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 Th17:
for a, b, k be Real st a >= 0 & b >= 0 & k > 0 holds
  max(a,b) to_power k <= a to_power k + b to_power k
proof
   let a,b,k be Real;
   assume A1: a >= 0 & b >= 0 & k > 0;
   per cases;
   suppose a <> 0 & b <> 0; then
A2: a to_power k >= 0 & b to_power k >= 0 by A1,POWER:34;
    max(a,b) = a or max(a,b) = b by XXREAL_0:def 10;
    hence max(a,b) to_power k <= a to_power k + b to_power k by A2,XREAL_1:40;
   end;
   suppose A3: a = 0; then
    a to_power k = 0 by A1,POWER:def 2;
    hence max(a,b) to_power k <= a to_power k + b to_power k
      by A1,A3,XXREAL_0:def 10;
   end;
   suppose A4: b =0; then
    b to_power k = 0 by A1,POWER:def 2;
    hence (max(a,b)) to_power k <= a to_power k + b to_power k
      by A1,A4,XXREAL_0:def 10;
   end;
end;
