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 Th33:
  (ex p,q st u = p~ & v = q~ & p`2 = q`1 & w = [p`1,q`2]~)& (ex p,
  q st u = p~ & v = q~ & p`2 = q`1 & w9 = [p`1,q`2]~) implies w = w9
proof
  given p,q such that
A1: u = p~ and
A2: v = q~ and
A3: p`2 = q`1 and
A4: w = [p`1,q`2]~;
  given p9,q9 such that
A5: u = p9~ and
A6: v = q9~ and
A7: p9`2 = q9`1 and
A8: w9 = [p9`1,q9`2]~;
  q ## q9 by A2,A6,Th28;
  then
A9: q`1,q`2 @@ q9`1,q9`2;
  p ## p9 by A1,A5,Th28;
  then p`1,p`2 @@ p9`1,p9`2;
  then p`1,q`2 @@ p9`1,q9`2 by A3,A7,A9,Th18;
  hence thesis by A4,A8,Th19,Th27;
end;
