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;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;

theorem Th123:
  for f, g, h being Function st f c= g holds f +* h c= g +* h
proof
  let f, g, h be Function;
  assume
A1: f c= g;
  now
    dom (f +* h) = dom f \/ dom h & dom (g +* h) = dom g \/ dom h by Def1;
    hence dom (f +* h) c= dom (g +* h) by A1,RELAT_1:11,XBOOLE_1:9;
    let x be object;
    assume x in dom (f +* h);
    then
A2: x in dom f or x in dom h by Th12;
    per cases;
    suppose
A3:   x in dom h;
      hence (f +* h).x = h.x by Th13
        .= (g +* h).x by A3,Th13;
    end;
    suppose
A4:   not x in dom h;
      then (f +* h).x = f.x & (g +* h).x = g.x by Th11;
      hence (f +* h).x = (g +* h).x by A1,A2,A4,GRFUNC_1:2;
    end;
  end;
  hence thesis by GRFUNC_1:2;
end;
