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 Th60:
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
ex r1 be Real st r1 = Integral(M,(abs f) to_power m) &
ex r2 be Real st r2 = Integral(M,(abs g) to_power n) &
Integral(M,abs(f(#)g)) <= r1 to_power (1/m) * r2 to_power (1/n)
proof
    let m,n be positive Real;
    assume
A1:  1/m +1/n =1 & f in Lp_Functions(M,m) & g in Lp_Functions(M,n);
then A2: m > 1 & n > 1 by Th1;
    consider f1 be PartFunc of X,REAL such that
A3:  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 A1;
    consider EDf be Element of S such that
A4:  M.EDf` =0 & dom f1 = EDf & f1 is EDf-measurable by A3;
    consider g1 be PartFunc of X,REAL such that
A5:  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 A1;
    consider EDg be Element of S such that
A6:  M.EDg` =0 & dom g1 = EDg & g1 is EDg-measurable by A5;
    set u =(abs f1) to_power m;
    set v =(abs g1) to_power n;
A7: 0 <= Integral(M,u) & 0 <= Integral(M,v) by A3,A5,A1,Th49;
    reconsider s1 = Integral(M,u), s2 = Integral(M,v) as Element of REAL
by A3,A5,A1,Th49;
A8: dom f1 = dom (abs f1) & dom g1 = dom (abs g1) by VALUED_1:def 11;
    reconsider Nf = EDf`, Ng = EDg` as Element of S by MEASURE1:34;
    set t1 = s1 to_power (1/m);
    set t2 = s2 to_power (1/n);
    set E = EDf /\ EDg;
A9:E` = EDf` \/ EDg` by XBOOLE_1:54;
    Nf is measure_zero of M & Ng is measure_zero of M
       by A4,A6,MEASURE1:def 7; then
A10:E` is measure_zero of M by A9,MEASURE1:37;
A11:dom (f1(#)g1) = EDf /\ EDg  by A4,A6,VALUED_1:def 4;
    f1 is E-measurable & g1 is E-measurable
         by A4,A6,MESFUNC6:16,XBOOLE_1:17; then
A12:f1(#)g1 is E-measurable by A4,A6,MESFUN7C:31;
A13:f1(#)g1 in L1_Functions M by A1,A3,A5,Th59; then
A14:ex fg1 be PartFunc of X,REAL st
     fg1=f1(#)g1 & ex ND be Element of S st M.ND=0 & dom fg1 = ND` &
     fg1 is_integrable_on M; then
A15:Integral(M,abs(f1(#)g1)) in REAL & abs(f1(#)g1) is_integrable_on M
        by LPSPACE1:44;
    per cases by A3,A5,A1,Th49;
    suppose A16: s1 = 0 & s2 >= 0;
     f1 in Lp_Functions (M,m) by A3; then
     f1 a.e.= X-->0,M  by A16,Th57; then
     consider Nf1 be Element of S such that
A17:  M.Nf1 = 0 & f1|Nf1` = (X-->0)|Nf1`;
     reconsider Z = (E \ Nf1)` as Element of S by MEASURE1:34;
A18: (E \ Nf1)` = E` \/ Nf1 by SUBSET_1:14;
     Nf1 is measure_zero of M by A17,MEASURE1:def 7; then
     Z is measure_zero of M by A10,A18,MEASURE1:37; then
A19: M.Z = 0 by MEASURE1:def 7;
     dom (X-->0) = X by FUNCOP_1:13; then
A20: dom ((X-->0)|Z`) = Z` by RELAT_1:62;
A21: dom ((f1(#)g1)|Z`) = Z` by A11,RELAT_1:62,XBOOLE_1:36;
     for x be object st x in dom((f1(#)g1)|Z`) holds
       ((f1(#)g1)|Z`).x = ((X-->0)|Z`).x
     proof
      let x be object;
      assume A22: x in dom ((f1(#)g1)|Z`); then
      x in X & not x in Nf1 by A21,XBOOLE_0:def 5; then
      x in Nf1` by XBOOLE_0:def 5; then
      f1.x = (f1|Nf1`).x & (X-->0).x = ((X-->0)|Nf1`).x by FUNCT_1:49; then
A23:  f1.x = 0 by A17,A22,FUNCOP_1:7;
A24:  dom ((f1(#)g1)|Z`) c= dom(f1(#)g1) by RELAT_1:60;
      ((f1(#)g1)|Z`).x = (f1(#)g1).x by A22,FUNCT_1:47
