reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th38:
  for A being Subset of REAL, B being Subset of R^1 st A = B holds
  A is open iff B is open
proof
  let A be Subset of REAL, B be Subset of R^1;
  assume
A1: A = B;
  hereby
    assume A is open;
    then A in Family_open_set RealSpace by JORDAN5A:5;
    then A in the topology of TopSpaceMetr RealSpace by TOPMETR:12;
    hence B is open by A1,PRE_TOPC:def 2,TOPMETR:def 6;
  end;
  assume B is open;
  then B in the topology of R^1 by PRE_TOPC:def 2;
  then A in Family_open_set RealSpace by A1,TOPMETR:12,def 6;
  hence thesis by JORDAN5A:5;
end;
