theorem Th34:
  ex b st u = [a,b]~
proof
  consider p such that
A1: u = p~ by Def7;
  consider b such that
A2: p`1,p`2 @@ a,b by Th15;
  [p`1,p`2] ## [a,b] by A2;
  then p ## [a,b];
  then p~ = [a,b]~ by Th27;
  hence thesis by A1;
end;
