theorem Th51:
   |.a1*b2-a2*b1.|<>0 & b2 <> 0 & a2/b2 is rational implies
   ex x,y be Element of INT st
   |.LF(a1,b1,c1).(x,y).|*|.LF(a2,b2,c2).(x,y).|<=|.a1*b2-a2*b1.|/4
   proof
     assume
A1:  |.a1*b2-a2*b1.|<>0 & b2 <> 0 & a2/b2 is rational;
A4:  |.a2*b1-a1*b2.| = |.-(a1*b2-a2*b1).| .=|.a1*b2-a2*b1.| by COMPLEX1:52;
     then ex x,y be Element of INT st
      |.LF(a2,b2,c2).(x,y).|*|.LF(a1,b1,c1).(x,y).|<=|.a2*b1-a1*b2.|/4
        by A1,Th50;
    hence thesis by A4;
   end;
