reserve A for set, x,y,z for object,
  k for Element of NAT;
reserve n for Nat,
  x for object;

theorem
  for M being FinSequence st M <> {} holds
    M = Del(M,len M) ^ <*M.(len M)*>
proof
  let M be FinSequence;
  assume M <> {};
  then consider q be FinSequence,x being object such that
A1: M = q ^ <*x*> by FINSEQ_1:46;
A2: len M = len q + len <*x*> by A1,FINSEQ_1:22
    .= len q + 1 by FINSEQ_1:39;
  then
A3: len Del(M,len M) = len q by Th11;
A4: dom q = Seg len q by FINSEQ_1:def 3;
A5: now
    let i be Nat;
    assume
A6: i in dom q;
    then i<=len q by A4,FINSEQ_1:1;
    then i in NAT & i<len M by A2,NAT_1:13,ORDINAL1:def 12;
    hence Del(M,len M).i = M.i by FINSEQ_3:110
      .= q.i by A1,A6,FINSEQ_1:def 7;
  end;
  M.(len M) = x by A1,A2,FINSEQ_1:42;
  hence thesis by A1,A3,A5,FINSEQ_2:9;
end;
