let a, b, c be real number ; :: thesis: ( ( for x being real number holds ((a * (x ^2 )) + (b * x)) + c >= 0 ) & a > 0 implies delta a,b,c <= 0 )
assume that
A1: for x being real number 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, Th6;
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_1:88;
then - (delta a,b,c) >= - 0 ;
hence delta a,b,c <= 0 by XREAL_1:26; :: thesis: verum