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 + m implies ex c1, c2 st len c1 = n & len c2 = m & b = c1 ^ c2
proof
  assume len b = n + m;
  then consider c19, c29 being XFinSequence such that
A1: len c19 = n & len c29 = m and
A2: b = c19 ^ c29 by Lm1;
  reconsider c2 = c29 as Element of E^omega by A2,Th5;
  reconsider c1 = c19 as Element of E^omega by A2,Th5;
  take c1, c2;
  thus thesis by A1,A2;
end;
