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 Th13:
  w1^v1 = w2^v2 implies (ex u st w1^u = w2 & v1 = u^v2) or ex u st
  w2^u = w1 & v2 = u^v1
proof
A1: len w1 < len w2 or len w1 >= len w2;
  assume w1^v1 = w2^v2;
  hence thesis by A1,Th12;
end;
