theorem
  for f1,k st k<len f1 holds (f1/^k).len (f1/^k)=f1.len f1 &
    (f1/^k)/.len (f1/^k)=f1/.len f1
proof
  let f1,k;
  assume
A1: k<len f1;
  then k+1<=len f1 by NAT_1:13;
  then k+1-k<=len f1-k by XREAL_1:9;
  then
A2: 1<=len (f1/^k) by A1,RFINSEQ:def 1;
  then len (f1/^k) in Seg len (f1/^k);
  then
A3: len (f1/^k) in dom (f1/^k) by FINSEQ_1:def 3;
  len (f1/^k)+k=len f1-k+k by A1,RFINSEQ:def 1
    .=len f1;
  hence
A4: (f1/^k).len (f1/^k)=f1.len f1 by A1,A3,RFINSEQ:def 1;
A5: (f1/^k)/.len (f1/^k)=(f1/^k).len (f1/^k) by A2,FINSEQ_4:15;
  0+1<=len f1 by A1,NAT_1:13;
  hence thesis by A4,A5,FINSEQ_4:15;
end;
