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 Th6:
for a,b be Real, f st f is nonnegative & a > 0 & b > 0 holds
  (f to_power a) to_power b = f to_power (a*b)
proof
   let a,b be Real;
   let f;
   assume A1: f is nonnegative & a > 0 & b > 0;
A2:dom (f to_power a) = dom f &
   dom((f to_power a) to_power b) = dom (f to_power a) &
   dom(f to_power (a*b)) = dom f by MESFUN6C:def 4;
   for x be object st x in dom((f to_power a) to_power b) holds
     ((f to_power a) to_power b).x = (f to_power (a*b)).x
   proof
    let x be object;
    assume A3: x in dom ((f to_power a) to_power b); then
A4: ((f to_power a) to_power b).x
       = ((f to_power a).x) to_power b by MESFUN6C:def 4
      .= ((f.x) to_power a) to_power b by A2,A3,MESFUN6C:def 4;
A5: (f to_power (a*b)).x = (f.x) to_power (a*b) by A2,A3,MESFUN6C:def 4; then
A6: f.x > 0 implies ((f to_power a) to_power b).x = (f to_power (a*b)).x
      by A4,POWER:33;
    now assume A7: f.x = 0; then
     ((f to_power a) to_power b).x = 0 to_power b by A1,A4,POWER:def 2; then
     ((f to_power a) to_power b).x = 0 by A1,POWER:def 2;
     hence ((f to_power a) to_power b).x = (f to_power (a*b)).x
       by A1,A7,A5,POWER:def 2;
    end;
    hence thesis by A6,A1,MESFUNC6:51;
   end;
   hence thesis by A2,FUNCT_1:2;
end;
