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 Th45:
  for X,Y,Z being non empty TopSpace st the carrier of Y = the
carrier of Z & the topology of Z c= the topology of Y for f being Function of X
  ,Y, g being Function of X,Z st f = g holds for x being Point of X holds f
  is_continuous_at x implies g is_continuous_at x
proof
  let X,Y,Z be non empty TopSpace;
  assume that
A1: the carrier of Y = the carrier of Z and
A2: the topology of Z c= the topology of Y;
  let f be Function of X,Y,g be Function of X,Z;
  assume
A3: f = g;
  let x be Point of X;
  assume
A4: f is_continuous_at x;
  for G being Subset of Z st G is open & g.x in G ex H being Subset of X
  st H is open & x in H & g.:H c= G
  proof
    let G be Subset of Z;
    reconsider F = G as Subset of Y by A1;
    assume that
A5: G is open and
A6: g.x in G;
    G in the topology of Z by A5;
    then F is open by A2;
    then consider H being Subset of X such that
A7: H is open & x in H & f.:H c= F by A3,A4,A6,Th43;
    take H;
    thus thesis by A3,A7;
  end;
  hence thesis by Th43;
end;
