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;
