 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;

theorem
  f=v & g=u implies f+g=v+u
proof
  assume
A1: f=v & g=u;
  reconsider v2=v, u2=u as VECTOR of CLSp_L1Funct M by TARSKI:def 3;
  thus v+u=v2+u2 by ZFMISC_1:87,FUNCT_1:49
    .=f+g by A1,Th19;
end;
