reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;
reserve F,G for Cardinal-Function;
reserve A,B for set;

theorem Th68:
  for f9,g9 being Function st dom g misses dom f9 \ dom g9 &
  f9 in sproduct f & g9 in sproduct g holds f9+*g9 in sproduct(f+*g)
proof
  let f9,g9 be Function such that
A1: dom g misses dom f9 \ dom g9 and
A2: f9 in sproduct f and
A3: g9 in sproduct g;
  set h = f9+*g9;
A4: dom f9 c= dom f by A2,Th49;
A5: dom g9 c= dom g by A3,Th49;
  then
A6: dom f9 \/ dom g9 c= dom f \/ dom g by A4,XBOOLE_1:13;
A7: dom h = dom f9 \/ dom g9 by FUNCT_4:def 1;
  then
A8: dom h c= dom(f+*g) by A6,FUNCT_4:def 1;
  for x being object holds x in dom h implies h.x in (f+*g).x
  proof let x be object;
    assume
A9: x in dom h;
    then x in dom(f+*g) by A8;
    then
A10: x in dom f \/ dom g by FUNCT_4:def 1;
    x in dom f9 \ dom g9 \/ dom g9 by A7,A9,XBOOLE_1:39;
    then
A11: x in dom f9 \ dom g9 or x in dom g9 by XBOOLE_0:def 3;
    now per cases;
      case
A12:    x in dom g;
        then h.x = g9.x by A1,A7,A9,A11,FUNCT_4:def 1,XBOOLE_0:3;
        hence h.x in g.x by A1,A3,A11,A12,Th49,XBOOLE_0:3;
      end;
      case not x in dom g;
        then
A13:    not x in dom g9 by A5;
        then
A14:    h.x = f9.x by A7,A9,FUNCT_4:def 1;
        x in dom f9 by A7,A9,A13,XBOOLE_0:def 3;
        hence h.x in f.x by A2,A14,Th49;
      end;
    end;
    hence thesis by A10,FUNCT_4:def 1;
  end;
  hence thesis by A8,Def9;
end;
