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);

theorem Th29:
f = v & g = u implies f+g = v+u
proof
   reconsider v2=v, u2=u as VECTOR of RLSp_PFunct(X) by TARSKI:def 3;
   reconsider h = v2+u2 as Element of PFuncs(X,REAL);
   reconsider v2,u2 as Element of PFuncs(X,REAL);
   assume A1:f=v & g=u;
A2:dom h= dom v2 /\ dom u2 &
   for x being Element of X st x in dom h holds h.x = v2.x + u2.x
      by LPSPACE1:6;
   for x be object st x in dom h holds h.x = f.x + g.x by A1,LPSPACE1:6; then
   h= f+g by A1,A2,VALUED_1:def 1;
   hence thesis by LPSPACE1:4;
end;
