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)];