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 Th24:
for k be Real st k > 0 for f,g be PartFunc of X,REAL holds
  for x be Element of X st x in dom f /\ dom g holds
   (abs(f + g) to_power k).x <=
    ((2 to_power k)(#)((abs f) to_power k + (abs g) to_power k)).x
proof
   let k be Real;
   assume A1: k > 0;
   let f,g be PartFunc of X,REAL;
   let x be Element of X;
   assume A2: x in dom f /\ dom g;
A3:dom(f + g) = dom f /\ dom g by VALUED_1:def 1; then
   dom abs(f + g) = dom f /\ dom g by VALUED_1:def 11; then
   x in dom (abs(f + g) to_power k) by A2,MESFUN6C:def 4; then
A4:(abs(f + g) to_power k).x = ((abs(f + g)).x) to_power k by MESFUN6C:def 4
    .= |.(f+g).x qua Complex.| to_power k by VALUED_1:18
    .= |.f.x + g.x qua Complex.| to_power k by A3,A2,VALUED_1:def 1;
   dom (abs f) = dom f & dom (abs g) = dom g by VALUED_1:def 11; then
   x in dom (abs f) & x in dom (abs g) by A2,XBOOLE_0:def 4; then
A5:x in dom ((abs f) to_power k) & x in dom ((abs g) to_power k)
      by MESFUN6C:def 4;
   |.f.x qua Complex.| to_power k = ((abs f).x) to_power k &
   |.g.x qua Complex.| to_power k = ((abs g).x) to_power k by VALUED_1:18; then
A6:|.f.x qua Complex.| to_power k = ((abs f) to_power k).x &
   |.g.x qua Complex.| to_power k = ((abs g) to_power k).x
     by A5,MESFUN6C:def 4;
   dom ((abs f) to_power k + (abs g) to_power k) =
    dom ((abs f) to_power k) /\ dom ((abs g) to_power k) by VALUED_1:def 1;then
   x in dom((abs f) to_power k + (abs g) to_power k) by A5,XBOOLE_0:def 4; then
   (2 to_power k)*(|.f.x qua Complex.| to_power k
     + |.g.x qua Complex.| to_power k)
    = (2 to_power k) * ((abs f) to_power k + (abs g) to_power k).x
      by A6,VALUED_1:def 1
   .= ((2 to_power k)(#)((abs f) to_power k + (abs g) to_power k)).x
      by VALUED_1:6;
   hence thesis by A1,A4,Th21;
end;
