 reserve a,b,r for Complex;
 reserve V for ComplexLinearSpace;
reserve A,B for non empty set;
reserve f,g,h for Element of PFuncs(A,COMPLEX);
reserve u,v,w for VECTOR of CLSp_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,A,B for Element of S,
  f,g,h,f1,g1 for PartFunc of X,COMPLEX;
reserve v,u for VECTOR of CLSp_L1Funct M;
reserve v,u for VECTOR of CLSp_AlmostZeroFunct M;
reserve x for Point of Pre-L-CSpace M;
reserve x,y for Point of L-1-CSpace M;

theorem Th45:
  (f in x & g in y implies f+g in x+y) & (f in x implies a(#)f in a*x)
proof
  set C = CCosetSet M;
  hereby
    reconsider x1=x,y1=y as Point of Pre-L-CSpace M;
    assume that
A1: f in x and
A2: g in y;
    y1 in the carrier of Pre-L-CSpace M;
    then
A3: y1 in C by Def19;
    then consider b be PartFunc of X,COMPLEX such that
A4: y1=a.e-Ceq-class(b,M) and
A5: b in L1_CFunctions M;
A6: b in y1 by A4,A5,Th31;
    ex r be PartFunc of X,COMPLEX st g=r & r in L1_CFunctions M & b in
    L1_CFunctions M & b a.e.cpfunc= r,M by A2,A4;
    then
A7: a.e-Ceq-class(b,M) = a.e-Ceq-class(g,M) by Th32;
    x1 in the carrier of Pre-L-CSpace M;
    then
A8: x1 in C by Def19;
    then consider a be PartFunc of X,COMPLEX such that
A9: x1=a.e-Ceq-class(a,M) and
A10: a in L1_CFunctions M;
    a in x1 by A9,A10,Th31;
    then (addCCoset M).(x1,y1) = a.e-Ceq-class(a+b,M) by A8,A3,A6,Def16;
    then
A11: x1+y1 = a.e-Ceq-class(a+b,M) by Def19;
    ex r be PartFunc of X,COMPLEX st f=r & r in L1_CFunctions M & a in
    L1_CFunctions M & a a.e.cpfunc= r,M by A1,A9;
    then a.e-Ceq-class(a,M) = a.e-Ceq-class(f,M) by Th32;
    then
    a.e-Ceq-class(a+b,M) = a.e-Ceq-class(f+g,M) by A1,A2,A9,A10,A4,A5,A7,Th34;
    hence f+g in x+y by A1,A2,A9,A4,A11,Th17,Th31;
  end;
  hereby
    reconsider x1=x as Point of Pre-L-CSpace M;
    x1 in the carrier of Pre-L-CSpace M;
    then
A12: x1 in C by Def19;
    then consider f1 be PartFunc of X,COMPLEX such that
A13: x1=a.e-Ceq-class(f1,M) and
A14: f1 in L1_CFunctions M;
    (lmultCCoset M).(a,x1) = a.e-Ceq-class(a(#)f1,M) by A13,A14,Th31,A12,Def18;
    then
A15: a*x1 = a.e-Ceq-class(a(#)f1,M) by Def19;
    assume
A16: f in x;
    then ex r be PartFunc of X,COMPLEX st f=r & r in L1_CFunctions M & f1 in
    L1_CFunctions M & f1 a.e.cpfunc= r,M by A13;
    then a.e-Ceq-class(f1,M) = a.e-Ceq-class(f,M) by Th32;
    then a.e-Ceq-class(a(#)f1,M) = a.e-Ceq-class(a(#)f,M) by A16,A13,A14,
    Th35;
    hence a(#)f in a*x by A16,A13,A15,Th18,Th31;
  end;
end;
