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 Th57:
  x, y -->. z, TS iff <*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS)
proof
  thus x, y -->. z, TS implies <*[x, y], [z, <%>E]*> is RedSequence of
  ==>.-relation(TS)
  proof
    assume x, y -->. z, TS;
    then [[x, y], [z, <%>E]] in ==>.-relation(TS) by Th37;
    hence thesis by REWRITE1:7;
  end;
  assume <*[x, y], [z, <%>E]*> is RedSequence of ==>.-relation(TS);
  then [[x, y], [z, <%>E]] in ==>.-relation(TS) by Th8;
  hence thesis by Th37;
end;
