
theorem
  for f being FinSequence st 2 <= len f holds (f /^ 1 ).len(f /^ 1) = f.len(f)
  proof
    let f be FinSequence;
    assume
A1: 2 <= len f;
    set g = f /^ 1;
A2: 1 <= len f by A1,XXREAL_0:2; then
A3: len g + 1 = len f - 1 + 1 by RFINSEQ:def 1
             .= len f;
    then 2 - 1 <= len g + 1 - 1 by A1,XREAL_1:13; then
    len g in Seg len g by FINSEQ_1:3;
    then len g in dom g by FINSEQ_1:def 3;
    hence thesis by A3,A2,RFINSEQ:def 1;
  end;
