
theorem Th7:
  for A being Subset of I[01], a, b being Real st a < b & A
  = [.a,b.[ holds [.a,b.] c= the carrier of I[01]
proof
  let A be Subset of I[01], a, b be Real;
  assume
A1: a < b;
A2: ].a,b.[ c= [. a,b .[ by XXREAL_1:22;
  assume A = [.a,b.[;
  then
A3: ].a,b.[ c= [. 0,1 .] by A2,BORSUK_1:40,XBOOLE_1:1;
  then
A4: b <= 1 by A1,XXREAL_1:51;
  0 <= a by A1,A3,XXREAL_1:51;
  hence thesis by A4,BORSUK_1:40,XXREAL_1:34;
end;
