element of bool ExtREAL
      st b1 is bounded_above & b2 is bounded_above
   holds sup (b1 + b2) <= (sup b1) + sup b2;

:: SUPINF_2:th 29
theorem
for b1, b2 being non empty Element of bool ExtREAL
      st b1 is bounded_below & b2 is bounded_below
   holds (inf b1) + inf b2 <= inf (b1 + b2);

:: SUPINF_2:th 30
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being Element of ExtREAL holds
      b2 is majorant of b1
   iff
      - b2 is minorant of - b1;

:: SUPINF_2:th 31
theorem
for b1 being non empty Element of bool ExtREAL
for b2 being Element of ExtREAL holds
      b2 is minorant of b1
   iff
      - b2 is majorant of - b1;

:: SUPINF_2:th 32
theorem
for b1 being non empty Element of bool ExtREAL holds
   inf - b1 = - sup b1;

:: SUPINF_2:th 33
theorem
for b1 being non empty Element of bool ExtREAL holds
   sup - b1 = - inf b1;

:: SUPINF_2:funcnot 10 => SUPINF_2:func 9
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  redefine func rng a3 -> non empty Element of bool ExtREAL;
end;

:: SUPINF_2:funcnot 11 => SUPINF_2:func 10
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  func sup A3 -> Element of ExtREAL equals
    sup rng a3;
end;

:: SUPINF_2:def 7
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
   sup b3 = sup rng b3;

:: SUPINF_2:funcnot 12 => SUPINF_2:func 11
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  func inf A3 -> Element of ExtREAL equals
    inf rng a3;
end;

:: SUPINF_2:def 8
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
   inf b3 = inf rng b3;

:: SUPINF_2:funcnot 13 => SUPINF_2:func 12
definition
  let a1 be Relation-like Function-like ext-real-valued set;
  let a2 be set;
  redefine func a1 . a2 -> Element of ExtREAL;
end;

:: SUPINF_2:funcnot 14 => SUPINF_2:func 13
definition
  let a1 be non empty set;
  let a2, a3 be non empty Element of bool ExtREAL;
  let a4 be Function-like quasi_total Relation of a1,a2;
  let a5 be Function-like quasi_total Relation of a1,a3;
  func A4 + A5 -> Function-like quasi_total Relation of a1,a2 + a3 means
    for b1 being Element of a1 holds
       it . b1 = (a4 . b1) + (a5 . b1);
end;

:: SUPINF_2:def 9
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
for b6 being Function-like quasi_total Relation of b1,b2 + b3 holds
      b6 = b4 + b5
   iff
      for b7 being Element of b1 holds
         b6 . b7 = (b4 . b7) + (b5 . b7);

:: SUPINF_2:th 34
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3 holds
   rng (b4 + b5) c= (rng b4) + rng b5;

:: SUPINF_2:th 35
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
   st (-infty in b2 implies not +infty in b3) & (+infty in b2 implies not -infty in b3)
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st (sup b4 = +infty implies sup b5 <> -infty) &
         (sup b4 = -infty implies sup b5 <> +infty)
   holds sup (b4 + b5) <= (sup b4) + sup b5;

:: SUPINF_2:th 36
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
   st (-infty in b2 implies not +infty in b3) & (+infty in b2 implies not -infty in b3)
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st (inf b4 = +infty implies inf b5 <> -infty) &
         (inf b4 = -infty implies inf b5 <> +infty)
   holds (inf b4) + inf b5 <= inf (b4 + b5);

:: SUPINF_2:funcnot 15 => SUPINF_2:func 14
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  func - A3 -> Function-like quasi_total Relation of a1,- a2 means
    for b1 being Element of a1 holds
       it . b1 = - (a3 . b1);
end;

:: SUPINF_2:def 10
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Function-like quasi_total Relation of b1,- b2 holds
      b4 = - b3
   iff
      for b5 being Element of b1 holds
         b4 . b5 = - (b3 . b5);

:: SUPINF_2:th 37
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
   rng - b3 = - rng b3;

:: SUPINF_2:th 38
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
   inf - b3 = - sup b3 & sup - b3 = - inf b3;

:: SUPINF_2:attrnot 1 => SUPINF_2:attr 1
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  attr a3 is bounded_above means
    sup a3 < +infty;
end;

