Article EXTREAL2, MML version 4.99.1005

:: EXTREAL2:th 2
theorem
for b1 being Element of ExtREAL
      st b1 <> +infty & b1 <> -infty
   holds ex b2 being Element of ExtREAL st
      b1 + b2 = 0;

:: EXTREAL2:th 3
theorem
for b1 being Element of ExtREAL
      st b1 <> +infty & b1 <> -infty & b1 <> 0
   holds ex b2 being Element of ExtREAL st
      b1 * b2 = 1.;

:: EXTREAL2:th 4
theorem
for b1 being Element of ExtREAL holds
   1. * b1 = b1 & (R_EAL 1) * b1 = b1;

:: EXTREAL2:th 5
theorem
for b1 being Element of ExtREAL holds
   0. - b1 = - b1;

:: EXTREAL2:th 7
theorem
for b1, b2 being Element of ExtREAL
      st 0 <= b1 & 0 <= b2
   holds 0 <= b1 + b2;

:: EXTREAL2:th 9
theorem
for b1, b2 being Element of ExtREAL
      st b1 <= 0 & b2 <= 0
   holds b1 + b2 <= 0;

:: EXTREAL2:th 11
theorem
for b1, b2, b3 being Element of ExtREAL
      st b1 <> +infty & b1 <> -infty & b2 + b1 = b3
   holds b2 = b3 - b1;

:: EXTREAL2:th 12
theorem
for b1 being Element of ExtREAL
      st b1 <> +infty & b1 <> -infty & b1 <> 0
   holds b1 * (1. / b1) = 1. & (1. / b1) * b1 = 1.;

:: EXTREAL2:th 13
theorem
for b1 being Element of ExtREAL holds
   b1 - b1 = 0;

:: EXTREAL2:th 14
theorem
for b1, b2 being Element of ExtREAL holds
- (b1 + b2) = (- b2) - b1;

:: EXTREAL2:th 15
theorem
for b1, b2 being Element of ExtREAL holds
- (b1 - b2) = (- b1) + b2 & - (b1 - b2) = b2 - b1;

:: EXTREAL2:th 16
theorem
for b1, b2 being Element of ExtREAL holds
- ((- b1) + b2) = b1 - b2 &
 - ((- b1) + b2) = b1 + - b2;

:: EXTREAL2:th 17
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty & 0 < b2 & b2 < +infty or b1 = -infty & b2 < 0 & -infty < b2)
   holds b1 / b2 = +infty;

:: EXTREAL2:th 18
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty & b2 < 0 & -infty < b2 or b1 = -infty & 0 < b2 & b2 < +infty)
   holds b1 / b2 = -infty;

:: EXTREAL2:th 19
theorem
for b1, b2 being Element of ExtREAL
      st -infty < b1 & b1 < +infty & b1 <> 0
   holds (b2 * b1) / b1 = b2 & b2 * (b1 / b1) = b2;

:: EXTREAL2:th 20
theorem
1. < +infty & -infty < 1.;

:: EXTREAL2:th 21
theorem
for b1 being Element of ExtREAL
   st (b1 = +infty or b1 = -infty)
for b2 being Element of ExtREAL
      st b2 in REAL
   holds b1 + b2 <> 0;

:: EXTREAL2:th 22
theorem
for b1 being Element of ExtREAL
   st (b1 = +infty or b1 = -infty)
for b2 being Element of ExtREAL holds
   b1 * b2 <> 1.;

:: EXTREAL2:th 23
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & b1 + b2 < +infty
   holds b1 <> +infty & b2 <> +infty;

:: EXTREAL2:th 24
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & -infty < b1 + b2
   holds b1 <> -infty & b2 <> -infty;

:: EXTREAL2:th 25
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & b1 - b2 < +infty
   holds b1 <> +infty & b2 <> -infty;

:: EXTREAL2:th 26
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & -infty < b1 - b2
   holds b1 <> -infty & b2 <> +infty;

:: EXTREAL2:th 27
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & b1 + b2 < b3
   holds b1 <> +infty & b2 <> +infty & b3 <> -infty & b1 < b3 - b2;

:: EXTREAL2:th 28
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & b3 < b1 - b2
   holds b3 <> +infty & b2 <> +infty & b1 <> -infty & b3 + b2 < b1;

:: EXTREAL2:th 29
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & b1 - b2 < b3
   holds b1 <> +infty & b2 <> -infty & b3 <> -infty & b1 < b3 + b2;

:: EXTREAL2:th 30
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & b3 < b1 + b2
   holds b3 <> +infty & b2 <> -infty & b1 <> -infty & b3 - b2 < b1;

