reserve x,y,X for set,
  i,j,k,m,n for Nat,
  p for FinSequence of X,
  ii for Integer;
reserve G for Graph,
  pe,qe for FinSequence of the carrier' of G,
  p,q for oriented Chain of G,
  W for Function,
  U,V,e,ee for set,
  v1,v2,v3,v4 for Vertex of G;
reserve G for finite Graph,
  P,Q for oriented Chain of G,
  v1,v2,v3 for Vertex of G;
reserve G for finite oriented Graph,
  P,Q for oriented Chain of G,
  W for Function of (the carrier' of G), Real>=0,
  v1,v2,v3,v4 for Vertex of G;
reserve f,g,h for Element of REAL*,
  r for Real;

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;
