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 Th21:
for a,b,k be Real st k > 0 holds (|.a+b qua Complex.|) to_power k
    <= (2 to_power k)*(|.a qua Complex.| to_power k +
          (|.b qua Complex.|) to_power k)
proof
   let a,b,k be Real;
   assume A1: k > 0; then
A2:(|.a+b qua Complex.|) to_power k
        <= (2*max(|.a qua Complex.|,|.b qua Complex.|)) to_power k by Th16;
A3:|.a.| >= 0 & |.b.| >= 0 by COMPLEX1:46; then
A4:(max(|.a qua Complex.|,|.b qua Complex.|)) to_power k
     <= |.a qua Complex.| to_power k + |.b qua Complex.| to_power k by A1,Th17;
   max(|.a.|,|.b.|) = |.a.| or max(|.a.|,|.b.|)= |.b.| by XXREAL_0:16; then
A5:(2*max(|.a qua Complex.|,|.b qua Complex.|)) to_power k
     = (2 to_power k)*(max(|.a qua Complex.|,|.b qua Complex.|)) to_power k
         by A1,A3,Th5;
   (2 to_power k) > 0 by POWER:34; then
   (2 to_power k) *(max(|.a qua Complex.|,|.b qua Complex.|)) to_power k
    <= (2 to_power k)*(|.a qua Complex.| to_power k
     + |.b qua Complex.| to_power k) by A4,XREAL_1:64;
   hence thesis by A2,A5,XXREAL_0:2;
end;
