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 Th18:
for f be PartFunc of X,REAL, a,b be Real st b > 0 holds
(|.a qua Complex.| to_power b)(#)((abs f) to_power b) = (abs(a(#)f)) to_power b
proof
   let f be PartFunc of X,REAL;
   let a,b be Real;
   assume A1: b >0;
A2:dom((|.a qua Complex.| to_power b)(#)((abs f) to_power b))
   = dom((abs f) to_power b) &
   dom(a(#)f) = dom f by VALUED_1:def 5;
A3:dom((abs f) to_power b) = dom (abs f) &
   dom abs(a(#)f) = dom ((abs (a(#)f)) to_power b) by MESFUN6C:def 4;
A4:dom (abs f) = dom f & dom abs(a(#)f) = dom (a(#)f) by VALUED_1:def 11;
   for x be Element of X st
    x in dom ((|.a qua Complex.| to_power b)(#)((abs f) to_power b)) holds
    ((|.a qua Complex.| to_power b)(#)((abs f) to_power b)).x
      = ((abs(a(#)f)) to_power b).x
   proof
    let x be Element of X;
    assume
A5: x in dom ((|.a qua Complex.| to_power b)(#)((abs f) to_power b));
A6:|.f.x.| >= 0 & |.a.| >= 0 by COMPLEX1:46;
    ((|.a qua Complex.| to_power b)(#)((abs f) to_power b)).x
     = (|.a qua Complex.| to_power b) * ((abs f) to_power b).x
          by A5,VALUED_1:def 5
    .= (|.a qua Complex.| to_power b) * (((abs f).x) to_power b)
    by A2,A5,MESFUN6C:def 4
    .= (|.a qua Complex.| to_power b) * ((|.f.x qua Complex.|) to_power b)
           by VALUED_1:18
    .= (|.a qua Complex.| * |.f.x qua Complex.|) to_power b by A1,A6,Th5
    .= |.a * f.x qua Complex.| to_power b by COMPLEX1:65
    .= |.(a(#)f).x qua Complex.| to_power b by VALUED_1:6
    .= ((abs(a(#)f)).x) to_power b by VALUED_1:18;
    hence thesis by A2,A3,A4,A5,MESFUN6C:def 4;
   end;
   hence thesis by A2,A3,A4,PARTFUN1:5;
end;
