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 Th4:
  a ^ b = <%x%> iff a = <%>E & b = <%x%> or b = <%>E & a = <%x%>
proof
  thus a ^ b = <%x%> implies a = <%>E & b = <%x%> or b = <%>E & a = <%x%>
  proof
    assume
A1: a ^ b = <%x%>;
    then len (a ^ b) = 1 by AFINSQ_1:34;
    then len a + len b = 1 by AFINSQ_1:17;
    then len a = 1 & b = <%>E or a = <%>E & len b = 1 by Th3;
    hence thesis by A1;
  end;
  assume a = <%>E & b = <%x%> or b = <%>E & a = <%x%>;
  hence thesis;
end;
