reserve a, b, r, s for Real;

theorem Th43:
  for X being Subset of Closed-Interval-TSpace(r,s), Y being
  Subset of REAL st X = Y holds X is connected iff Y is interval
proof
  let X be Subset of Closed-Interval-TSpace(r,s), Y be Subset of REAL such
  that
A1: X = Y;
  reconsider Z = X as Subset of R^1 by A1,TOPMETR:17;
  hereby
    assume X is connected;
    then Z is connected by CONNSP_1:23;
    hence Y is interval by A1;
  end;
  assume Y is interval;
  then Z is connected by A1;
  hence thesis by CONNSP_1:23;
end;
