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 Th7:
for a,b be Real, f st f is nonnegative & a > 0 & b > 0 holds
  (f to_power a)(#)(f 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 b) = dom f by MESFUN6C:def 4;
A3:dom((f to_power a)(#)(f to_power b))
    = dom (f to_power a) /\ dom (f to_power b) by VALUED_1:def 4; then
A4:dom ((f to_power a)(#)(f to_power b)) = dom(f to_power (a+b))
     by A2,MESFUN6C:def 4;
   for x be object st x in dom((f to_power a)(#)(f to_power b)) holds
    ((f to_power a)(#)(f to_power b)).x = (f to_power (a+b)).x
   proof
    let x be object;
    assume A5: x in dom((f to_power a)(#)(f to_power b));
    then (f to_power a).x = ((f.x) to_power a) &
    (f to_power b).x = ((f.x) to_power b) by A2,A3,MESFUN6C:def 4; then
A6: ((f to_power a)(#)(f to_power b)).x
     = ((f.x) to_power a) * ((f.x) to_power b) by A5,VALUED_1:def 4;
A7: (f to_power (a+b)).x = (f.x) to_power (a+b) by A4,A5,MESFUN6C:def 4;
then
A8: f.x > 0 implies ((f to_power a)(#)(f to_power b)).x = (f to_power (a+b)).x
      by A6,POWER:27;
    now assume A9: f.x = 0; then
     ((f to_power a)(#)(f to_power b)).x = 0 * (0 to_power b)
        by A1,A6,POWER:def 2;
     hence ((f to_power a)(#)(f to_power b)).x = (f to_power (a+b)).x
        by A7,A1,A9,POWER:def 2;
    end;
    hence thesis by A1,A8,MESFUNC6:51;
   end;
   hence thesis by A4,FUNCT_1:2;
end;
