theorem Th8:
  rng seq c= dom h implies (g(#)h)/*seq = g(#)(h/*seq)
proof
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom (g(#)h) by VALUED_1:def 5;
  now
    let n be Element of NAT;
A3: seq.n in rng seq by VALUED_0:28;
    thus ((g(#)h)/*seq).n = (g(#)h)/.(seq.n) by A2,FUNCT_2:109
      .= g * (h/.(seq.n)) by A2,A3,CFUNCT_1:4
      .= g *((h/*seq).n) by A1,FUNCT_2:109
      .= (g(#)(h/*seq)).n by VALUED_1:6;
  end;
  hence thesis by FUNCT_2:63;
end;