:: SUPINF_2:dfs 10
definiens
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
To prove
     a3 is bounded_above
it is sufficient to prove
  thus sup a3 < +infty;

:: SUPINF_2:def 11
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded_above(b1, b2)
   iff
      sup b3 < +infty;

:: SUPINF_2:attrnot 2 => SUPINF_2:attr 2
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  attr a3 is bounded_below means
    -infty < inf a3;
end;

:: SUPINF_2:dfs 11
definiens
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
To prove
     a3 is bounded_below
it is sufficient to prove
  thus -infty < inf a3;

:: SUPINF_2:def 12
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded_below(b1, b2)
   iff
      -infty < inf b3;

:: SUPINF_2:attrnot 3 => SUPINF_2:attr 3
definition
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
  attr a3 is bounded means
    a3 is bounded_above(a1, a2) & a3 is bounded_below(a1, a2);
end;

:: SUPINF_2:dfs 12
definiens
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  let a3 be Function-like quasi_total Relation of a1,a2;
To prove
     a3 is bounded
it is sufficient to prove
  thus a3 is bounded_above(a1, a2) & a3 is bounded_below(a1, a2);

:: SUPINF_2:def 13
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded(b1, b2)
   iff
      b3 is bounded_above(b1, b2) & b3 is bounded_below(b1, b2);

:: SUPINF_2:condreg 1
registration
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  cluster Function-like quasi_total bounded -> bounded_above bounded_below (Relation of a1,a2);
end;

:: SUPINF_2:condreg 2
registration
  let a1 be non empty set;
  let a2 be non empty Element of bool ExtREAL;
  cluster Function-like quasi_total bounded_above bounded_below -> bounded (Relation of a1,a2);
end;

:: SUPINF_2:th 39
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded(b1, b2)
   iff
      sup b3 < +infty & -infty < inf b3;

:: SUPINF_2:th 40
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded_above(b1, b2)
   iff
      - b3 is bounded_below(b1, - b2);

:: SUPINF_2:th 41
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded_below(b1, b2)
   iff
      - b3 is bounded_above(b1, - b2);

:: SUPINF_2:th 42
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2 holds
      b3 is bounded(b1, b2)
   iff
      - b3 is bounded(b1, - b2);

:: SUPINF_2:th 43
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of b1 holds
   -infty <= b3 . b4 & b3 . b4 <= +infty;

:: SUPINF_2:th 44
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of b1
      st b2 c= REAL
   holds -infty < b3 . b4 & b3 . b4 < +infty;

:: SUPINF_2:th 45
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
for b4 being Element of b1 holds
   inf b3 <= b3 . b4 & b3 . b4 <= sup b3;

:: SUPINF_2:th 46
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
      st b2 c= REAL
   holds    b3 is bounded_above(b1, b2)
   iff
      sup b3 in REAL;

:: SUPINF_2:th 47
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
      st b2 c= REAL
   holds    b3 is bounded_below(b1, b2)
   iff
      inf b3 in REAL;

:: SUPINF_2:th 48
theorem
for b1 being non empty set
for b2 being non empty Element of bool ExtREAL
for b3 being Function-like quasi_total Relation of b1,b2
      st b2 c= REAL
   holds    b3 is bounded(b1, b2)
   iff
      inf b3 in REAL & sup b3 in REAL;

:: SUPINF_2:th 49
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
   st b2 c= REAL & b3 c= REAL
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st b4 is bounded_above(b1, b2) & b5 is bounded_above(b1, b3)
   holds b4 + b5 is bounded_above(b1, b2 + b3);

:: SUPINF_2:th 50
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
   st b2 c= REAL & b3 c= REAL
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st b4 is bounded_below(b1, b2) & b5 is bounded_below(b1, b3)
   holds b4 + b5 is bounded_below(b1, b2 + b3);

:: SUPINF_2:th 51
theorem
for b1 being non empty set
for b2, b3 being non empty Element of bool ExtREAL
   st b2 c= REAL & b3 c= REAL
for b4 being Function-like quasi_total Relation of b1,b2
for b5 being Function-like quasi_total Relation of b1,b3
      st b4 is bounded(b1, b2) & b5 is bounded(b1, b3)
   holds b4 + b5 is bounded(b1, b2 + b3);

