
theorem
  for p being non empty FinSequence holds <*p.1*>^'p = p
proof
  let p be non empty FinSequence;
  set o = <*p.1*>^'p;
A1: len o +1 = len <*p.1*> + len p by FINSEQ_6:139;
A2: len <*p.1*> = 1 by FINSEQ_1:39;
A3: now
    let k be Nat such that
A4: 1<=k and
A5: k <= len o;
    per cases;
    suppose
A6:   k <= len <*p.1*>;
      then k <= 1 by FINSEQ_1:39;
      then
A7:   k = 1 by A4,XXREAL_0:1;
      hence o.k = <*p.1*>.1 by A6,FINSEQ_6:140
        .= p.k by A7;
    end;
    suppose
      k > len <*p.1*>;
      then consider i being Element of NAT such that
A8:   k = len <*p.1*>+i and
A9:   1 <= i by FINSEQ_4:84;
      i < len p by A1,A2,A5,A8,NAT_1:13;
      hence o.k = p.k by A2,A8,A9,FINSEQ_6:141;
    end;
  end;
  len o +1 = 1 + len p by A1,FINSEQ_1:39;
  hence thesis by A3,FINSEQ_1:14;
end;
