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 Th54:
(f in x & g in y implies f+g in x+y) & (f in x implies a(#)f in a*x)
proof
   set C = CosetSet(M,k);
   hereby assume A1: f in x & g in y;
    x in the carrier of Pre-Lp-Space(M,k); then
A2: x in C by Def11; then
    consider a be PartFunc of X,REAL such that
A3:  x=a.e-eq-class_Lp(a,M,k) & a in Lp_Functions(M,k);
A4: a in x by A3,Th38;
    y in the carrier of Pre-Lp-Space(M,k); then
A5: y in C by Def11; then
    consider b be PartFunc of X,REAL such that
A6:  y=a.e-eq-class_Lp(b,M,k) & b in Lp_Functions(M,k);
    b in y by A6,Th38; then
    (addCoset(M,k)).(x,y) = a.e-eq-class_Lp(a+b,M,k) by A2,A5,A4,Def8;
    then
A7: x+y = a.e-eq-class_Lp(a+b,M,k) by Def11;
    ex r be PartFunc of X,REAL st f=r & r in Lp_Functions(M,k) &
      a a.e.= r,M by A1,A3; then
A8:a.e-eq-class_Lp(a,M,k) = a.e-eq-class_Lp(f,M,k) by Th42;
    ex r be PartFunc of X,REAL st g=r & r in Lp_Functions (M,k) &
      b a.e.= r,M by A1,A6; then
    a.e-eq-class_Lp(b,M,k) = a.e-eq-class_Lp(g,M,k) by Th42; then
    a.e-eq-class_Lp(a+b,M,k) = a.e-eq-class_Lp(f+g,M,k) by A1,A3,A6,A8,Th45;
    hence f+g in x+y by Th38,A7,Th25,A3,A1,A6;
   end;
   hereby assume A9:f in x;
    x in the carrier of Pre-Lp-Space (M,k); then
A10:x in C by Def11; then
    consider f1 be PartFunc of X,REAL such that
A11: x=a.e-eq-class_Lp(f1,M,k) & f1 in Lp_Functions (M,k);
    f1 in x by A11,Th38; then
    (lmultCoset(M,k)).(a,x) = a.e-eq-class_Lp(a(#)f1,M,k) by A10,Def10;
    then
A12:a*x = a.e-eq-class_Lp(a(#)f1,M,k) by Def11;
    ex r be PartFunc of X,REAL st f=r & r in Lp_Functions (M,k) &
      f1 a.e.= r,M by A9,A11; then
    a.e-eq-class_Lp(f1,M,k) = a.e-eq-class_Lp(f,M,k) by Th42; then
    a.e-eq-class_Lp(a(#)f1,M,k) = a.e-eq-class_Lp(a(#)f,M,k) by A11,A9,Th47;
    hence a(#)f in a*x by A12,Th26,A9,A11,Th38;
   end;
end;
