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

theorem Th31:
  for f be Function, d,i,j be object st i <> j holds (f+*(i,d)).j = f.j
proof
  let f be Function, d,i,j be object such that
A1: i <> j;
A2: not j in dom(i.-->d) by A1,TARSKI:def 1;
  per cases;
  suppose
    i in dom f;
    hence (f+*(i,d)).j = (f+*(i.-->d)).j by Def2
      .= f.j by A2,FUNCT_4:11;
  end;
  suppose
    not i in dom f;
    hence thesis by Def2;
  end;
end;
