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);

theorem Th41:
f a.e.= g,M implies a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k)
proof
   assume
A1: f a.e.= g,M;
   now let x be object;
    assume x in a.e-eq-class_Lp(f,M,k); then
    consider r be PartFunc of X,REAL such that
A2:  x=r & r in Lp_Functions(M,k) & f a.e.= r,M;
    r a.e.= f,M by A2; then
    r a.e.= g,M by A1,LPSPACE1:30; then
    g a.e.= r,M;
    hence x in a.e-eq-class_Lp(g,M,k) by A2;
   end; then
A3:a.e-eq-class_Lp(f,M,k) c= a.e-eq-class_Lp(g,M,k);
   now let x be object;
    assume x in a.e-eq-class_Lp(g,M,k); then
    consider r be PartFunc of X,REAL such that
A4:  x=r & r in Lp_Functions(M,k) & g a.e.= r,M;
    r a.e.= g,M & g a.e.= f,M by A1,A4; then
    r a.e.= f,M by LPSPACE1:30; then
    f a.e.= r,M;
    hence x in a.e-eq-class_Lp(f,M,k) by A4;
   end; then
   a.e-eq-class_Lp(g,M,k) c= a.e-eq-class_Lp(f,M,k);
   hence a.e-eq-class_Lp(f,M,k) = a.e-eq-class_Lp(g,M,k) by A3;
end;
