
theorem Th4: :: BORSUK_4:48
  for a,b being Real,B being Subset of I[01] st
  B=[.a,b.] holds B is closed
proof
  let a,b be Real,B be Subset of I[01];
  assume
A1: B=[.a,b.];
  reconsider B2=B as Subset of R^1 by BORSUK_1:40,TOPMETR:17,XBOOLE_1:1;
A2: B2 is closed by A1,TREAL_1:1;
  reconsider I1=[.0,1.] as Subset of R^1 by TOPMETR:17;
A3: [#](R^1|I1)=the carrier of I[01] by BORSUK_1:40,PRE_TOPC:def 5;
A4: I[01]=R^1|I1 by TOPMETR:19,20;
  B=B2 /\ [#](R^1|I1) by A3,XBOOLE_1:28;
  hence thesis by A2,A4,PRE_TOPC:13;
end;
