reserve x, x1, x2, y, y1, y2, z, z1, z2 for object, X, X1, X2 for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u9, u1, u2, v, v1, v2, w, w1, w2 for Element of E^omega;
reserve F, F1, F2 for Subset of E^omega;
reserve i, k, l, n for Nat;

theorem Th10:
  w = u^v & u <> <%>E & v <> <%>E implies len u < len w & len v < len w
proof
  assume that
A1: w = u^v and
A2: u <> <%>E and
A3: v <> <%>E;
A4: len w = len u + len v by A1,AFINSQ_1:17;
  then len w + len u > len u + len v + 0 by A2,XREAL_1:8;
  hence thesis by A3,A4,XREAL_1:6,8;
end;
