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 Th52:
   |.a1*b2-a2*b1.|<>0 & b1 = 0 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
     set Delta = |.a1*b2-a2*b1.|;
     assume that
A1:  |.a1*b2-a2*b1.|<>0 and
A2:  b1 = 0;
A4:  b2 <> 0 by A1,A2;
     per cases;
     suppose
A5:    a2/b2 is irrational & ZeroPointSet(LF(a2,b2,c2))={};
       for eps holds 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
         let eps;
         ex x,y be Element of INT st
         |.LF(a2,b2,c2).(x,y).|*|.LF(a1,b1,c1).(x,y).|<|.a1*b2-a2*b1.|/4 &
         |.LF(a2,b2,c2).(x,y).| < eps by A1,A5,Th47;
         hence thesis;
       end;
       hence thesis;
     end;
     suppose ZeroPointSet(LF(a2,b2,c2))<>{};
       hence thesis by Th49;
     end;
     suppose a2/b2 is rational & ZeroPointSet(LF(a2,b2,c2))={};
       hence thesis by A1,A4,Th51;
     end;
   end;
