reserve A,B,C for Ordinal;
reserve a,b,c,d for natural Ordinal;
reserve l,m,n for natural Ordinal;
reserve i,j,k for Element of omega;
reserve x,y,z for Element of RAT+;
reserve i,j,k for natural Ordinal;
reserve r,s,t for Element of RAT+;

theorem
  r < s implies ex t st r < t & t < s
proof
  assume
A1: r < s;
  then consider x such that
A2: s = r+x by Def13;
  consider y such that
A3: x = y+y by Th60;
  take t = r+y;
A4: s = t+y by A2,A3,Th51;
  then
A5: t <=' s;
  r <=' t;
  hence r < t by A1,A4,Th66;
  r <> s by A1;
  then s <> t by A4,Th62;
  hence thesis by A5,Th66;
end;
