reserve k,m,n for Nat,
  a, b, c for object,
  x, y, X, Y, Z for set,
  D for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R, P1, P2, Q1, Q2, R1, R2 for FinSequence-membered set;
reserve S, T for non empty FinSequence-membered set;
reserve A for Function of P, NAT;
reserve U, V, W for Subset of P*;
reserve k,l,m,n,i,j for Nat,
  a, b, c for object,
  x, y, z, X, Y, Z for set,
  D, D1, D2 for non empty set;
reserve p, q, r, s, t, u, v for FinSequence;
reserve P, Q, R for FinSequence-membered set;
reserve B, C for antichain;
reserve S, T for Polish-language;
reserve A for Polish-arity-function of T;
reserve U, V, W for Polish-language of T;

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;
