theorem Th63:
  for S for q being Element of S^^1 holds decomp(S, 1, q) = <*q*>
proof
  let S;
  let q be Element of S^^1;
  set w = decomp(S, 1, q);
  1 in Seg 1 by FINSEQ_1:2, TARSKI:def 1;
  then consider k such that
    A2: 1 = k+1 and
    A3: w.1 = S-head((S^^k)-tail q) by Def32;
  w.1 = S-head q by A2, A3, Th58 .= q;
  hence thesis by Th62, FINSEQ_1:40;
end;
