reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem
  dom f misses dom g implies (f +* g)|(dom f) = f
proof
  assume dom f misses dom g;
  then
A1: dom f \ dom g = dom f by XBOOLE_1:83;
  dom((f +* g)|(dom f)) = dom(f +* g) /\ dom f by RELAT_1:61
    .= (dom f \/ dom g) /\ dom f by Def1
    .= dom f by XBOOLE_1:21;
  hence thesis by A1,Th24,GRFUNC_1:3;
end;
