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 Th25:
  for TS being transition-system over F holds x, u ==>. y, v, TS
  implies x, u^w ==>. y, v^w, TS
proof
  let TS be transition-system over F;
  assume x, u ==>. y, v, TS;
  then consider u1 such that
A1: x, u1 -->. y, TS and
A2: u = u1^v;
  u^w = u1^(v^w) by A2,AFINSQ_1:27;
  hence thesis by A1;
end;
