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 Th48:
  ZeroPointSet( LF(a1,b1,c1))<>{} 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 ZeroPointSet( LF(a1,b1,c1))<>{}; then
     consider p be object such that
A2:  p in ZeroPointSet( LF(a1,b1,c1)) by XBOOLE_0:def 1;
     LF(a1,b1,c1).p = 0 & p in dom LF(a1,b1,c1) by A2,Th13; then
     consider p1,p2 be object such that
A4:  p1 in INT & p2 in INT & p = [p1,p2] by ZFMISC_1:def 2;
     reconsider x = p1,y=p2 as Element of INT by A4;
     LF(a1,b1,c1).(x,y)=0 by A2,Th13,A4; then
     |.LF(a1,b1,c1).(x,y).|*|.LF(a2,b2,c2).(x,y).|<=|.a1*b2-a2*b1.|/4;
     hence thesis;
   end;
