let a, b be real number ; :: thesis: ( 0 <= a & 0 <= b implies sqrt (a * b) = (sqrt a) * (sqrt b) )
assume that
A1: 0 <= a and
A2: 0 <= b ; :: thesis: sqrt (a * b) = (sqrt a) * (sqrt b)
A3: 0 <= sqrt a by A1, Def4;
A4: 0 <= sqrt b by A2, Def4;
(sqrt (a * b)) ^2 = a * b by A1, A2, Def4
.= ((sqrt a) ^2 ) * b by A1, Def4
.= ((sqrt a) ^2 ) * ((sqrt b) ^2 ) by A2, Def4
.= ((sqrt a) * (sqrt b)) ^2 ;
hence sqrt (a * b) = sqrt (((sqrt a) * (sqrt b)) ^2 ) by A1, A2, Def4
.= (sqrt a) * (sqrt b) by A3, A4, Def4 ;
:: thesis: verum