theorem Th22:
  for F,G being Element of Funcs(REAL*,REAL*),f being Element of
REAL*, i be Nat holds (repeat (F*G)).(i+1).f = F.(G.((repeat (F*G)).
  i.f))
proof
  let F,G be Element of Funcs(REAL*,REAL*),f be Element of REAL*,i;
  set Fi=(repeat (F*G)).i, ff=Fi.f, FFi=(F*G)*Fi;
A1: dom (F*G) = REAL* by Lm5;
A2: dom FFi=REAL* by Lm5;
  thus (repeat (F*G)).(i+1).f=FFi.f by Def2
    .=(F*G).ff by A2,FUNCT_1:12
    .=F.(G.ff) by A1,FUNCT_1:12;
end;