                      .= f1.x * g1.x by A22,A24,VALUED_1:def 4
                      .= (Z`-->0).x by A22,A21,A23,FUNCOP_1:7
                      .= (X/\Z`-->0).x by XBOOLE_1:28;
      hence thesis by FUNCOP_1:12;
     end; then
     (f1(#)g1)|Z` =(X-->0)|Z` by A20,A21,FUNCT_1:def 11; then
A25: f1(#)g1 a.e.= X-->0,M by A19;
     X-->0 in L1_Functions M by Th56; then
     Integral(M,abs(f1(#)g1)) = Integral(M,abs(X-->0))
       by A13,A25,LPSPACE1:45; then
A26: Integral(M,abs (f1(#)g1)) = 0 by LPSPACE1:54;
     t1 * t2 = 0 * t2 by A16,POWER:def 2;
     hence thesis by A3,A5,A26;
    end;
    suppose A27: s1 > 0 & s2 = 0;
     g1 in Lp_Functions(M,n) by A5; then
     g1 a.e.= X-->0,M  by A27,Th57; then
     consider Ng1 be Element of S such that
A28:  M.Ng1 = 0 & g1|Ng1` = (X-->0)|Ng1`;
     reconsider Z = (E \ Ng1)` as Element of S by MEASURE1:34;
A29: (E \ Ng1)` = E` \/ Ng1 by SUBSET_1:14;
     Ng1 is measure_zero of M by A28,MEASURE1:def 7; then
     Z is measure_zero of M by A10,A29,MEASURE1:37; then
A30: M.Z = 0 by MEASURE1:def 7;
     dom (X-->0) = X by FUNCOP_1:13; then
A31: dom ((X-->0)|Z`) = Z` by RELAT_1:62;
A32: dom ((f1(#)g1)|Z`) = Z` by A11,RELAT_1:62,XBOOLE_1:36;
     for x be object st x in dom((f1(#)g1)|Z`) holds
       ((f1(#)g1)|Z`).x = ((X-->0)|Z`).x
     proof
      let x be object;
      assume A33: x in dom((f1(#)g1)|Z`); then
      x in X & not x in Ng1 by A32,XBOOLE_0:def 5; then
      x in Ng1` by XBOOLE_0:def 5; then
      g1.x = (g1|Ng1`).x & (X-->0).x = ((X-->0)|Ng1`).x by FUNCT_1:49; then
A34:  g1.x = 0 by A28,A33,FUNCOP_1:7;
A35:  dom ((f1(#)g1)|Z`) c= dom(f1(#)g1) by RELAT_1:60;
      ((f1(#)g1)|Z`).x = (f1(#)g1).x by A33,FUNCT_1:47
                      .= f1.x * g1.x by A33,A35,VALUED_1:def 4
                      .= (Z`-->0).x by A33,A32,A34,FUNCOP_1:7
                      .= (X/\Z`-->0).x by XBOOLE_1:28;
      hence thesis by FUNCOP_1:12;
     end; then
     (f1(#)g1)|Z` =(X-->0)|Z` by A31,A32,FUNCT_1:def 11; then
A36: f1(#)g1 a.e.= X-->0,M by A30;
     X-->0 in L1_Functions M by Th56; then
     Integral(M,abs(f1(#)g1))=Integral(M,abs(X-->0))
         by A13,A36,LPSPACE1:45; then
A37: Integral(M,abs(f1(#)g1))=0 by LPSPACE1:54;
     t1 * t2 = t1 * 0 by A27,POWER:def 2;
     hence thesis by A3,A5,A37;
    end;
    suppose A38: s1 <> 0 & s2 <> 0; then
A39: t1 > 0 & t2 >0 by A7,POWER:34; then
A40: |.1/(t1*t2).| = 1/(t1*t2) by ABSVALUE:def 1;
     set w = (1/(t1*t2))(#)(f1(#)g1);
     set F = (1/m)(#)((1/t1(#)(abs f1)) to_power m);
     set G = (1/n)(#)((1/t2(#)(abs g1)) to_power n);
     set z = F + G;
A41: dom (1/t1(#)(abs f1)) = dom abs f1 &
     dom (1/t2(#)(abs g1)) = dom abs g1 by VALUED_1:def 5;
     dom F = dom ((1/t1(#)(abs f1)) to_power m) &
     dom G = dom ((1/t2(#)(abs g1)) to_power n ) by VALUED_1:def 5; then
A42: dom F = dom abs f1 & dom G = dom abs g1 by A41,MESFUN6C:def 4; then
A43: dom z = dom abs f1 /\ dom abs g1 by VALUED_1:def 1;
     ((1/t1)(#)(abs f1)) to_power m = ((1/t1) to_power m)(#)u &
     ((1/t2)(#)(abs g1)) to_power n = ((1/t2) to_power n)(#)v
