Article REAL_1, MML version 4.99.1005

:: REAL_1:condreg 1
registration
  cluster -> real (Element of REAL);
end;

:: REAL_1:modenot 1
definition
  mode Real is Element of REAL;
end;

:: REAL_1:exreg 1
registration
  cluster complex ext-real positive real Element of REAL;
end;

:: REAL_1:funcnot 1 => REAL_1:func 1
definition
  let a1 be Element of REAL;
  redefine func - a1 -> Element of REAL;
  involutiveness;
::  for a1 being Element of REAL holds
::     - - a1 = a1;
end;

:: REAL_1:funcnot 2 => REAL_1:func 2
definition
  let a1 be Element of REAL;
  redefine func a1 " -> Element of REAL;
  involutiveness;
::  for a1 being Element of REAL holds
::     a1 " " = a1;
end;

:: REAL_1:funcnot 3 => REAL_1:func 3
definition
  let a1 be real set;
  let a2 be Element of REAL;
  redefine func a1 + a2 -> Element of REAL;
  commutativity;
::  for a1 being real set
::  for a2 being Element of REAL holds
::     a1 + a2 = a2 + a1;
end;

:: REAL_1:funcnot 4 => REAL_1:func 4
definition
  let a1 be real set;
  let a2 be Element of REAL;
  redefine func a1 * a2 -> Element of REAL;
  commutativity;
::  for a1 being real set
::  for a2 being Element of REAL holds
::     a1 * a2 = a2 * a1;
end;

:: REAL_1:funcnot 5 => REAL_1:func 5
definition
  let a1 be real set;
  let a2 be Element of REAL;
  redefine func a1 - a2 -> Element of REAL;
end;

:: REAL_1:funcnot 6 => REAL_1:func 6
definition
  let a1 be real set;
  let a2 be Element of REAL;
  redefine func a1 / a2 -> Element of REAL;
end;

:: REAL_1:funcnot 7 => REAL_1:func 7
definition
  let a1 be Element of REAL;
  let a2 be real set;
  redefine func a1 + a2 -> Element of REAL;
  commutativity;
::  for a1 being Element of REAL
::  for a2 being real set holds
::     a1 + a2 = a2 + a1;
end;

:: REAL_1:funcnot 8 => REAL_1:func 8
definition
  let a1 be Element of REAL;
  let a2 be real set;
  redefine func a1 * a2 -> Element of REAL;
  commutativity;
::  for a1 being Element of REAL
::  for a2 being real set holds
::     a1 * a2 = a2 * a1;
end;

:: REAL_1:funcnot 9 => REAL_1:func 9
definition
  let a1 be Element of REAL;
  let a2 be real set;
  redefine func a1 - a2 -> Element of REAL;
end;

:: REAL_1:funcnot 10 => REAL_1:func 10
definition
  let a1 be Element of REAL;
  let a2 be real set;
  redefine func a1 / a2 -> Element of REAL;
end;

:: REAL_1:th 25
theorem
for b1 being real set holds
   b1 - 0 = b1;

:: REAL_1:th 26
theorem
- 0 = 0;

:: REAL_1:prednot 1 => not XXREAL_0:pred 1
definition
  let a1, a2 be ext-real set;
  pred A2 < A1 means
    (a2 <= a1) implies a2 = a1;
  reflexivity;
::  for a1 being ext-real set holds
::     a1 <= a1;
  connectedness;
::  for a1, a2 being ext-real set
::        st a2 < a1
::     holds a2 <= a1;
end;

:: REAL_1:dfs 1
definiens
  let a1, a2 be real set;
To prove
     a1 <= a2
it is sufficient to prove
  thus (a2 <= a1) implies a2 = a1;

:: REAL_1:def 5
theorem
for b1, b2 being real set holds
   b1 <= b2
iff
   (b2 <= b1 implies b2 = b1);

:: REAL_1:th 73
theorem
for b1, b2, b3 being real set
      st 0 < b1
   holds    b2 < b3
   iff
      b2 / b1 < b3 / b1;

:: REAL_1:th 74
theorem
for b1, b2, b3 being real set
      st b1 < 0
   holds    b2 < b3
   iff
      b3 / b1 < b2 / b1;

:: REAL_1:sch 1
scheme REAL_1:sch 1
ex b1 being Element of bool REAL st
   for b2 being Element of REAL holds
         b2 in b1
      iff
         P1[b2]


:: REAL_1:th 92
theorem
for b1, b2, b3, b4 being real set holds
(b1 <= b2 & b3 <= b4 implies b1 - b4 <= b2 - b3) &
 ((b1 < b2 & b3 <= b4 or b1 <= b2 & b3 < b4) implies b1 - b4 < b2 - b3);