reserve r for Real;
reserve a, b for Real;
reserve T for non empty TopSpace;
reserve A for non empty SubSpace of T;
reserve P,Q for Subset of T,
  p for Point of T;
reserve M for non empty MetrSpace,
  p for Point of M;
reserve A for non empty SubSpace of M;
reserve F,G for Subset-Family of M;

theorem Th19:
  a <= b implies for P being Subset of R^1 st P = [. a,b .] holds
  Closed-Interval-TSpace(a,b) = R^1|P
proof
  assume
A1: a <= b;
  let P be Subset of R^1;
  assume P = [. a,b .];
  then [#](Closed-Interval-TSpace(a,b)) = P by A1,Th18;
  hence thesis by PRE_TOPC:def 5;
end;
