Article TOPMETR2, MML version 4.99.1005
:: TOPMETR2:th 1
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds [.b1,b2.] /\ [.b2,b3.] = {b2};
:: TOPMETR2:th 2
theorem
for b1, b2 being Relation-like Function-like set
st b1 is one-to-one &
b2 is one-to-one &
(for b3, b4 being set
st b3 in proj1 b2 & b4 in (proj1 b1) \ proj1 b2
holds b2 . b3 <> b1 . b4)
holds b1 +* b2 is one-to-one;
:: TOPMETR2:th 3
theorem
for b1, b2 being Relation-like Function-like set
st b1 .: ((proj1 b1) /\ proj1 b2) c= proj2 b2
holds (proj2 b1) \/ proj2 b2 = proj2 (b1 +* b2);
:: TOPMETR2:th 4
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3 being Element of the carrier of b1
for b4, b5 being SubSpace of b1
for b6 being Function-like quasi_total Relation of the carrier of b4,the carrier of b2
for b7 being Function-like quasi_total Relation of the carrier of b5,the carrier of b2
st ([#] b4) \/ [#] b5 = [#] b1 &
([#] b4) /\ [#] b5 = {b3} &
b4 is compact &
b5 is compact &
b1 is being_T2 &
b6 is continuous(b4, b2) &
b7 is continuous(b5, b2) &
b6 . b3 = b7 . b3
holds b6 +* b7 is Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2;
:: TOPMETR2:th 5
theorem
for b1, b2 being non empty TopSpace-like TopStruct
for b3, b4 being SubSpace of b2
for b5, b6 being Element of the carrier of b2
for b7 being Function-like quasi_total Relation of the carrier of b3,the carrier of b1
for b8 being Function-like quasi_total Relation of the carrier of b4,the carrier of b1
st ([#] b3) \/ [#] b4 = [#] b2 &
([#] b3) /\ [#] b4 = {b5,b6} &
b3 is compact &
b4 is compact &
b2 is being_T2 &
b7 is continuous(b3, b1) &
b8 is continuous(b4, b1) &
b7 . b5 = b8 . b5 &
b7 . b6 = b8 . b6
holds b7 +* b8 is Function-like quasi_total continuous Relation of the carrier of b2,the carrier of b1;
:: TOPMETR2:th 6
theorem
for b1 being TopSpace-like being_T2 TopStruct
for b2, b3 being Element of bool the carrier of b1
for b4 being Element of the carrier of b1
for b5 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 | b2
for b6 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 | b3
st b2 /\ b3 = {b4} & b5 is being_homeomorphism(I[01], b1 | b2) & b5 . 1 = b4 & b6 is being_homeomorphism(I[01], b1 | b3) & b6 . 0 = b4
holds ex b7 being Function-like quasi_total Relation of the carrier of I[01],the carrier of b1 | (b2 \/ b3) st
b7 is being_homeomorphism(I[01], b1 | (b2 \/ b3)) & b7 . 0 = b5 . 0 & b7 . 1 = b6 . 1;