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;

theorem Th33:
  (X-->0) + (X-->0) = X-->0 & a(#)(X-->0) = X-->0
proof
  set g1 = X-->0;
  set g2 = X-->0;
A1: now
    let x be Element of X;
    assume x in dom(a(#)g1);
    then (a(#)g1).x = a * g1.x by VALUED_1:def 5;
    then (a(#)g1).x = a * 0 by FUNCOP_1:7;
    hence (a(#)g1).x = (X --> 0).x by FUNCOP_1:7;
  end;
A2: dom(g1+g2) = dom g1 /\ dom g2 by VALUED_1:def 1;
  now
    let x be Element of X;
    assume x in dom(X --> 0);
    then (g1+g2).x = g1.x + g2.x by A2,VALUED_1:def 1;
    then (g1+g2).x = 0 + g2.x by FUNCOP_1:7;
    hence (g1+g2).x = (X --> 0).x;
  end;
  hence g1+g2 = X-->0 by A2,PARTFUN1:5;
  dom(a(#)g1) = dom(X-->0) by VALUED_1:def 5;
  hence thesis by A1,PARTFUN1:5;
end;
