theorem
  s ==>. s, S implies ex v, w, s1 st s = v^s1^w & s1 -->. s1, S
proof
  given v, w, s1, t1 such that
A1: s = v^s1^w and
A2: s = v^t1^w and
A3: s1 -->. t1, S;
  v^s1 = v^t1 by A1,A2,AFINSQ_1:28;
  then s1 = t1 by AFINSQ_1:28;
  hence thesis by A1,A3;
end;
