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
  |.a1*b2-a2*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
     assume
A1:  |.a1*b2-a2*b1.|<>0;
     per cases;
       suppose
A2:     a1/b1 is irrational & ZeroPointSet(LF(a1,b1,c1))={};
        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(a1,b1,c1).(x,y).|*|.LF(a2,b2,c2).(x,y).|<|.a1*b2-a2*b1.|/4 &
         |.LF(a1,b1,c1).(x,y).| < eps by A1,A2,Th46;
         hence thesis;
         end;
         hence thesis;
       end;
       suppose ZeroPointSet(LF(a1,b1,c1))<>{};
         hence thesis by Th48;
       end;
       suppose a1/b1 is rational & ZeroPointSet(LF(a1,b1,c1))={};
         hence thesis by A1,Th52,Th50;
       end;
    end;
