:: TOPMETR2 semantic presentation

theorem Th1: :: TOPMETR2:1
for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies [.b1,b2.] /\ [.b2,b3.] = {b2} )
proof end;

theorem Th2: :: TOPMETR2:2
for b1, b2 being Function holds
( b1 is one-to-one & b2 is one-to-one & ( for b3, b4 being set holds
not ( b3 in dom b2 & b4 in (dom b1) \ (dom b2) & not b2 . b3 <> b1 . b4 ) ) implies b1 +* b2 is one-to-one )
proof end;

Lemma2: for b1, b2 being Function holds
( b1 .: ((dom b1) /\ (dom b2)) c= rng b2 implies (rng b1) \ (rng b2) c= rng (b1 +* b2) )
proof end;

theorem Th3: :: TOPMETR2:3
for b1, b2 being Function holds
( b1 .: ((dom b1) /\ (dom b2)) c= rng b2 implies (rng b1) \/ (rng b2) = rng (b1 +* b2) )
proof end;

theorem Th4: :: TOPMETR2:4
for b1, b2 being non empty TopSpace
for b3 being Point of b1
for b4, b5 being SubSpace of b1
for b6 being Function of b4,b2
for b7 being Function of b5,b2 holds
( ([#] b4) \/ ([#] b5) = [#] b1 & ([#] b4) /\ ([#] b5) = {b3} & b4 is compact & b5 is compact & b1 is_T2 & b6 is continuous & b7 is continuous & b6 . b3 = b7 . b3 implies b6 +* b7 is continuous Function of b1,b2 )
proof end;

theorem Th5: :: TOPMETR2:5
for b1, b2 being non empty TopSpace
for b3, b4 being SubSpace of b2
for b5, b6 being Point of b2
for b7 being Function of b3,b1
for b8 being Function of b4,b1 holds
( ([#] b3) \/ ([#] b4) = [#] b2 & ([#] b3) /\ ([#] b4) = {b5,b6} & b3 is compact & b4 is compact & b2 is_T2 & b7 is continuous & b8 is continuous & b7 . b5 = b8 . b5 & b7 . b6 = b8 . b6 implies b7 +* b8 is continuous Function of b2,b1 )
proof end;

theorem Th6: :: TOPMETR2:6
for b1 being being_T2 TopSpace
for b2, b3 being Subset of b1
for b4 being Point of b1
for b5 being Function of I[01] ,(b1 | b2)
for b6 being Function of I[01] ,(b1 | b3) holds
not ( b2 /\ b3 = {b4} & b5 is_homeomorphism & b5 . 1 = b4 & b6 is_homeomorphism & b6 . 0 = b4 & ( for b7 being Function of I[01] ,(b1 | (b2 \/ b3)) holds
not ( b7 is_homeomorphism & b7 . 0 = b5 . 0 & b7 . 1 = b6 . 1 ) ) )
proof end;