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;
reserve v,u for VECTOR of RLSp_LpFunct(M,k);
reserve v,u for VECTOR of RLSp_AlmostZeroLpFunct(M,k);
reserve x for Point of Pre-Lp-Space(M,k);
reserve x,y for Point of Lp-Space(M,k);

theorem Th61:
for m be positive Real for r1,r2,r3 be Real st 1 <= m &
 f in Lp_Functions(M,m) & g in Lp_Functions(M,m) &
 r1 = Integral(M,(abs f) to_power m) &
 r2 = Integral(M,(abs g) to_power m) &
 r3 = Integral(M,(abs (f+g)) to_power m) holds
 r3 to_power (1/m) <= r1 to_power (1/m) + r2 to_power (1/m)
proof
   let m be positive Real;
   let r1,r2,r3 be Real;
   assume
A1: 1 <= m & f in Lp_Functions(M,m) & g in Lp_Functions(M,m) &
    r1 = Integral(M,(abs f) to_power m) &
    r2 = Integral(M,(abs g) to_power m) &
    r3 = Integral(M,(abs (f+g)) to_power m);
   per cases;
   suppose A2: m=1; then
A3: r1 = Integral(M,(abs f)) & r2 = Integral(M,(abs g)) &
    r3 = Integral(M,(abs (f+g))) by A1,Th8;
A4:ex f1 be PartFunc of X,REAL st
     f=f1 & ex ND be Element of S st M.ND` =0 & dom f1 = ND &
     f1 is ND-measurable & (abs f1) to_power m is_integrable_on M by A1;
A5:ex g1 be PartFunc of X,REAL st
     g=g1 & ex ND be Element of S st M.ND` =0 & dom g1 = ND &
     g1 is ND-measurable & (abs g1) to_power m is_integrable_on M by A1;
    then (abs f) is_integrable_on M &
    (abs g) is_integrable_on M by A2,A4,Th8; then
    f is_integrable_on M & g is_integrable_on M by A4,A5,MESFUNC6:94; then
    Integral(M,abs(f+g)) <= Integral(M,abs f) + Integral(M,abs g)
       by LPSPACE1:55; then
A6:  r3 <= r1 + r2 by A3,XXREAL_3:def 2;
    r1 to_power (1/m) = r1 & r2 to_power (1/m) = r2 by A2,POWER:25;
    hence thesis by A6,A2,POWER:25;
   end;
   suppose A7: m <> 1;
    set n1= 1 - 1/m;
   1 < m by A1,A7,XXREAL_0:1; then
   1/m < 1 by XREAL_1:189; then
   0 < n1 by XREAL_1:50; then
   reconsider n=1/n1 as positive Real;
   1/m +1/n =1; then
   ex rr1,rr2,rr3 be Real st rr1 = Integral(M,(abs f) to_power m) &
    rr2 = Integral(M,(abs g) to_power m) &
    rr3 = Integral(M,(abs (f+g)) to_power m) &
    rr3 to_power (1/m) <= rr1 to_power (1/m) + rr2 to_power (1/m) by A1,Lm5;
   hence thesis by A1;
   end;
end;
