reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th3:
  for x,a being set, f being Function st a in dom f holds (commute
  (x .--> f)).a = x .--> f.a
proof
  let x,a be set;
  let f be Function;
  set g = x .--> f;
A1: dom g = {x};
A2: f in Funcs(dom f, rng f) by FUNCT_2:def 2;
  rng g = {f} by FUNCOP_1:8;
  then rng g c= Funcs(dom f, rng f) by A2,ZFMISC_1:31;
  then
A3: g in Funcs({x}, Funcs(dom f, rng f)) by A1,FUNCT_2:def 2;
A4: g.x = f by FUNCOP_1:72;
A5: x in {x} by TARSKI:def 1;
  assume
A6: a in dom f;
  then
A7: (commute g).a.x = f.a by A3,A4,A5,FUNCT_6:56;
  dom ((commute g).a) = {x} by A3,A6,A4,A5,FUNCT_6:56;
  hence thesis by A7,DICKSON:1;
end;