:: EXTREAL2:th 31
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & (b2 = +infty implies b3 <> +infty) & (b2 = -infty implies b3 <> -infty) & b1 + b2 <= b3
   holds b2 <> +infty & b1 <= b3 - b2;

:: EXTREAL2:th 32
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & (b2 = +infty implies b3 <> +infty) & b1 <= b3 - b2
   holds b2 <> +infty & b1 + b2 <= b3;

:: EXTREAL2:th 33
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & (b2 = +infty implies b3 <> -infty) & (b2 = -infty implies b3 <> +infty) & b1 - b2 <= b3
   holds b2 <> -infty & b1 <= b3 + b2;

:: EXTREAL2:th 34
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & (b2 = -infty implies b3 <> +infty) & b1 <= b3 + b2
   holds b2 <> -infty & b1 - b2 <= b3;

:: EXTREAL2:th 40
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & (b2 = +infty implies b3 <> -infty) & (b2 = -infty implies b3 <> +infty) & (b1 = +infty implies b3 <> +infty) & (b1 = -infty implies b3 <> -infty)
   holds (b1 - b2) - b3 = b1 - (b2 + b3);

:: EXTREAL2:th 41
theorem
for b1, b2, b3 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty) & (b2 = +infty implies b3 <> +infty) & (b2 = -infty implies b3 <> -infty) & (b1 = +infty implies b3 <> -infty) & (b1 = -infty implies b3 <> +infty)
   holds (b1 - b2) + b3 = b1 - (b2 - b3);

:: EXTREAL2:th 42
theorem
for b1, b2 being Element of ExtREAL
      st b1 * b2 <> +infty & b1 * b2 <> -infty & b1 is not Element of REAL
   holds b2 is Element of REAL;

:: EXTREAL2:th 43
theorem
for b1, b2 being Element of ExtREAL holds
   (0 < b1 & 0 < b2 or b1 < 0 & b2 < 0)
iff
   0 < b1 * b2;

:: EXTREAL2:th 44
theorem
for b1, b2 being Element of ExtREAL holds
   (0 < b1 & b2 < 0 or b1 < 0 & 0 < b2)
iff
   b1 * b2 < 0;

:: EXTREAL2:th 45
theorem
for b1, b2 being Element of ExtREAL holds
   ((b1 < 0 implies 0 < b1) & (b2 < 0 implies 0 < b2) or (0 < b1 implies b1 < 0) & (0 < b2 implies b2 < 0))
iff
   0 <= b1 * b2;

:: EXTREAL2:th 46
theorem
for b1, b2 being Element of ExtREAL holds
   ((0 < b1 implies b1 < 0) & (b2 < 0 implies 0 < b2) or (b1 < 0 implies 0 < b1) & (0 < b2 implies b2 < 0))
iff
   b1 * b2 <= 0;

:: EXTREAL2:th 47
theorem
for b1, b2 being Element of ExtREAL holds
(b1 <= - b2 implies b2 <= - b1) & (- b1 <= b2 implies - b2 <= b1);

:: EXTREAL2:th 49
theorem
for b1 being Element of ExtREAL
for b2 being Element of REAL
      st b1 = b2
   holds |.b1.| = abs b2;

:: EXTREAL2:th 50
theorem
for b1 being Element of ExtREAL
      st |.b1.| <> b1
   holds |.b1.| = - b1;

:: EXTREAL2:th 51
theorem
for b1 being Element of ExtREAL holds
   0 <= |.b1.|;

:: EXTREAL2:th 52
theorem
for b1 being Element of ExtREAL
      st b1 <> 0
   holds 0 < |.b1.|;

:: EXTREAL2:th 53
theorem
for b1 being Element of ExtREAL holds
      b1 = 0
   iff
      |.b1.| = 0;

:: EXTREAL2:th 54
theorem
for b1 being Element of ExtREAL
      st |.b1.| = - b1 & b1 <> 0
   holds b1 < 0;

:: EXTREAL2:th 55
theorem
for b1 being Element of ExtREAL
      st b1 <= 0
   holds |.b1.| = - b1;

:: EXTREAL2:th 56
theorem
for b1, b2 being Element of ExtREAL holds
|.b1 * b2.| = |.b1.| * |.b2.|;

:: EXTREAL2:th 57
theorem
for b1 being Element of ExtREAL holds
   - |.b1.| <= b1 & b1 <= |.b1.|;

:: EXTREAL2:th 58
theorem
for b1, b2 being Element of ExtREAL
      st |.b1.| < b2
   holds - b2 < b1 & b1 < b2;

:: EXTREAL2:th 59
theorem
for b1, b2 being Element of ExtREAL
      st - b1 < b2 & b2 < b1
   holds 0 < b1 & |.b2.| < b1;

