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 Th32:
  ID(M) = [b,b]~
proof
  p in ID(M) iff p in [b,b]~
  proof
    thus p in ID(M) implies p in [b,b]~
    proof
      assume p in ID(M);
      then ex q st p = q & q`1 = q`2;
      then
A1:   p`1,p`2 @@ b,b;
      p ## [b,b] by A1;
      hence thesis;
    end;
    thus p in [b,b]~ implies p in ID(M)
    proof
      assume p in [b,b]~;
      then
A2:   p ## [b,b] by Th26;
      p`1,p`2 @@ b,b by A2;
      then p`1 = p`2 by Th11,Th12;
      hence thesis;
    end;
  end;
  then for p being object holds p in ID(M) iff p in [b,b]~;
  hence thesis by TARSKI:2;
end;
