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 Th75:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S holds
  Pre-L-Space M = Pre-Lp-Space(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 = CosetSet M &
   the addF of Pre-L-Space M = addCoset M &
   0.(Pre-L-Space M) = zeroCoset M &
   the Mult of Pre-L-Space M = lmultCoset M by LPSPACE1:def 18;
   CosetSet M = CosetSet(M,1) & addCoset M = addCoset(M,1) &
   zeroCoset M = zeroCoset(M,1) & lmultCoset M = lmultCoset(M,1)
      by Th71,Th72,Th73,Th74;
   hence thesis by A1,Def11;
end;
