theorem
 f c= g & not x in dom f implies f c= g+*(x,y)
 proof assume that
A1: f c= g;
  assume not x in dom f;
  then
A2: f c= f +*(x .--> y) by FUNCT_4:107;
  per cases;
  suppose x in dom g;
   then g+*(x,y) = g+*(x.-->y) by Def2;
   then f +*(x .--> y) c= g+*(x,y) by A1,FUNCT_4:123;
  hence f c= g+*(x,y) by A2;
  end;
  suppose not x in dom g;
  hence f c= g+*(x,y) by A1,Def2;
  end;
 end;
