
theorem
  for n be Nat holds Rev (0*n) = 0*n
proof
  let n be Nat;
A1: now
    let k be Nat;
    assume
A2: k in dom 0*n;
    then k in Seg len 0*n by FINSEQ_1:def 3;
    then
A3: k in Seg n by CARD_1:def 7;
    then n - k + 1 in Seg n by FINSEQ_5:2;
    then
A4: len 0*n - k + 1 in Seg n by CARD_1:def 7;
    thus (Rev 0*n).k = (0*n).(len 0*n - k + 1) by A2,FINSEQ_5:58
      .= 0 by A4,FUNCOP_1:7
      .= (0*n).k by A3,FUNCOP_1:7;
  end;
  dom Rev (0*n) = Seg len Rev (0*n) by FINSEQ_1:def 3
    .= Seg len 0*n by FINSEQ_5:def 3
    .= dom 0*n by FINSEQ_1:def 3;
  hence thesis by A1,FINSEQ_1:13;
end;
