reserve X for non empty set,
        x for Element of X,
        S for SigmaField of X,
        M for sigma_Measure of S,
        f,g,f1,g1 for PartFunc of X,REAL,
        l,m,n,n1,n2 for Nat,
        a,b,c for Real;
reserve k for positive Real;

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;
