let z, s be Element of COMPLEX ; :: thesis: ( z ^2 = s & Im s = 0 & Re s > 0 & not z = sqrt (Re s) implies z = - (sqrt (Re s)) )
assume A1: z ^2 = s ; :: thesis: ( not Im s = 0 or not Re s > 0 or z = sqrt (Re s) or z = - (sqrt (Re s)) )
assume A2: ( Im s = 0 & Re s > 0 ) ; :: thesis: ( z = sqrt (Re s) or z = - (sqrt (Re s)) )
then ( 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, Th28;
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 A2, SQUARE_1:89;
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 A2, SQUARE_1:89;
then ( z = (sqrt (Re s)) + ((sqrt ((0 + ((Re s) - (Re s))) / 2)) * <i> ) or z = (- (sqrt (Re s))) + (- ((sqrt ((0 + ((Re s) - (Re s))) / 2)) * <i> )) ) by A2, SQUARE_1:89;
hence ( z = sqrt (Re s) or z = - (sqrt (Re s)) ) by SQUARE_1:82; :: thesis: verum