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 Th76:
  X1 is SubSpace of X0 implies for A being Subset of X0, x0 being
  Point of X0, x1 being Point of X1 st A c= the carrier of X1 & A is
  a_neighborhood of x0 & x0 = x1 holds g is_continuous_at x0 iff g|X1
  is_continuous_at x1
proof
  assume
A1: X1 is SubSpace of X0;
  let A be Subset of X0, x0 be Point of X0, x1 be Point of X1 such that
A2: A c= the carrier of X1 and
A3: A is a_neighborhood of x0 and
A4: x0 = x1;
  thus g is_continuous_at x0 implies g|X1 is_continuous_at x1 by A1,A4,Th74;
  thus g|X1 is_continuous_at x1 implies g is_continuous_at x0
  proof
    assume
A5: g|X1 is_continuous_at x1;
    for G being Subset of Y st G is open & g.x0 in G ex H being Subset of
    X0 st H is open & x0 in H & g.:H c= G
    proof
      let G be Subset of Y such that
A6:   G is open and
A7:   g.x0 in G;
      (g|X1).x1 in G by A1,A4,A7,Th65;
      then consider H1 being Subset of X1 such that
A8:   H1 is open and
A9:   x1 in H1 and
A10:  (g|X1).:H1 c= G by A5,A6,Th43;
      consider V being Subset of X0 such that
A11:  V is open and
A12:  V c= A and
A13:  x0 in V by A3,CONNSP_2:6;
      reconsider V1 = V as Subset of X1 by A2,A12,XBOOLE_1:1;
A14:  H1 /\ V1 c= V by XBOOLE_1:17;
      then reconsider H = H1 /\ V1 as Subset of X0 by XBOOLE_1:1;
A15:  for z being Point of Y holds z in g.:H implies z in G
      proof
        set f = g|X1;
        let z be Point of Y;
        assume z in g.:H;
        then consider y being Point of X0 such that
A16:    y in H and
A17:    z = g.y by FUNCT_2:65;
        y in V by A14,A16;
        then y in A by A12;
        then
A18:    z = f.y by A1,A2,A17,Th65;
        H1 /\ V1 c= H1 by XBOOLE_1:17;
        then z in f.:H1 by A16,A18,FUNCT_2:35;
        hence thesis by A10;
      end;
      take H;
      V1 is open by A1,A11,TOPS_2:25;
      then H1 /\ V1 is open by A8;
      hence thesis by A1,A4,A9,A11,A13,A14,A15,SUBSET_1:2,TSEP_1:9
,XBOOLE_0:def 4;
    end;
    hence thesis by Th43;
  end;
end;
