theorem Th30:
  s ==>. t, ==>.-relation(S) implies s ==>. t, S
proof
  assume s ==>. t, ==>.-relation(S);
  then consider v, w, s1, t1 such that
A1: s = v^s1^w & t = v^t1^w and
A2: s1 -->. t1, ==>.-relation(S);
  [s1, t1] in ==>.-relation(S) by A2;
  then s1 ==>. t1, S by Def6;
  hence thesis by A1,Th13;
end;
