reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;
reserve S for non empty TopStruct;
reserve f for Function of T,S;
reserve SS for non empty TopSpace;
reserve f for Function of TS,SS;
reserve T, S for non empty TopSpace,
  p for Point of T;

theorem
  for T1,T2 being SubSpace of T,
      f being Function of T1,S, g being Function of T2,S
  st ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = {p} &
  T1 is compact & T2 is compact & T is T_2 & f is continuous & g is continuous
  & f.p = g.p holds f+*g is continuous Function of T,S
proof
  let T1,T2 be SubSpace of T, f be Function of T1,S, g be Function of T2,S;
  assume
A1: ([#] T1) \/ ([#] T2) = [#] T & ([#] T1) /\ ([#] T2) = {p} &
    T1 is compact & T2 is compact & T is T_2 & f is continuous &
    g is continuous;
  assume f.p = g.p;
  then f|{p} tolerates g|{p} by PARTFUN1:81;
  hence thesis by A1,Lm1;
end;
