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 Th59:
for m,n be positive Real st
  1/m +1/n =1 & f in Lp_Functions(M,m) & g in Lp_Functions(M,n) holds
    f(#)g in L1_Functions M & f(#)g is_integrable_on M
proof
    let m,n be positive Real;
    assume that
A1:  1/m +1/n =1 and
A2:  f in Lp_Functions(M,m) & g in Lp_Functions(M,n);
A3: m > 1 & n > 1 by A1,Th1;
    consider f1 be PartFunc of X,REAL such that
A4:  f=f1 & ex NDf be Element of S st M.NDf` =0 & dom f1 = NDf &
     f1 is NDf-measurable & (abs f1) to_power m is_integrable_on M by A2;
    consider EDf be Element of S such that
A5:  M.EDf` =0 & dom f1 = EDf & f1 is EDf-measurable by A4;
    consider g1 be PartFunc of X,REAL such that
A6:  g=g1 & ex NDg be Element of S st M.NDg` =0 & dom g1 = NDg &
     g1 is NDg-measurable & (abs g1) to_power n is_integrable_on M by A2;
    consider EDg be Element of S such that
A7:  M.EDg` =0 & dom g1 = EDg & g1 is EDg-measurable by A6;
    set u =(abs f1) to_power m;
    set v =(abs g1) to_power n;
    set w = f1(#)g1;
    set z = (1/m)(#)u + (1/n)(#)v;
A8: dom f1 = dom(abs f1) & dom g1 = dom(abs g1) by VALUED_1:def 11; then
A9: dom u = dom f1 & dom v = dom g1 by MESFUN6C:def 4; then
A10: dom w = dom u /\ dom v by VALUED_1:def 4;
    set Nf = EDf`;
    set Ng = EDg`;
    set E = EDf /\ EDg;
    reconsider Nf,Ng as Element of S by MEASURE1:34;
    dom u = Nf` & dom v = Ng` by A5,A7,A8,MESFUN6C:def 4; then
    u in L1_Functions M & v in L1_Functions M by A4,A5,A6,A7; then
    (1/m)(#)u in L1_Functions M & (1/n)(#)v in L1_Functions M
       by LPSPACE1:24; then
    z in L1_Functions M by LPSPACE1:23; then
A11:ex h be PartFunc of X,REAL st z = h &
     ex ND be Element of S st M.ND=0 & dom h = ND` & h is_integrable_on M;
    dom((1/m)(#)u) = dom u & dom((1/n)(#)v) = dom v by VALUED_1:def 5; then
A12:dom z = dom u /\ dom v by VALUED_1:def 1;
A13:E` = EDf` \/ EDg` by XBOOLE_1:54;
    Nf is measure_zero of M & Ng is measure_zero of M
      by A5,A7,MEASURE1:def 7; then
    Nf \/ Ng is measure_zero of M by MEASURE1:37; then
A14:M.E` = 0 by A13,MEASURE1:def 7;
    f1 is E-measurable & g1 is E-measurable
       by A5,A7,MESFUNC6:16,XBOOLE_1:17; then
A15:w is E-measurable by A5,A7,MESFUN7C:31;
    for x be Element of X st x in dom w holds |.w.x qua Complex.| <= z.x
    proof
     let x be Element of X;
     assume A16: x in dom w;
     abs(f1(#)g1) = (abs f1)(#)(abs g1) by RFUNCT_1:24; then
     abs(f1(#)g1).x = (abs f1).x * (abs g1).x by VALUED_1:5; then
A17: |.(f1(#)g1).x.| = (abs f1).x * (abs g1).x by VALUED_1:18;
A18: (abs f1).x >= 0 & (abs g1).x >= 0 by MESFUNC6:51;
 x in dom u & x in dom v by A16,A10,XBOOLE_0:def 4;
     then ((abs f1).x) to_power m /m = (1/m)*(((abs f1) to_power m).x) &
     ((abs g1).x) to_power n /n = (1/n)*(((abs g1) to_power n).x)
          by MESFUN6C:def 4; then
     ((abs f1).x) to_power m /m = ((1/m)(#)((abs f1) to_power m)).x &
     ((abs g1).x) to_power n /n = ((1/n)(#)((abs g1) to_power n)).x
          by VALUED_1:6; then
     |.w.x.| <= ((1/m)(#)u).x + ((1/n)(#)v).x by A1,A3,A17,A18,HOLDER_1:5;
     hence thesis by A16,A10,A12,VALUED_1:def 1;
    end;then
A19:w is_integrable_on M by A5,A7,A9,A10,A11,A15,A12,MESFUNC6:96;
    set ND = E`;
    reconsider ND as Element of S by MEASURE1:34;
    dom w = ND` by A5,A7,VALUED_1:def 4;
    hence thesis by A4,A6,A14,A19;
end;