:: SUPINF_2:th 52
theorem
ex b1 being Function-like quasi_total Relation of NAT,ExtREAL st
   b1 is one-to-one & NAT = rng b1 & rng b1 is non empty Element of bool ExtREAL;

:: SUPINF_2:attrnot 4 => CARD_4:attr 1
notation
  let a1 be non empty set;
  let a2 be Element of bool a1;
  synonym denumerable for countable;
end;

:: SUPINF_2:attrnot 5 => SUPINF_2:attr 4
definition
  let a1 be non empty set;
  let a2 be Element of bool a1;
  redefine attr a2 is denumerable means
    (a2 is not empty) implies ex b1 being Function-like quasi_total Relation of NAT,a1 st
       a2 = rng b1;
end;

:: SUPINF_2:dfs 13
definiens
  let a1 be non empty set;
  let a2 be Element of bool a1;
To prove
     a1 is countable
it is sufficient to prove
  thus (a2 is not empty) implies ex b1 being Function-like quasi_total Relation of NAT,a1 st
       a2 = rng b1;

:: SUPINF_2:def 14
theorem
for b1 being non empty set
for b2 being Element of bool b1 holds
      b2 is countable
   iff
      (b2 is not empty implies ex b3 being Function-like quasi_total Relation of NAT,b1 st
         b2 = rng b3);

:: SUPINF_2:exreg 1
registration
  cluster non empty ext-real-membered countable Element of bool ExtREAL;
end;

:: SUPINF_2:modenot 1
definition
  mode Denum_Set_of_R_EAL is non empty countable Element of bool ExtREAL;
end;

:: SUPINF_2:attrnot 6 => SUPINF_2:attr 5
definition
  let a1 be set;
  attr a1 is nonnegative means
    for b1 being Element of ExtREAL
          st b1 in a1
       holds 0. <= b1;
end;

:: SUPINF_2:dfs 14
definiens
  let a1 be set;
To prove
     a1 is nonnegative
it is sufficient to prove
  thus for b1 being Element of ExtREAL
          st b1 in a1
       holds 0. <= b1;

:: SUPINF_2:def 15
theorem
for b1 being set holds
      b1 is nonnegative
   iff
      for b2 being Element of ExtREAL
            st b2 in b1
         holds 0. <= b2;

:: SUPINF_2:exreg 2
registration
  cluster non empty ext-real-membered countable nonnegative Element of bool ExtREAL;
end;

:: SUPINF_2:modenot 2
definition
  mode Pos_Denum_Set_of_R_EAL is non empty countable nonnegative Element of bool ExtREAL;
end;

:: SUPINF_2:modenot 3 => SUPINF_2:mode 1
definition
  let a1 be non empty countable Element of bool ExtREAL;
  mode Num of A1 -> Function-like quasi_total Relation of NAT,ExtREAL means
    a1 = rng it;
end;

:: SUPINF_2:dfs 15
definiens
  let a1 be non empty countable Element of bool ExtREAL;
  let a2 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a2 is Num of a1
it is sufficient to prove
  thus a1 = rng a2;

:: SUPINF_2:def 16
theorem
for b1 being non empty countable Element of bool ExtREAL
for b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b2 is Num of b1
   iff
      b1 = rng b2;

:: SUPINF_2:th 53
theorem
for b1 being non empty countable Element of bool ExtREAL
for b2 being Num of b1 holds
   ex b3 being Function-like quasi_total Relation of NAT,ExtREAL st
      b3 . 0 = b2 . 0 &
       (for b4 being Element of NAT
       for b5 being Element of ExtREAL
             st b5 = b3 . b4
          holds b3 . (b4 + 1) = b5 + (b2 . (b4 + 1)));

:: SUPINF_2:funcnot 16 => SUPINF_2:func 15
definition
  let a1 be non empty countable Element of bool ExtREAL;
  let a2 be Num of a1;
  func Ser(A1,A2) -> Function-like quasi_total Relation of NAT,ExtREAL means
    it . 0 = a2 . 0 &
     (for b1 being Element of NAT
     for b2 being Element of ExtREAL
           st b2 = it . b1
        holds it . (b1 + 1) = b2 + (a2 . (b1 + 1)));
end;

