theorem Th19:
for f be PartFunc of X,REAL, a,b be Real st a > 0 & b > 0 holds
(a to_power b)(#)((abs f) to_power b) = (a(#)(abs f)) to_power b
proof
   let f be PartFunc of X,REAL;
   let a,b be Real;
   assume A1: a > 0 & b > 0; then
A2:|.a.| = a by COMPLEX1:43; then
   (a to_power b)(#)((abs f) to_power b)
     = (abs(a(#)f)) to_power b by A1,Th18;
   hence thesis by A2,RFUNCT_1:25;
end;