        by A39,Th19; then
A44: (1/t1(#)(abs f1)) to_power m is_integrable_on M &
     (1/t2(#)(abs g1)) to_power n is_integrable_on M
        by A3,A5,MESFUNC6:102;then
A45: F is_integrable_on M & G is_integrable_on M by MESFUNC6:102; then
A46: z is_integrable_on M by MESFUNC6:100;
A47:dom w = dom (f1(#)g1) by VALUED_1:def 5; then
A48:dom w = dom f1 /\ dom g1 by VALUED_1:def 4;
    dom((1/(t1*t2))(#)abs(f1(#)g1)) = dom abs(f1(#)g1) by VALUED_1:def 5; then
A49:dom((1/(t1*t2))(#)abs(f1(#)g1)) = dom (f1(#)g1) by VALUED_1:def 11;
A50:w is E-measurable by A11,A12,MESFUNC6:21;
    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 A51: x in dom w;
     (abs f1).x >= 0 & (abs g1).x >= 0 by MESFUNC6:51;then
A52: (1/t1*(abs f1).x) * (1/t2*(abs g1).x) <=
     (1/t1*(abs f1).x) to_power m /m + (1/t2*(abs g1).x)to_power n /n
        by A1,A2,A39,HOLDER_1:5;
     dom((abs f1)(#)(abs g1)) = dom abs f1 /\ dom abs g1
        by VALUED_1:def 4; then
A53: ((abs f1)(#)(abs g1)).x = (abs f1).x * (abs g1).x
       by A8,A48,A51,VALUED_1:def 4;
A54: ((1/t1*(abs f1).x) * (1/t2*(abs g1).x))
        = ((1/t1)*(1/t2)*(abs f1).x)*((abs g1).x)
       .= (1/(t1*t2))*(abs f1).x * (abs g1).x by XCMPLX_1:102
       .= (1/(t1*t2))*(((abs f1)(#)(abs g1)).x) by A53
       .= (1/(t1*t2))*abs(f1(#)g1).x  by RFUNCT_1:24
       .= ((1/(t1*t2))(#)abs(f1(#)g1)).x by A47,A51,A49,VALUED_1:def 5
       .= (abs w).x by A40,RFUNCT_1:25;
A55: 1/t1*(abs f1).x = (1/t1(#)(abs f1)).x &
     1/t2*(abs g1).x = (1/t2(#)(abs g1)).x by VALUED_1:6;
     dom((1/t1(#)(abs f1)) to_power m) = dom f1 &
     dom((1/t2(#)(abs g1)) to_power n) = dom g1 by A8,A41,MESFUN6C:def 4; then
     x in dom ((1/t1(#)(abs f1)) to_power m) &
     x in dom ((1/t2(#)(abs g1)) to_power n) by A48,A51,XBOOLE_0:def 4; then
 ((1/t1(#)(abs f1)).x) to_power m = ((1/t1(#)(abs f1)) to_power m).x &
     ((1/t2(#)(abs g1)).x) to_power n = ((1/t2(#)(abs g1)) to_power n).x
         by MESFUN6C:def 4;
     then ((1/t1(#)(abs f1)).x) to_power m /m = F.x &
     ((1/t2(#)(abs g1)).x) to_power n /n = G.x by VALUED_1:6; then
     (1/t1*(abs f1).x) to_power m /m + (1/t2*(abs g1).x)to_power n /n = z.x
        by A8,A48,A51,A43,A55,VALUED_1:def 1;
     hence thesis by A52,A54,VALUED_1:18;
    end; then
A56:Integral(M,abs w) <= Integral(M,z) by A4,A6,A46,A8,A48,A43,A50,MESFUNC6:96;
 consider E1 be Element of S such that
A57: E1 = dom F /\ dom G & Integral(M,F+G) =Integral(M,F|E1)+Integral(M,G|E1)
        by A45,MESFUNC6:101;
    EDf = X /\ EDf & EDg = X /\ EDg by XBOOLE_1:28; then
A58:EDf = X \ Nf & EDg = X \ Ng by XBOOLE_1:48;
A59:EDf \ E = EDf \ EDg by XBOOLE_1:47
           .= (X \ Nf) \ X \/ (X \ Nf) /\ Ng by A58,XBOOLE_1:52
           .= X \(Nf \/ X) \/ (X \ Nf) /\ Ng by XBOOLE_1:41
           .= X \ X \/(X \ Nf)/\ Ng by XBOOLE_1:12
           .= {} \/ (X \ Nf) /\ Ng by XBOOLE_1:37;
A60:EDg \ E = EDg \ EDf by XBOOLE_1:47
           .= (X \ Ng) \ X \/ (X \ Ng) /\ Nf by A58,XBOOLE_1:52
           .= X \(Ng \/ X) \/ (X \ Ng) /\ Nf by XBOOLE_1:41
           .= X \ X \/(X \ Ng)/\ Nf by XBOOLE_1:12
