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 Th15:
  for TS being transition-system over X holds x, y -->. z, TS
  implies x in TS & y in X & z in TS & x in dom dom (the Tran of TS) & y in rng
  dom (the Tran of TS) & z in rng (the Tran of TS)
proof
  let TS be transition-system over X;
  assume x, y -->. z, TS;
  then
A1: [[x, y], z] in the Tran of TS;
  then [x, y] in [: the carrier of TS, X :] by ZFMISC_1:87;
  hence x in the carrier of TS & y in X & z in the carrier of TS by A1,
ZFMISC_1:87;
  [x, y] in dom (the Tran of TS) by A1,XTUPLE_0:def 12;
  hence x in dom dom (the Tran of TS) & y in rng dom (the Tran of TS) by
XTUPLE_0:def 12,def 13;
  thus z in rng (the Tran of TS) by A1,XTUPLE_0:def 13;
end;
