let s, z be Complex; :: thesis: ( z ^2 = s & Im s = 0 & Re s < 0 & not z = (sqrt (- (Re s))) * <i> implies z = - ((sqrt (- (Re s))) * <i>) )
assume A1: z ^2 = s ; :: thesis: ( not Im s = 0 or not Re s < 0 or z = (sqrt (- (Re s))) * <i> or z = - ((sqrt (- (Re s))) * <i>) )
assume that
A2: Im s = 0 and
A3: Re s < 0 ; :: thesis: ( z = (sqrt (- (Re s))) * <i> or z = - ((sqrt (- (Re s))) * <i>) )
( z = (sqrt (((Re s) + (sqrt (((Re s) ^2) + 0))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2) + 0))) / 2)) * <i>) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2) + 0))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2) + (0 ^2)))) / 2))) * <i>) ) by A1, A2, Th19;
then ( z = (sqrt (((Re s) + (- (Re s))) / 2)) + ((sqrt (((- (Re s)) + (sqrt (((Re s) ^2) + 0))) / 2)) * <i>) or z = (- (sqrt (((Re s) + (sqrt (((Re s) ^2) + 0))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2) + 0))) / 2))) * <i>) ) by A3, SQUARE_1:23;
then ( z = (sqrt (((Re s) + (- (Re s))) / 2)) + ((sqrt (((- (Re s)) + (- (Re s))) / 2)) * <i>) or z = (- (sqrt (((Re s) + (- (Re s))) / 2))) + ((- (sqrt (((- (Re s)) + (sqrt (((Re s) ^2) + 0))) / 2))) * <i>) ) by A3, SQUARE_1:23;
then ( z = (sqrt (((Re s) + (- (Re s))) / 2)) + ((sqrt (((- (Re s)) + (- (Re s))) / 2)) * <i>) or z = (- (sqrt (((Re s) + (- (Re s))) / 2))) + ((- (sqrt (((- (Re s)) + (- (Re s))) / 2))) * <i>) ) by A3, SQUARE_1:23;
hence ( z = (sqrt (- (Re s))) * <i> or z = - ((sqrt (- (Re s))) * <i>) ) by SQUARE_1:17; :: thesis: verum