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

theorem
  for X being set, a being set, f being Function st dom f = X \/ {a}
  holds f = f|X +* (a .--> f.a)
proof
  let X be set, a be set, f be Function;
  assume
A1: dom f = X \/ {a};
  a in {a} by TARSKI:def 1;
  then
A2: a in dom f by A1,XBOOLE_0:def 3;
  thus f = f|X +* f|{a} by A1,FUNCT_4:70
    .= f|X +* (a .--> f.a) by A2,Th6;
end;
