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;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;
reserve X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;

theorem Th111:
  for X1, X2 being non empty SubSpace of X, g being Function of
X1 union X2,Y for x1 being Point of X1, x2 being Point of X2, x being Point of
  X1 union X2 st x = x1 & x = x2 holds g is_continuous_at x iff g|X1
  is_continuous_at x1 & g|X2 is_continuous_at x2
proof
  let X1, X2 be non empty SubSpace of X, g be Function of X1 union X2,Y;
  let x1 be Point of X1, x2 be Point of X2, x be Point of X1 union X2 such
  that
A1: x = x1 and
A2: x = x2;
A3: X2 is SubSpace of X1 union X2 by TSEP_1:22;
A4: X1 is SubSpace of X1 union X2 by TSEP_1:22;
  hence g is_continuous_at x implies g|X1 is_continuous_at x1 & g|X2
  is_continuous_at x2 by A1,A2,A3,Th74;
  thus g|X1 is_continuous_at x1 & g|X2 is_continuous_at x2 implies g
  is_continuous_at x
  proof
    assume that
A5: g|X1 is_continuous_at x1 and
A6: g|X2 is_continuous_at x2;
    for G being a_neighborhood of g.x ex H being a_neighborhood of x st g
    .:H c= G
    proof
      let G be a_neighborhood of g.x;
      g.x = (g|X1).x1 by A1,A4,Th65;
      then consider H1 being a_neighborhood of x1 such that
A7:   (g|X1).:H1 c= G by A5;
      g.x = (g|X2).x2 by A2,A3,Th65;
      then consider H2 being a_neighborhood of x2 such that
A8:   (g|X2).:H2 c= G by A6;
      the carrier of X2 c= the carrier of X1 union X2 by A3,TSEP_1:4;
      then reconsider S2 = H2 as Subset of X1 union X2 by XBOOLE_1:1;
      g.:S2 c= G by A3,A8,Th68;
      then
A9:   S2 c= g"G by FUNCT_2:95;
      the carrier of X1 c= the carrier of X1 union X2 by A4,TSEP_1:4;
      then reconsider S1 = H1 as Subset of X1 union X2 by XBOOLE_1:1;
      consider H being a_neighborhood of x such that
A10:  H c= H1 \/ H2 by A1,A2,Th16;
      take H;
      g.:S1 c= G by A4,A7,Th68;
      then S1 c= g"G by FUNCT_2:95;
      then S1 \/ S2 c= g"G by A9,XBOOLE_1:8;
      then H c= g"G by A10,XBOOLE_1:1;
      hence thesis by FUNCT_2:95;
    end;
    hence thesis;
  end;
end;
