reserve r1,r2,r3 for non negative Real;
reserve n,m1 for Nat;
reserve s for Real;
reserve cn,cd,i1,j1 for Integer;
reserve r for irrational Real;
reserve q for Rational;
reserve c0,c1,c2,u,a0,b0 for Real;
reserve a,b for Real;
reserve n for Integer;
reserve a1,a2,b1,b2,c1,c2 for Element of REAL;
reserve eps for positive Real;
reserve r1 for non negative Real;
reserve q,q1 for Element of RAT;

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;
