
theorem Th23:
  for A being Subset of REAL, B being Subset of R^1 st A = B holds
  A is closed iff B is closed
proof
  let A be Subset of REAL, B be Subset of R^1 such that
A1: A = B;
  thus A is closed implies B is closed
  proof
    assume A is closed;
    then A`` is closed;
    then A` is open by RCOMP_1:def 5;
    then A` in the topology of R^1 by Lm8,Th5;
   hence [#]R^1 \ B is open by A1,TOPMETR:17;
  end;
  assume B is closed;
  then B` in the topology of R^1 by PRE_TOPC:def 2;
  then A` is open by A1,Lm8,Th5,TOPMETR:17;
  then A`` is closed by RCOMP_1:def 5;
  hence thesis;
end;
