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;

theorem Th57:
  for g being Function of X0,Y ex h being Function of X,Y st h|X0 = g
proof
  let g be Function of X0,Y;
  now
    per cases;
    suppose
A1:   the TopStruct of X = the TopStruct of X0;
      then reconsider h = g as Function of X,Y;
      take h;
      thus h|X0 = g by A1,Th54;
    end;
    suppose
A2:   the TopStruct of X <> the TopStruct of X0;
      Y is SubSpace of Y by TSEP_1:2;
      then reconsider B = the carrier of Y as non empty Subset of Y by TSEP_1:1
;
      set y = the Element of B;
      reconsider A0 = the carrier of X0 as Subset of X by TSEP_1:1;
A3:   X is SubSpace of X by TSEP_1:2;
      then reconsider A = the carrier of X as non empty Subset of X by TSEP_1:1
;
      reconsider A1 = A \ A0 as Subset of X;
A4:   A0 misses A1 by XBOOLE_1:79;
      A0 <> A by A2,A3,TSEP_1:5;
      then reconsider A1 as non empty Subset of A by XBOOLE_1:37;
      reconsider g1 = A1 --> y as Function of A1,B;
      reconsider A0 as non empty Subset of A;
      reconsider g0 = g as Function of A0,B;
      set G = g0 union g1;
      the carrier of X = A1 \/ A0 by XBOOLE_1:45;
      then reconsider h = G as Function of X,Y;
      take h;
      thus h|X0 = g by A4,Th1;
    end;
  end;
  hence thesis;
end;
