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 Th76:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds
 L-1-Norm M = Lp-Norm(M,1)
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S;
A1:the carrier of Pre-L-Space M = the carrier of Pre-Lp-Space(M,1) by Th75;
   now let x be Element of the carrier of Pre-L-Space M;
    x in the carrier of Pre-L-Space M; then
 x in CosetSet M by LPSPACE1:def 18;
    then consider g be PartFunc of X,REAL such that
A2:  x = a.e-eq-class(g,M) & g in L1_Functions M;
    consider a be PartFunc of X,REAL such that
A3:  a in x & (L-1-Norm M).x = Integral(M,|.a.|) by LPSPACE1:def 19;
A4: ex p be PartFunc of X,REAL st
     a = p & p in L1_Functions M & g in L1_Functions M & g a.e.= p,M
       by A2,A3;
    consider b be PartFunc of X,REAL such that
A5:  b in x & ex r be Real st r = Integral(M,(|.b.|) to_power 1) &
      (Lp-Norm(M,1)).x = r to_power (1/1) by A1,Def12;
A6: ex q be PartFunc of X,REAL st b = q & q in L1_Functions M
      & g in L1_Functions M & g a.e.= q,M by A2,A5;
    a a.e.= g,M by A4; then
    a a.e.= b,M by A6,LPSPACE1:30; then
A7: Integral(M,|.a.|) = Integral(M,|.b.|) by A2,A3,A5,LPSPACE1:45;
    (|.b.|) to_power 1 = |.b.| by Th8;
    hence (L-1-Norm M).x = (Lp-Norm(M,1)).x by A3,A5,A7,POWER:25;
   end;
   hence thesis by A1,FUNCT_2:63;
end;
