theorem
  for B, C, p st p is (B^C)-headed holds p is B-headed
proof
  let B, C, p;
  assume p is (B^C)-headed;
  then consider q, r such that
    A1: q in B^C and
    A2: p = q^r;
  consider s, t such that
    A4: q = s^t and
    A5: s in B and
        t in C by A1, Def2;
  p = s^(t^r) by A2, A4, FINSEQ_1:32;
  hence thesis by A5;
end;
