let a, b, c be Real; :: thesis: ( ( for x being Real holds ((a * (x ^2)) + (b * x)) + c < 0 ) & a < 0 implies delta (a,b,c) < 0 )
assume that
A1: for x being Real holds ((a * (x ^2)) + (b * x)) + c < 0 and
A2: a < 0 ; :: thesis: delta (a,b,c) < 0
((a * ((- (b / (2 * a))) ^2)) + (b * (- (b / (2 * a))))) + c < 0 by A1;
then ((((2 * a) * (- (b / (2 * a)))) + b) ^2) - (delta (a,b,c)) > 0 by A2, Th9;
then A3: (((- ((2 * a) * (b / (2 * a)))) + b) ^2) - (delta (a,b,c)) > 0 ;
2 * a <> 0 by A2;
then (((- b) + b) ^2) - (delta (a,b,c)) > 0 by A3, XCMPLX_1:87;
then - (delta (a,b,c)) > - 0 ;
hence delta (a,b,c) < 0 by XREAL_1:24; :: thesis: verum