reserve a,x,y for object, A,B for set,
  l,m,n for Nat;

theorem Th29:
  for f be Function, d,i be object holds dom(f+*(i,d)) = dom f
proof
  let f be Function, x,i be object;
  per cases;
  suppose
A1: i in dom f;
    then
A2: {i} c= dom f by ZFMISC_1:31;
    thus dom(f+*(i,x)) = dom(f+*(i.-->x)) by A1,Def2
      .= dom f \/ dom(i.-->x) by FUNCT_4:def 1
      .= dom f \/ {i}
      .= dom f by A2,XBOOLE_1:12;
  end;
  suppose
    not i in dom f;
    hence thesis by Def2;
  end;
end;
