Article ARYTM_1, MML version 4.99.1005

:: ARYTM_1:th 1
theorem
for b1, b2 being Element of REAL+
      st b1 + b2 = b2
   holds b1 = {};

:: ARYTM_1:th 2
theorem
for b1, b2 being Element of REAL+
      st b1 *' b2 = {} & b1 <> {}
   holds b2 = {};

:: ARYTM_1:th 3
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b2 <=' b3
   holds b1 <=' b3;

:: ARYTM_1:th 4
theorem
for b1, b2 being Element of REAL+
      st b1 <=' b2 & b2 <=' b1
   holds b1 = b2;

:: ARYTM_1:th 5
theorem
for b1, b2 being Element of REAL+
      st b1 <=' b2 & b2 = {}
   holds b1 = {};

:: ARYTM_1:th 6
theorem
for b1, b2 being Element of REAL+
      st b1 = {}
   holds b1 <=' b2;

:: ARYTM_1:th 7
theorem
for b1, b2, b3 being Element of REAL+ holds
   b1 <=' b2
iff
   b1 + b3 <=' b2 + b3;

:: ARYTM_1:th 8
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b1 *' b3 <=' b2 *' b3;

:: ARYTM_1:funcnot 1 => ARYTM_1:func 1
definition
  let a1, a2 be Element of REAL+;
  func A1 -' A2 -> Element of REAL+ means
    it + a2 = a1
    if a2 <=' a1
    otherwise it = {};
end;

:: ARYTM_1:def 1
theorem
for b1, b2, b3 being Element of REAL+ holds
(b2 <=' b1 implies    (b3 = b1 -' b2
 iff
    b3 + b2 = b1)) &
 (b2 <=' b1 or    (b3 = b1 -' b2
 iff
    b3 = {}));

:: ARYTM_1:th 9
theorem
for b1, b2 being Element of REAL+
      st b2 < b1
   holds b1 -' b2 <> {};

:: ARYTM_1:th 10
theorem
for b1, b2 being Element of REAL+
      st b1 <=' b2 & b2 -' b1 = {}
   holds b1 = b2;

:: ARYTM_1:th 11
theorem
for b1, b2 being Element of REAL+ holds
b1 -' b2 <=' b1;

:: ARYTM_1:th 12
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b1 <=' b3
   holds b2 + (b3 -' b1) = (b2 -' b1) + b3;

:: ARYTM_1:th 13
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b3 + (b2 -' b1) = (b3 + b2) -' b1;

:: ARYTM_1:th 14
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b3 <=' b1
   holds (b2 -' b1) + b3 = b2 -' (b1 -' b3);

:: ARYTM_1:th 15
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b1 <=' b3
   holds (b3 -' b1) + b2 = (b2 -' b1) + b3;

:: ARYTM_1:th 16
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b3 -' b2 <=' b3 -' b1;

:: ARYTM_1:th 17
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b1 -' b3 <=' b2 -' b3;

:: ARYTM_1:funcnot 2 => ARYTM_1:func 2
definition
  let a1, a2 be Element of REAL+;
  func A1 - A2 -> set equals
    a1 -' a2
    if a2 <=' a1
    otherwise [{},a2 -' a1];
end;

:: ARYTM_1:def 2
theorem
for b1, b2 being Element of REAL+ holds
(b2 <=' b1 implies b1 - b2 = b1 -' b2) &
 (b2 <=' b1 or b1 - b2 = [{},b2 -' b1]);

:: ARYTM_1:th 18
theorem
for b1 being Element of REAL+ holds
   b1 - b1 = {};

:: ARYTM_1:th 19
theorem
for b1, b2 being Element of REAL+
      st b1 = {} & b2 <> {}
   holds b1 - b2 = [{},b2];

:: ARYTM_1:th 20
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b3 + (b2 -' b1) = (b3 + b2) - b1;

:: ARYTM_1:th 21
theorem
for b1, b2, b3 being Element of REAL+
      st b2 < b1
   holds b3 - (b1 -' b2) = (b3 + b2) - b1;

:: ARYTM_1:th 22
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b3 < b1
   holds b2 - (b1 -' b3) = (b2 -' b1) + b3;

:: ARYTM_1:th 23
theorem
for b1, b2, b3 being Element of REAL+
      st b2 < b1 & b3 < b1
   holds b2 - (b1 -' b3) = b3 - (b1 -' b2);

:: ARYTM_1:th 24
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b2 - (b1 + b3) = (b2 -' b1) - b3;

:: ARYTM_1:th 25
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2 & b3 <=' b2
   holds (b2 -' b3) - b1 = (b2 -' b1) - b3;

:: ARYTM_1:th 26
theorem
for b1, b2, b3 being Element of REAL+
      st b1 <=' b2
   holds b3 *' (b2 -' b1) = (b3 *' b2) - (b3 *' b1);

:: ARYTM_1:th 27
theorem
for b1, b2, b3 being Element of REAL+
      st b2 < b1 & b3 <> {}
   holds [{},b3 *' (b1 -' b2)] = (b3 *' b2) - (b3 *' b1);

:: ARYTM_1:th 28
theorem
for b1, b2, b3 being Element of REAL+
      st b1 -' b2 <> {} & b2 <=' b1 & b3 <> {}
   holds (b3 *' b2) - (b3 *' b1) = [{},b3 *' (b1 -' b2)];