let z1, z3, z be Element of COMPLEX ; :: thesis: ( z1 <> 0 & Polynom z1,0 ,z3,0 ,z = 0 implies for s being Element of COMPLEX holds
( not s = - (z3 / z1) or z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) ) )
assume that
A1: z1 <> 0 and
A2: Polynom z1,0 ,z3,0 ,z = 0 ; :: thesis: for s being Element of COMPLEX holds
( not s = - (z3 / z1) or z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
let s be Element of COMPLEX ; :: thesis: ( not s = - (z3 / z1) or z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
0 = ((z1 * (z ^2 )) + z3) * z by A2;
then A3: ( Polynom z1,0 ,z3,z = 0 or z = 0 ) ;
assume s = - (z3 / z1) ; :: thesis: ( z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) )
hence ( z = 0 or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) * <i> ) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2))) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2 ) + ((Im s) ^2 )))) / 2)) * <i> ) ) by A1, A3, Th36; :: thesis: verum