theorem
  f in x implies x= a.e-Ceq-class(f,M) & ||.x.|| = Integral(M,abs f)
proof
  assume
A1: f in x;
  reconsider y=x as Point of Pre-L-CSpace M;
  y in the carrier of Pre-L-CSpace M;
  then y in CCosetSet M by Def19;
  then consider g be PartFunc of X,COMPLEX such that
A2: y=a.e-Ceq-class(g,M) & g in L1_CFunctions M;
  g in y by A2,Th31;
  then f a.e.cpfunc= g,M by A1,Th39;
  hence thesis by A1,A2,Th32,Th43;
end;
