theorem Th55:
f in x implies x= a.e-eq-class_Lp(f,M,k) & ( ex r be Real st
     0 <=r & r = Integral(M,(abs f) to_power k) & ||.x.|| = r to_power (1/k) )
proof
   assume A1:f in x;
   x in the carrier of Pre-Lp-Space(M,k); then
   x in CosetSet(M,k) by Def11; then
   consider g be PartFunc of X,REAL such that
A2: x=a.e-eq-class_Lp(g,M,k) & g in Lp_Functions(M,k);
   g in x by A2,Th38; then
   f a.e.= g,M & f in Lp_Functions(M,k) & g in Lp_Functions(M,k) by A1,Th50;
   hence thesis by Th53,A1,A2,Th42;
end;