:: SUPINF_2:def 17
theorem
for b1 being non empty countable Element of bool ExtREAL
for b2 being Num of b1
for b3 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b3 = Ser(b1,b2)
   iff
      b3 . 0 = b2 . 0 &
       (for b4 being Element of NAT
       for b5 being Element of ExtREAL
             st b5 = b3 . b4
          holds b3 . (b4 + 1) = b5 + (b2 . (b4 + 1)));

:: SUPINF_2:th 54
theorem
for b1 being non empty countable nonnegative Element of bool ExtREAL
for b2 being Num of b1
for b3 being Element of NAT holds
   0. <= b2 . b3;

:: SUPINF_2:th 55
theorem
for b1 being non empty countable nonnegative Element of bool ExtREAL
for b2 being Num of b1
for b3 being Element of NAT holds
   (Ser(b1,b2)) . b3 <= (Ser(b1,b2)) . (b3 + 1) &
    0. <= (Ser(b1,b2)) . b3;

:: SUPINF_2:th 56
theorem
for b1 being non empty countable nonnegative Element of bool ExtREAL
for b2 being Num of b1
for b3, b4 being Element of NAT holds
(Ser(b1,b2)) . b3 <= (Ser(b1,b2)) . (b3 + b4);

:: SUPINF_2:modenot 4 => SUPINF_2:mode 2
definition
  let a1 be non empty countable Element of bool ExtREAL;
  mode Set_of_Series of A1 -> non empty Element of bool ExtREAL means
    ex b1 being Num of a1 st
       it = rng Ser(a1,b1);
end;

:: SUPINF_2:dfs 17
definiens
  let a1 be non empty countable Element of bool ExtREAL;
  let a2 be non empty Element of bool ExtREAL;
To prove
     a2 is Set_of_Series of a1
it is sufficient to prove
  thus ex b1 being Num of a1 st
       a2 = rng Ser(a1,b1);

:: SUPINF_2:def 18
theorem
for b1 being non empty countable Element of bool ExtREAL
for b2 being non empty Element of bool ExtREAL holds
      b2 is Set_of_Series of b1
   iff
      ex b3 being Num of b1 st
         b2 = rng Ser(b1,b3);

:: SUPINF_2:funcnot 17 => SUPINF_2:func 16
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  redefine func rng a1 -> non empty Element of bool ExtREAL;
end;

:: SUPINF_2:funcnot 18 => SUPINF_2:func 17
definition
  let a1 be non empty countable nonnegative Element of bool ExtREAL;
  let a2 be Num of a1;
  func SUM(A1,A2) -> Element of ExtREAL equals
    sup rng Ser(a1,a2);
end;

:: SUPINF_2:def 19
theorem
for b1 being non empty countable nonnegative Element of bool ExtREAL
for b2 being Num of b1 holds
   SUM(b1,b2) = sup rng Ser(b1,b2);

:: SUPINF_2:prednot 1 => SUPINF_2:pred 1
definition
  let a1 be non empty countable nonnegative Element of bool ExtREAL;
  let a2 be Num of a1;
  pred A1 is_sumable A2 means
    SUM(a1,a2) in REAL;
end;

:: SUPINF_2:dfs 19
definiens
  let a1 be non empty countable nonnegative Element of bool ExtREAL;
  let a2 be Num of a1;
To prove
     a1 is_sumable a2
it is sufficient to prove
  thus SUM(a1,a2) in REAL;

:: SUPINF_2:def 20
theorem
for b1 being non empty countable nonnegative Element of bool ExtREAL
for b2 being Num of b1 holds
      b1 is_sumable b2
   iff
      SUM(b1,b2) in REAL;

:: SUPINF_2:th 57
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
   rng b1 is non empty countable Element of bool ExtREAL;

:: SUPINF_2:funcnot 19 => SUPINF_2:func 18
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  redefine func rng a1 -> non empty countable Element of bool ExtREAL;
end;

:: SUPINF_2:funcnot 20 => SUPINF_2:func 19
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  func Ser A1 -> Function-like quasi_total Relation of NAT,ExtREAL means
    for b1 being Num of rng a1
          st b1 = a1
       holds it = Ser(rng a1,b1);
end;

:: SUPINF_2:def 21
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL holds
   b2 = Ser b1
iff
   for b3 being Num of rng b1
         st b3 = b1
      holds b2 = Ser(rng b1,b3);