:: EXTREAL2:th 60
theorem
for b1, b2 being Element of ExtREAL holds
   - b1 <= b2 & b2 <= b1
iff
   |.b2.| <= b1;

:: EXTREAL2:th 61
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty)
   holds |.b1 + b2.| <= |.b1.| + |.b2.|;

:: EXTREAL2:th 62
theorem
for b1 being Element of ExtREAL
      st -infty < b1 & b1 < +infty & b1 <> 0
   holds |.b1.| * |.1. / b1.| = 1.;

:: EXTREAL2:th 63
theorem
for b1 being Element of ExtREAL
      st (b1 = +infty or b1 = -infty)
   holds |.b1.| * |.1. / b1.| = 0;

:: EXTREAL2:th 64
theorem
for b1 being Element of ExtREAL
      st b1 <> 0
   holds |.1. / b1.| = 1. / |.b1.|;

:: EXTREAL2:th 65
theorem
for b1, b2 being Element of ExtREAL
      st (b1 <> -infty & b1 <> +infty or b2 <> -infty & b2 <> +infty) &
         b2 <> 0
   holds |.b1 / b2.| = |.b1.| / |.b2.|;

:: EXTREAL2:th 66
theorem
for b1 being Element of ExtREAL holds
   |.b1.| = |.- b1.|;

:: EXTREAL2:th 67
theorem
|.+infty.| = +infty & |.-infty.| = +infty;

:: EXTREAL2:th 68
theorem
for b1, b2 being Element of ExtREAL
      st (b1 is Element of REAL or b2 is Element of REAL)
   holds |.b1.| - |.b2.| <= |.b1 - b2.|;

:: EXTREAL2:th 69
theorem
for b1, b2 being Element of ExtREAL
      st (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty)
   holds |.b1 - b2.| <= |.b1.| + |.b2.|;

:: EXTREAL2:th 70
theorem
for b1 being Element of ExtREAL holds
   |.|.b1.|.| = |.b1.|;

:: EXTREAL2:th 71
theorem
for b1, b2, b3, b4 being Element of ExtREAL
      st (b1 = +infty implies b2 <> -infty) & (b1 = -infty implies b2 <> +infty) & |.b1.| <= b3 & |.b2.| <= b4
   holds |.b1 + b2.| <= b3 + b4;

:: EXTREAL2:th 72
theorem
for b1, b2 being Element of ExtREAL
      st (b1 is Element of REAL or b2 is Element of REAL)
   holds |.|.b1.| - |.b2.|.| <= |.b1 - b2.|;

:: EXTREAL2:th 73
theorem
for b1, b2 being Element of ExtREAL
      st 0 <= b1 * b2
   holds |.b1 + b2.| = |.b1.| + |.b2.|;

:: EXTREAL2:th 80
theorem
for b1, b2 being Element of ExtREAL
      st b1 <> +infty & b2 <> +infty & (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty)
   holds min(b1,b2) = ((b1 + b2) - |.b1 - b2.|) / R_EAL 2;

:: EXTREAL2:th 91
theorem
for b1, b2 being Element of ExtREAL
      st b1 <> -infty & b2 <> -infty & (b1 = +infty implies b2 <> +infty) & (b1 = -infty implies b2 <> -infty)
   holds max(b1,b2) = ((b1 + b2) + |.b1 - b2.|) / R_EAL 2;

:: EXTREAL2:funcnot 1 => EXTREAL2:func 1
definition
  let a1, a2 be Element of ExtREAL;
  redefine func max(a1,a2) -> Element of ExtREAL;
  commutativity;
::  for a1, a2 being Element of ExtREAL holds
::  max(a1,a2) = max(a2,a1);
  idempotence;
::  for a1 being Element of ExtREAL holds
::     max(a1,a1) = a1;
end;

:: EXTREAL2:funcnot 2 => EXTREAL2:func 2
definition
  let a1, a2 be Element of ExtREAL;
  redefine func min(a1,a2) -> Element of ExtREAL;
  commutativity;
::  for a1, a2 being Element of ExtREAL holds
::  min(a1,a2) = min(a2,a1);
  idempotence;
::  for a1 being Element of ExtREAL holds
::     min(a1,a1) = a1;
end;

:: EXTREAL2:th 100
theorem
for b1, b2 being Element of ExtREAL holds
(min(b1,b2)) + max(b1,b2) = b1 + b2;

:: EXTREAL2:th 101
theorem
for b1 being Element of ExtREAL
      st |.b1.| = +infty & b1 <> +infty
   holds b1 = -infty;

:: EXTREAL2:th 102
theorem
for b1 being Element of ExtREAL holds
      |.b1.| < +infty
   iff
      b1 in REAL;