reserve i for Integer,
  a, b, r, s for Real;

theorem Th21:
  for f being Function of R^1,R^1, g being PartFunc of REAL,REAL,
x being Point of R^1 st f = g & g is_continuous_in x holds f is_continuous_at x
proof
  let f be Function of R^1,R^1, g be PartFunc of REAL,REAL, x be Point of R^1
  such that
A1: f = g and
A2: g is_continuous_in x;
  let G be a_neighborhood of f.x;
  consider Z being Neighbourhood of g.x such that
A3: Z c= G by A1,Th20;
  consider N being Neighbourhood of x such that
A4: g.:N c= Z by A2,FCONT_1:5;
  reconsider H = N as a_neighborhood of x by Th19,TOPMETR:17;
  take H;
  thus thesis by A1,A3,A4;
end;
