
theorem Th14:
  for a, b, c, d being Real, f being Function of
  Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(c,d), g being PartFunc of
REAL, REAL st f is continuous one-to-one & a < b & c < d & f = g & f.a = c & f.
  b = d holds g| [.a,b.] is continuous
proof
  let a, b, c, d be Real;
  let f be Function of Closed-Interval-TSpace(a,b), Closed-Interval-TSpace(c,d
  );
  let g be PartFunc of REAL, REAL;
  assume that
A1: f is continuous one-to-one and
A2: a < b and
A3: c < d & f = g & f.a = c & f.b = d;
  for x0 being Real st x0 in dom(g| [.a,b.]) holds g| [.a,b.]
  is_continuous_in x0
  proof
    let x0 be Real;
    assume x0 in dom(g| [.a,b.]);
    then x0 in [.a,b.] by RELAT_1:57;
    then reconsider x1=x0 as Point of Closed-Interval-TSpace(a,b) by A2,
TOPMETR:18;
    f is_continuous_at x1 & x0 is Real by A1,TMAP_1:44;
    hence thesis by A1,A2,A3,Th13;
  end;
  hence thesis by FCONT_1:def 2;
end;
