theorem Th57:
  for S, m, n, s st m+1 <= n & s in S^^n
      holds s is (S^^m)-headed & (S^^m)-tail s is S-headed
proof
  let S, m, n, s;
  assume that
  A1: m+1 <= n and
  A2: s in S^^n;
  consider l such that A3: m+1+l = n by A1, NAT_1:10;
  A4: m+(1+l) = n by A3;
  then S^^n is (S^^m)-headed by Th51;
  hence s is (S^^m)-headed by A2;
  set u = (S^^m)-tail s;
  s in (S^^m)^(S^^(1+l)) by A2, A4, Th10;
  then u in S^^(1+l) by Th54;
  then A6: u in (S^^1)^(S^^l) by Th10;
  S^(S^^l) is S-headed;
  hence thesis by A6;
end;
