theorem Th69:
  ex x st a,x _|_ p,q & LIN p,q,x
proof
A1: now
    assume p<>q;
    then consider x such that
A2: p,q // p,x & p,q _|_ x,a by Def7;
    take x;
    thus a,x _|_ p,q & LIN p,q,x by A2,Th61;
  end;
  now
    assume
A3: p=q;
    take x=a;
    p,q // p,a by A3,Th58;
    hence a,x _|_ p,q & LIN p,q,x by Th58;
  end;
  hence thesis by A1;
end;
