reserve a,b,r for Real;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,REAL);
reserve u,v,w for VECTOR of RLSp_PFunctA;
reserve X for non empty set,
  x for Element of X,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E,E1,E2 for Element of S,
  f,g,h,f1,g1 for PartFunc of X ,REAL;
reserve v,u for VECTOR of RLSp_L1Funct M;
reserve v,u for VECTOR of RLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-Space M;
reserve x,y for Point of L-1-Space M;

theorem Th51:
  (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;
  hereby
    reconsider x1=x,y1=y as Point of Pre-L-Space M;
    assume that
A1: f in x and
A2: g in y;
    y1 in the carrier of Pre-L-Space M;
    then
A3: y1 in C by Def18;
    then consider b be PartFunc of X,REAL such that
A4: y1=a.e-eq-class(b,M) and
A5: b in L1_Functions M;
A6: b in y1 by A4,A5,Th38;
    ex r be PartFunc of X,REAL st g=r & r in L1_Functions M & b in
    L1_Functions M & b a.e.= r,M by A2,A4;
    then
A7: a.e-eq-class(b,M) = a.e-eq-class(g,M) by Th39;
    x1 in the carrier of Pre-L-Space M;
    then
A8: x1 in C by Def18;
    then consider a be PartFunc of X,REAL such that
A9: x1=a.e-eq-class(a,M) and
A10: a in L1_Functions M;
    a in x1 by A9,A10,Th38;
    then (addCoset M).(x1,y1) = a.e-eq-class(a+b,M) by A8,A3,A6,Def15;
    then
A11: x1+y1 = a.e-eq-class(a+b,M) by Def18;
    ex r be PartFunc of X,REAL st f=r & r in L1_Functions M & a in
    L1_Functions M & a a.e.= r,M by A1,A9;
    then a.e-eq-class(a,M) = a.e-eq-class(f,M) by Th39;
    then
    a.e-eq-class(a+b,M) = a.e-eq-class(f+g,M) by A1,A2,A9,A10,A4,A5,A7,Th41;
    hence f+g in x+y by A1,A2,A9,A4,A11,Th23,Th38;
  end;
  hereby
    reconsider x1=x as Point of Pre-L-Space M;
    x1 in the carrier of Pre-L-Space M;
    then
A12: x1 in C by Def18;
    then consider f1 be PartFunc of X,REAL such that
A13: x1=a.e-eq-class(f1,M) and
A14: f1 in L1_Functions M;
    f1 in x1 by A13,A14,Th38;
    then (lmultCoset M).(a,x1) = a.e-eq-class(a(#)f1,M) by A12,Def17;
    then
A15: a*x1 = a.e-eq-class(a(#)f1,M) by Def18;
    assume
A16: f in x;
    then ex r be PartFunc of X,REAL st f=r & r in L1_Functions M & f1 in
    L1_Functions M & f1 a.e.= r,M by A13;
    then a.e-eq-class(f1,M) = a.e-eq-class(f,M) by Th39;
    then a.e-eq-class(a(#)f1,M) = a.e-eq-class(a(#)f,M) by A16,A13,A14,Th42;
    hence a(#)f in a*x by A16,A13,A15,Th24,Th38;
  end;
end;
