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;
reserve TS for non empty transition-system over F;
reserve s, s9, s1, s2, t, t1, t2 for Element of TS;
reserve S for Subset of TS;

theorem
  x, u ==>* y, TS & y, v ==>* z, TS implies x, u^v ==>* z, TS
proof
  assume that
A1: x, u ==>* y, TS and
A2: y, v ==>* z, TS;
  x, u^v^{} ==>* y, v, TS by A1,Th96;
  then
A3: x, u^v^<%>E ==>* y, v^{}, TS;
  y, v ==>* z, <%>E, TS by A2;
  then x, u^v^{} ==>* z, <%>E, TS by A3,Th83;
  hence thesis;
end;
