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 Th92:
  not <%>E in rng dom (the Tran of TS) implies (x, <%e%> ==>* y,
  <%>E, TS implies x, <%e%> ==>. y, <%>E, TS)
proof
  assume
A1: not <%>E in rng dom (the Tran of TS);
  assume x, <%e%> ==>* y, <%>E, TS;
  then ==>.-relation(TS) reduces [x, <%e%>], [y, <%>E];
  then [[x, <%e%>], [y, <%>E]] in ==>.-relation(TS) by A1,Th79;
  hence thesis by Def4;
end;
