reserve i,j,k,x for object;

theorem Th6:
  for A,B,C being set holds rng FuncComp(Funcs(A,B),Funcs(B,C)) c= Funcs(A,C)
proof
  let A,B,C be set;
  let i be object;
  set F = FuncComp(Funcs(A,B),Funcs(B,C));
  assume i in rng F;
  then consider j being object such that
A1: j in dom F and
A2: i = F.j by FUNCT_1:def 3;
  consider f,g being Function such that
A3: j = [g,f] and
A4: F.j = g*f by A1,Def9;
  g in Funcs(B,C) & f in Funcs(A,B) by A1,A3,ZFMISC_1:87;
  hence thesis by A2,A4,FUNCT_2:128;
end;
