reserve MS for non empty MidStr;
reserve a, b for Element of MS;
reserve M for MidSp;
reserve a,b,c,d,a9,b9,c9,d9,x,y,x9 for Element of M;
reserve p,q,r,p9,q9 for Element of [:the carrier of M,the carrier of M:];
reserve u,v,w,u9,w9 for Vector of M;

theorem Th31:
  ex u st for p holds p in u iff p`1 = p`2
proof
  set a = the Element of M;
  take [a,a]~;
  let p;
  p`1,p`2 @@ a,a iff p ## [a,a];
  hence thesis by Th13,Th26;
end;