:: SUPINF_2:attrnot 7 => SUPINF_2:attr 6
definition
  let a1 be Relation-like set;
  attr a1 is nonnegative means
    proj2 a1 is nonnegative;
end;

:: SUPINF_2:dfs 21
definiens
  let a1 be Relation-like set;
To prove
     a1 is nonnegative
it is sufficient to prove
  thus proj2 a1 is nonnegative;

:: SUPINF_2:def 22
theorem
for b1 being Relation-like set holds
      b1 is nonnegative
   iff
      proj2 b1 is nonnegative;

:: SUPINF_2:funcnot 21 => SUPINF_2:func 20
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  func SUM A1 -> Element of ExtREAL equals
    sup rng Ser a1;
end;

:: SUPINF_2:def 23
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
   SUM b1 = sup rng Ser b1;

:: SUPINF_2:th 58
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is nonnegative
   iff
      for b3 being Element of b1 holds
         0. <= b2 . b3;

:: SUPINF_2:th 59
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
for b2 being Element of NAT
      st b1 is nonnegative
   holds (Ser b1) . b2 <= (Ser b1) . (b2 + 1) &
    0. <= (Ser b1) . b2;

:: SUPINF_2:th 60
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2, b3 being Element of NAT holds
(Ser b1) . b2 <= (Ser b1) . (b2 + b3);

:: SUPINF_2:th 61
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative &
      (for b3 being Element of NAT holds
         b1 . b3 <= b2 . b3)
for b3 being Element of NAT holds
   (Ser b1) . b3 <= (Ser b2) . b3;

:: SUPINF_2:th 62
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is nonnegative &
         (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3)
   holds SUM b1 <= SUM b2;

:: SUPINF_2:th 63
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
   (Ser b1) . 0 = b1 . 0 &
    (for b2 being Element of NAT
    for b3 being Element of ExtREAL
          st b3 = (Ser b1) . b2
       holds (Ser b1) . (b2 + 1) = b3 + (b1 . (b2 + 1)));

:: SUPINF_2:th 64
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is nonnegative &
         (ex b2 being Element of NAT st
            b1 . b2 = +infty)
   holds SUM b1 = +infty;

:: SUPINF_2:attrnot 8 => SUPINF_2:attr 7
definition
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
  attr a1 is summable means
    SUM a1 in REAL;
end;

:: SUPINF_2:dfs 23
definiens
  let a1 be Function-like quasi_total Relation of NAT,ExtREAL;
To prove
     a1 is summable
it is sufficient to prove
  thus SUM a1 in REAL;

:: SUPINF_2:def 24
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL holds
      b1 is summable
   iff
      SUM b1 in REAL;

:: SUPINF_2:th 65
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is nonnegative &
         (ex b2 being Element of NAT st
            b1 . b2 = +infty)
   holds b1 is not summable;

:: SUPINF_2:th 66
theorem
for b1, b2 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is nonnegative &
         (for b3 being Element of NAT holds
            b1 . b3 <= b2 . b3) &
         b2 is summable
   holds b1 is summable;

:: SUPINF_2:th 67
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st b1 is nonnegative
for b2 being natural set
      st for b3 being Element of NAT
              st b2 <= b3
           holds b1 . b3 = 0.
   holds SUM b1 = (Ser b1) . b2;

:: SUPINF_2:th 68
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
   st for b2 being Element of NAT holds
        b1 . b2 in REAL
for b2 being Element of NAT holds
   (Ser b1) . b2 in REAL;

:: SUPINF_2:th 69
theorem
for b1 being Function-like quasi_total Relation of NAT,ExtREAL
      st b1 is nonnegative &
         (ex b2 being Element of NAT st
            (for b3 being Element of NAT
                   st b2 <= b3
                holds b1 . b3 = 0.) &
             (for b3 being Element of NAT
                   st b3 <= b2
                holds b1 . b3 <> +infty))
   holds b1 is summable;

:: SUPINF_2:th 70
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL holds
      b2 is nonnegative
   iff
      for b3 being set holds
         0. <= b2 . b3;

:: SUPINF_2:th 71
theorem
for b1 being set
for b2 being Function-like Relation of b1,ExtREAL
      st for b3 being set
              st b3 in dom b2
           holds 0. <= b2 . b3
   holds b2 is nonnegative;