reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;

theorem Th46:
  for X,Y,Z being non empty TopSpace st the carrier of X = the
carrier of Y & the topology of Y c= the topology of X for f being Function of X
  ,Z, g being Function of Y,Z st f = g holds for x being Point of X, y being
  Point of Y st x = y holds g is_continuous_at y implies f is_continuous_at x
proof
  let X,Y,Z be non empty TopSpace;
  assume that
A1: the carrier of X = the carrier of Y and
A2: the topology of Y c= the topology of X;
  let f be Function of X,Z, g be Function of Y,Z;
  assume
A3: f = g;
  let x be Point of X, y be Point of Y;
  assume
A4: x = y;
  assume
A5: g is_continuous_at y;
  for G being Subset of Z st G is open & f.x in G ex H being Subset of X
  st H is open & x in H & f.:H c= G
  proof
    let G be Subset of Z;
    assume G is open & f.x in G;
    then consider H being Subset of Y such that
A6: H is open and
A7: y in H & g.:H c= G by A3,A4,A5,Th43;
    reconsider F = H as Subset of X by A1;
    take F;
    H in the topology of Y by A6;
    hence thesis by A2,A3,A4,A7;
  end;
  hence thesis by Th43;
end;
