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;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;

theorem Th74:
  for x0 being Point of X0, x1 being Point of X1 st x0 = x1 holds
  X1 is SubSpace of X0 & g is_continuous_at x0 implies g|X1 is_continuous_at x1
proof
  let x0 be Point of X0, x1 be Point of X1 such that
A1: x0 = x1;
  assume
A2: X1 is SubSpace of X0;
  assume
A3: g is_continuous_at x0;
  for G being Subset of Y st G is open & (g|X1).x1 in G ex H0 being Subset
  of X1 st H0 is open & x1 in H0 & (g|X1).:H0 c= G
  proof
    reconsider C = the carrier of X1 as Subset of X0 by A2,TSEP_1:1;
    let G be Subset of Y such that
A4: G is open and
A5: (g|X1).x1 in G;
    g.x0 in G by A1,A2,A5,Th65;
    then consider H being Subset of X0 such that
A6: H is open & x0 in H and
A7: g.:H c= G by A3,A4,Th43;
    reconsider H0 = H /\ C as Subset of X1 by XBOOLE_1:17;
    g.:H0 c= g.:H /\ g.:C & g.:H /\ g.:C c= g.:H by RELAT_1:121,XBOOLE_1:17;
    then g.:H0 c= g.:H by XBOOLE_1:1;
    then
A8: g.:H0 c= G by A7,XBOOLE_1:1;
    take H0;
    g|X1 = g|C by A2,Def5;
    then H0 = H /\ [#]X1 & (g|X1).:H0 c= g.:H0 by RELAT_1:128;
    hence thesis by A1,A2,A6,A8,TOPS_2:24,XBOOLE_0:def 4,XBOOLE_1:1;
  end;
  hence thesis by Th43;
end;
