let a, b, c be real number ; :: thesis: ( delta a,b,c < 0 implies for x being real number holds not Polynom a,b,c,x = 0 )
set e = a * c;
assume delta a,b,c < 0 ; :: thesis: for x being real number holds not Polynom a,b,c,x = 0
then (b ^2 ) - ((4 * a) * c) < 0 by QUIN_1:def 1;
then A1: ((b ^2 ) - (4 * (a * c))) * (4 " ) < 0 by XREAL_1:134;
given y being real number such that A2: Polynom a,b,c,y = 0 ; :: thesis: contradiction
set t = ((a ^2 ) * (y ^2 )) + ((a * b) * y);
a * (((a * (y ^2 )) + (b * y)) + c) = a * 0 by A2;
then ((((a ^2 ) * (y ^2 )) + ((a * b) * y)) + ((b ^2 ) / 4)) - (((b ^2 ) * (4 " )) - ((4 * (a * c)) * (4 " ))) = 0 ;
then A3: ((a * y) + (b / 2)) ^2 = ((b ^2 ) - (4 * (a * c))) * (4 " ) ;
then (a * y) + (b / 2) > 0 by A1, XREAL_1:135;
hence contradiction by A3, A1, XREAL_1:135; :: thesis: verum