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 Th16:
for a,b,k be Real st k > 0 holds
 |.a+b qua Complex.| to_power k
   <= (|.a qua Complex.| + |.b qua Complex.|) to_power k &
 (|.a qua Complex.| + |.b qua Complex.|) to_power k
   <= (2 * max(|.a qua Complex.|,|.b qua Complex.|)) to_power k &
 |.a+b qua Complex.| to_power k
    <= (2*max(|.a qua Complex.|,|.b qua Complex.|)) to_power k
proof
   let a,b,k be Real;
   assume A1: k > 0;
A2:|.a+b qua Complex.| <= |.a qua Complex.| + |.b qua Complex.| by ABSVALUE:9;
   |.a.| <= max(|.a.|,|.b.|) & |.b.| <= max(|.a.|,|.b.|) by XXREAL_0:25; then
A3:|.a qua Complex.| + |.b qua Complex.|
     <= max(|.a qua Complex.|,|.b qua Complex.|) +
    max(|.a qua Complex.|,|.b qua Complex.|) by XREAL_1:7; then
A4:|.a+b qua Complex.| <= 2*max(|.a qua Complex.|,|.b qua Complex.|)
    by A2,XXREAL_0:2;
   0 <= |.a+b qua Complex.| by COMPLEX1:46;
   hence thesis by A1,A2,A3,A4,HOLDER_1:3;
end;