           .= {} \/ (X \ Ng) /\ Nf by XBOOLE_1:37;
    set NF = EDf /\ Ng;
    set NG = EDg /\ Nf;
    Nf is measure_zero of M & Ng is measure_zero of M
        by A4,A6,MEASURE1:def 7; then
    NF is measure_zero of M & NG is measure_zero of M
        by MEASURE1:36,XBOOLE_1:17; then
A61:M.NF = 0 & M.NG = 0 by MEASURE1:def 7;
    E = EDf /\ E & E = EDg /\ E by XBOOLE_1:17,28; then
A62:E = EDf \ NF & E = EDg \ NG by A58,A59,A60,XBOOLE_1:48;
    R_EAL F is_integrable_on M by A45; then
    ex E being Element of S st
     E = dom R_EAL F & R_EAL F is E-measurable; then
A63:F is EDf-measurable by A42,A8,A4;
    R_EAL G is_integrable_on M by A45; then
    ex E being Element of S st
     E = dom R_EAL G & R_EAL G is E-measurable; then
A64:G is EDg-measurable by A42,A8,A6;
    (1/t1) to_power m = t1 to_power (-m) by A38,A7,POWER:32,34; then
    (1/t1) to_power m = s1 to_power((1/m)*(-m)) by A7,A38,POWER:33; then
    (1/t1) to_power m = s1 to_power (-1*(1/m)*m); then
    (1/t1) to_power m = s1 to_power (-1) by XCMPLX_1:106; then
    (1/t1) to_power m = (1/s1) to_power 1 by A7,A38,POWER:32; then
A65:(1/t1) to_power m = 1/s1 by POWER:25;
A66:(1/s1 qua ExtReal) * s1 = 1 &
     (1/s2 qua ExtReal) * s2 = 1
       by A38,XCMPLX_1:106;
A67:(1/t2) to_power n = t2 to_power (-n) by A38,A7,POWER:32,34
     .= s2 to_power((1/n)*(-n)) by A7,A38,POWER:33
     .= s2 to_power (-1*(1/n)*n)
     .= s2 to_power (-1) by XCMPLX_1:106
     .= (1/s2) to_power 1 by A7,A38,POWER:32
     .= 1/s2 by POWER:25;
A68:Integral(M,F|E) = Integral(M,F) by A4,A8,A42,A62,A61,A63,MESFUNC6:89
  .= (1/m) *Integral(M,(1/t1(#)(abs f1)) to_power m) by A44,MESFUNC6:102
  .=  (1/m) *Integral(M,((1/t1) to_power m)(#)((abs f1) to_power m))
     by A39,Th19
  .=  (1/m) * (((1/t1) to_power m) * Integral(M,(abs f1) to_power m))
     by A3,MESFUNC6:102
  .= 1/m by A65,A66,XXREAL_3:81;
A69:
Integral(M,G|E) = Integral(M,G) by A6,A8,A42,A62,A61,A64,MESFUNC6:89
  .=  (1/n)*Integral(M,(1/t2(#)(abs g1)) to_power n) by A44,MESFUNC6:102
  .=  (1/n) * Integral(M,((1/t2) to_power n)(#)((abs g1) to_power n))
     by A39,Th19
  .=  (1/n) * (((1/t2) to_power n) * Integral(M,(abs g1) to_power n))
     by A5,MESFUNC6:102
  .= 1/n by A66,A67,XXREAL_3:81;
  reconsider n1=1/n, m1=1/m as Real;
A70:Integral(M,F+G) = Integral(M,F|E)+Integral(M,G|E)
       by A42,A4,A6,A8,A57
     .= m1+n1 by A69,A68,SUPINF_2:1
     .= jj by A1;
    abs w = |.1/(t1*t2) qua Complex.|(#)abs(f1(#)g1) by RFUNCT_1:25; then
    abs w = (1/(t1*t2))(#)abs(f1(#)g1) by A39,ABSVALUE:def 1; then
A71:Integral(M,abs w) = (1/(t1*t2)) * Integral(M,abs(f1(#)g1))
        by A15,MESFUNC6:102;
    reconsider c1 = Integral(M,abs(f1(#)g1)) as Element of REAL
by A14,LPSPACE1:44;
    (1/(t1*t2) qua ExtReal) * Integral(M,abs(f1(#)g1))
      = (1/(t1*t2) qua ExtReal) * c1;
     then
    (1/(t1*t2)) * Integral(M,abs(f1(#)g1)) = (1/(t1*t2)) * c1;
    then
    (t1*t2)*((1/(t1*t2)) * c1) <= (t1*t2)*1 by A39,A56,A71,A70,XREAL_1:64; then
A72:(t1*t2)*(1/(t1*t2)) * c1 <= (t1*t2);
    (t1*t2)*(1/(t1*t2)) = 1 by A39,XCMPLX_1:106;
    hence thesis by A3,A5,A72;
    end;
end;
