 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;

theorem Th19:
  f=v & g=u implies f+g=v+u
proof
  reconsider v2=v,u2=u as VECTOR of CLSp_PFunct(X) by TARSKI:def 3;
  reconsider h = v2+u2 as Element of PFuncs(X,COMPLEX);
  reconsider v3= v2,u3=u2 as Element of PFuncs(X,COMPLEX);
A1: dom h= dom v3 /\ dom u3 by Th4;
  assume
A2: f=v & g=u;
  then for x be object st x in dom h holds h.x = f.x + g.x by Th4;
  then h= f+g by A2,A1,VALUED_1:def 1;
  hence thesis by ZFMISC_1:87,FUNCT_1:49;
end;
