reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem
  len b = n + 1 implies ex c, e st len c = n & b = <%e%> ^ c
proof
  assume
A1: len b = n + 1;
  then b <> {};
  then consider c9 being XFinSequence, x being object such that
A2: b = <%x%> ^ c9 by Th8;
  <%x%> is Element of E^omega by A2,Th5;
  then reconsider e = x as Element of E by Th6;
  reconsider c = c9 as Element of E^omega by A2,Th5;
  take c, e;
  n + 1 = len c + len <%x%> by A1,A2,AFINSQ_1:17
    .= len c + 1 by AFINSQ_1:34;
  hence thesis by A2;
end;
