Article RCOMP_2, MML version 4.99.1005
:: RCOMP_2:th 2
theorem
for b1, b2, b3 being real set holds
b1 < b2 & b3 < b2
iff
max(b1,b3) < b2;
:: RCOMP_2:funcnot 1 => RCOMP_2:func 1
definition
let a1 be real set;
let a2 be ext-real set;
redefine func [.A1,A2.[ -> Element of bool REAL equals
{b1 where b1 is Element of REAL: a1 <= b1 & b1 < a2};
end;
:: RCOMP_2:def 1
theorem
for b1 being real set
for b2 being ext-real set holds
[.b1,b2.[ = {b3 where b3 is Element of REAL: b1 <= b3 & b3 < b2};
:: RCOMP_2:funcnot 2 => RCOMP_2:func 2
definition
let a1 be ext-real set;
let a2 be real set;
redefine func ].A1,A2.] -> Element of bool REAL equals
{b1 where b1 is Element of REAL: a1 < b1 & b1 <= a2};
end;
:: RCOMP_2:def 2
theorem
for b1 being ext-real set
for b2 being real set holds
].b1,b2.] = {b3 where b3 is Element of REAL: b1 < b3 & b3 <= b2};
:: RCOMP_2:th 3
theorem
for b1, b2, b3 being real set holds
b1 in [.b2,b3.[
iff
b2 <= b1 & b1 < b3;
:: RCOMP_2:th 4
theorem
for b1, b2, b3 being real set holds
b1 in ].b2,b3.]
iff
b2 < b1 & b1 <= b3;
:: RCOMP_2:th 5
theorem
for b1, b2 being real set
st b1 < b2
holds [.b1,b2.[ = ].b1,b2.[ \/ {b1};
:: RCOMP_2:th 6
theorem
for b1, b2 being real set
st b1 < b2
holds ].b1,b2.] = ].b1,b2.[ \/ {b2};
:: RCOMP_2:th 7
theorem
for b1 being real set holds
[.b1,b1.[ = {};
:: RCOMP_2:th 8
theorem
for b1 being real set holds
].b1,b1.] = {};
:: RCOMP_2:th 9
theorem
for b1, b2 being real set
st b1 <= b2
holds [.b2,b1.[ = {};
:: RCOMP_2:th 10
theorem
for b1, b2 being real set
st b1 <= b2
holds ].b2,b1.] = {};
:: RCOMP_2:th 11
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds [.b1,b2.[ \/ [.b2,b3.[ = [.b1,b3.[;
:: RCOMP_2:th 12
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds ].b1,b2.] \/ ].b2,b3.] = ].b1,b3.];
:: RCOMP_2:th 13
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b1 <= b3 & b2 <= b4 & b3 <= b4
holds [.b1,b4.] = ([.b1,b2.[ \/ [.b2,b3.]) \/ ].b3,b4.];
:: RCOMP_2:th 14
theorem
for b1, b2, b3, b4 being real set
st b1 < b2 & b1 < b3 & b2 < b4 & b3 < b4
holds ].b1,b4.[ = (].b1,b2.] \/ ].b2,b3.[) \/ [.b3,b4.[;
:: RCOMP_2:th 15
theorem
for b1, b2, b3, b4 being real set holds
[.b1,b2.[ /\ [.b3,b4.[ = [.max(b1,b3),min(b2,b4).[;
:: RCOMP_2:th 16
theorem
for b1, b2, b3, b4 being real set holds
].b1,b2.] /\ ].b3,b4.] = ].max(b1,b3),min(b2,b4).];
:: RCOMP_2:th 17
theorem
for b1, b2 being real set holds
].b1,b2.[ c= [.b1,b2.[ & ].b1,b2.[ c= ].b1,b2.] & [.b1,b2.[ c= [.b1,b2.] & ].b1,b2.] c= [.b1,b2.];
:: RCOMP_2:th 18
theorem
for b1, b2, b3, b4 being real set
st b1 in [.b2,b3.[ & b4 in [.b2,b3.[
holds [.b1,b4.] c= [.b2,b3.[;
:: RCOMP_2:th 19
theorem
for b1, b2, b3, b4 being real set
st b1 in ].b2,b3.] & b4 in ].b2,b3.]
holds [.b1,b4.] c= ].b2,b3.];
:: RCOMP_2:th 20
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds [.b1,b2.] \/ ].b2,b3.] = [.b1,b3.];
:: RCOMP_2:th 21
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds [.b1,b2.[ \/ [.b2,b3.] = [.b1,b3.];
:: RCOMP_2:th 22
theorem
for b1, b2, b3, b4 being real set
st [.b1,b2.[ meets [.b3,b4.[
holds b3 <= b2;
:: RCOMP_2:th 23
theorem
for b1, b2, b3, b4 being real set
st ].b1,b2.] meets ].b3,b4.]
holds b3 <= b2;
:: RCOMP_2:th 24
theorem
for b1, b2, b3, b4 being real set
st [.b1,b2.[ meets [.b3,b4.[
holds [.b1,b2.[ \/ [.b3,b4.[ = [.min(b1,b3),max(b2,b4).[;
:: RCOMP_2:th 25
theorem
for b1, b2, b3, b4 being real set
st ].b1,b2.] meets ].b3,b4.]
holds ].b1,b2.] \/ ].b3,b4.] = ].min(b1,b3),max(b2,b4).];
:: RCOMP_2:th 26
theorem
for b1, b2, b3, b4 being real set
st [.b1,b2.[ meets [.b3,b4.[
holds [.b1,b2.[ \ [.b3,b4.[ = [.b1,b3.[ \/ [.b4,b2.[;
:: RCOMP_2:th 27
theorem
for b1, b2, b3, b4 being real set
st ].b1,b2.] meets ].b3,b4.]
holds ].b1,b2.] \ ].b3,b4.] = ].b1,b3.] \/ ].b4,b2.];