let a1, a2, b1, b2, c1, c2 be Element of REAL ; :: thesis: ( |.((a1 * b2) - (a2 * b1)).| <> 0 implies ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4 )
assume A1: |.((a1 * b2) - (a2 * b1)).| <> 0 ; :: thesis: ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4

per cases ( ( a1 / b1 is irrational & ZeroPointSet (LF (a1,b1,c1)) = {} ) or ZeroPointSet (LF (a1,b1,c1)) <> {} or ( a1 / b1 is rational & ZeroPointSet (LF (a1,b1,c1)) = {} ) ) ;

suppose A2: ( a1 / b1 is irrational & ZeroPointSet (LF (a1,b1,c1)) = {} ) ; :: thesis: ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4

for eps being positive Real ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4

proof

let eps be positive Real; :: thesis: ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4
ex x, y being 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 ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4 ; :: thesis: verum

end;

hence ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4 ; :: thesis: verum

end;

suppose ZeroPointSet (LF (a1,b1,c1)) <> {} ; :: thesis: ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4

hence ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4 by Th48; :: thesis: verum

end;

suppose ( a1 / b1 is rational & ZeroPointSet (LF (a1,b1,c1)) = {} ) ; :: thesis: ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4

hence ex x, y being Element of INT st |.((LF (a1,b1,c1)) . (x,y)).| * |.((LF (a2,b2,c2)) . (x,y)).| <= |.((a1 * b2) - (a2 * b1)).| / 4 by A1, Th52, Th50; :: thesis: verum

end;

end;