reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th10:
  for X being TopSpace, Y being non empty TopSpace for A,B being
  closed Subset of X for f being continuous Function of X|A, Y for g being
  continuous Function of X|B, Y st f tolerates g holds f+*g is continuous
  Function of X|(A \/ B), Y
proof
  let X be TopSpace, Y be non empty TopSpace;
  let A,B be closed Subset of X;
  let f be continuous Function of X|A, Y;
  let g be continuous Function of X|B, Y such that
A1: f tolerates g;
A2: the carrier of X|(A \/ B) = A \/ B by PRE_TOPC:8;
  the carrier of X|B = B by PRE_TOPC:8;
  then
A3: dom g = B by FUNCT_2:def 1;
  the carrier of X|A = A by PRE_TOPC:8;
  then
A4: dom f = A by FUNCT_2:def 1;
A5: rng (f+*g) c= rng f \/ rng g by FUNCT_4:17;
  dom (f+*g) = dom f \/ dom g by FUNCT_4:def 1;
  then reconsider h = f+*g as Function of X|(A \/ B), Y by A5,A2,A4,A3,
FUNCT_2:2,XBOOLE_1:1;
  h is continuous
  proof
    let C be Subset of Y;
A6: [#](X|(A \/ B)) = A \/ B by PRE_TOPC:8;
    assume
A7: C is closed;
    then f"C is closed by PRE_TOPC:def 6;
    then consider C1 being Subset of X such that
A8: C1 is closed and
A9: C1 /\ [#](X|A) = f"C by PRE_TOPC:13;
    g"C is closed by A7,PRE_TOPC:def 6;
    then consider C2 being Subset of X such that
A10: C2 is closed and
A11: C2 /\ [#](X|B) = g"C by PRE_TOPC:13;
A12: C1/\A\/C2/\B is closed by A8,A10;
A13: [#](X|A) = A by PRE_TOPC:8;
A14: [#](X|B) = B by PRE_TOPC:8;
    h"C = (f"C)\/(g"C) by A1,Th1
      .= ((f"C)\/(g"C))/\(A\/B) by A13,A14,XBOOLE_1:13,28;
    hence thesis by A12,A9,A11,A6,A13,A14,PRE_TOPC:13;
  end;
  hence thesis;
end;
