theorem Th72:
  for Y, Z, D for p being FinSequence for f being Function of Y, D
        for g being Function of Z, D
      st rng p c= Y & rng p c= Z & for a st a in rng p holds f.a = g.a
    holds f * p = g * p
proof
  let Y, Z, D;
  let p be FinSequence;
  let f be Function of Y, D;
  let g be Function of Z, D;
  assume that
      A1: rng p c= Y and
      A2: rng p c= Z and
      A3: for a st a in rng p holds f.a = g.a;
  reconsider p1 = p as FinSequence of Y by A1, FINSEQ_1:def 4;
  reconsider q = f * p1 as FinSequence by FINSEQ_2:32;
  reconsider p2 = p as FinSequence of Z by A2, FINSEQ_1:def 4;
  reconsider r = g * p2 as FinSequence by FINSEQ_2:32;
  q = r
    proof
    thus len q = len p by FINSEQ_2:33
          .= len r by FINSEQ_2:33;
    let k;
    assume that
    A6: 1 <= k and
    A7: k <= len q;
    k <= len p by A7, FINSEQ_2:33;
    then k in Seg len p by A6, FINSEQ_1:1;
    then A8: k in dom p by FINSEQ_1:def 3;
    hence q.k = f.(p.k) by FUNCT_1:13
        .= g.(p.k) by A3, A8, FUNCT_1:3
        .= r.k by FUNCT_1:13, A8;
    end;
  hence thesis;
end;
