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 Th11:
 for x being object holds
  not x in dom g implies (f +* g).x = f.x
proof let x be object;
  assume
A1: not x in dom g;
  per cases;
  suppose
    x in dom f;
    then x in dom f \/ dom g by XBOOLE_0:def 3;
    hence thesis by A1,Def1;
  end;
  suppose
A2: not x in dom f;
    then not x in dom f \/ dom g by A1,XBOOLE_0:def 3;
    then not x in dom(f+*g) by Def1;
    hence (f+*g).x = {} by FUNCT_1:def 2
      .= f.x by A2,FUNCT_1:def 2;
  end;
end;
