theorem Th12:
  s ==>. t, S implies u^s ==>. u^t, S & s^u ==>. t^u, S
proof
  given v, w, s1, t1 such that
A1: s = v^s1^w and
A2: t = v^t1^w and
A3: s1 -->. t1, S;
A4: u^t = u^(v^t1)^w by A2,AFINSQ_1:27
    .= u^v^t1^w by AFINSQ_1:27;
  u^s = u^(v^s1)^w by A1,AFINSQ_1:27
    .= u^v^s1^w by AFINSQ_1:27;
  hence u^s ==>. u^t, S by A3,A4;
  s^u = v^s1^(w^u) & t^u = v^t1^(w^u) by A1,A2,AFINSQ_1:27;
  hence thesis by A3;
end;
