theorem Th18:
  for I,J being FinSequence-membered set
  for f being FinSequence holds I c= J iff f^^I c= f^^J
  proof
    let I,J be FinSequence-membered set;
    let f be FinSequence;
    thus I c= J implies f^^I c= f^^J
    proof assume
A1:   I c= J;
      let a; assume a in f^^I;
      then ex b being Element of I st a = f^b & b in I;
      hence a in f^^J by A1;
    end;
    assume
A2: f^^I c= f^^J;
    let a; assume
A3: a in I;
    then reconsider a as FinSequence;
    f^a in f^^I by A3;
    then f^a in f^^J by A2;
    then consider b being Element of J such that
A4: f^a = f^b & b in J;
    thus thesis by A4,FINSEQ_1:33;
  end;
