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 Th25:
  f=v & g=u implies f+g=v+u
proof
  reconsider v2=v as VECTOR of RLSp_PFunct(X) by TARSKI:def 3;
  reconsider u2=u as VECTOR of RLSp_PFunct(X) by TARSKI:def 3;
  reconsider h = v2+u2 as Element of PFuncs(X,REAL);
  reconsider v3= v2 as Element of PFuncs(X,REAL);
  reconsider u3= u2 as Element of PFuncs(X,REAL);
A1: dom h= dom v3 /\ dom u3 by Th6;
  assume
A2: f=v & g=u;
  then for x be object st x in dom h holds h.x = f.x + g.x by Th6;
  then h= f+g by A2,A1,VALUED_1:def 1;
  hence thesis by Th4;
end;
