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 Th21:
  for TS1 being transition-system over F1, TS2 being
transition-system over F2 st the Tran of TS1 = the Tran of TS2 & x1, x2 ==>. y1
  , y2, TS1 holds x1, x2 ==>. y1, y2, TS2
proof
  let TS1 be transition-system over F1, TS2 be transition-system over F2 such
  that
A1: the Tran of TS1 = the Tran of TS2 and
A2: x1, x2 ==>. y1, y2, TS1;
  consider v, w such that
A3: v = y2 & x1, w -->. y1, TS1 & x2 = w^v by A2;
  take v, w;
  thus thesis by A1,A3;
end;
