
theorem Th24:
  for A being Subset of REAL, B being Subset of I[01] st A = B
  holds A is closed iff B is closed
proof
  reconsider Z = the carrier of I[01] as Subset of R^1 by BORSUK_1:1;
  let A be Subset of REAL, B be Subset of I[01];
  assume
A1: A = B;
  the carrier of I[01] c= the carrier of R^1 by BORSUK_1:1;
  then reconsider C = A as Subset of R^1 by A1,XBOOLE_1:1;
  hereby
    assume A is closed;
    then
A2: C is closed by JORDAN5A:23;
    C /\ [#] I[01] = B by A1,XBOOLE_1:28;
    hence B is closed by A2,PRE_TOPC:13;
  end;
  assume
A3: B is closed;
  Z is closed by Lm1,BORSUK_1:def 11;
  then B is closed iff C is closed by A1,TSEP_1:8;
  hence thesis by A3,JORDAN5A:23;
end;
