theorem
  a,b // a9,b9 & c,d // b,a & c,d9 // b9,a9 implies d = d9
proof
  assume that
A1: a,b // a9,b9 and
A2: c,d // b,a and
A3: c,d9 // b9,a9;
  a9,b9 // a,b by A1,Th3;
  then b9,a9 // b,a by Th7;
  then c,d // b9,a9 by A2,Def1;
  then c,d // c,d9 by A3,Def1;
  hence thesis by Th4;
end;
