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;
reserve v,u for VECTOR of RLSp_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 RLSp_L1Funct M by TARSKI:def 3;
  thus v+u=v2+u2 by Th4
    .=f+g by A1,Th25;
end;
