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 Th31:
  [x, y] in ==>.-relation(TS) implies ex s, v, t, w st x = [s, v] & y = [t, w]
proof
  assume
A1: [x, y] in ==>.-relation(TS);
  then y in [: the carrier of TS, E^omega :] by ZFMISC_1:87;
  then
A2: ex y1, y2 being object
st y1 in the carrier of TS & y2 in E^omega & y = [y1, y2] by
ZFMISC_1:def 2;
  x in [: the carrier of TS, E^omega :] by A1,ZFMISC_1:87;
  then ex x1, x2 being object
st x1 in the carrier of TS & x2 in E^omega & x = [x1, x2] by
ZFMISC_1:def 2;
  hence thesis by A2;
end;
