reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th35:
  for X, Y be RealNormSpace for f,g,h be Point of
R_NormSpace_of_BoundedLinearOperators(X,Y) holds h = f+g iff for x be VECTOR of
  X holds h.x = f.x + g.x
proof
  let X, Y be RealNormSpace;
  let f,g,h be Point of R_NormSpace_of_BoundedLinearOperators(X,Y);
  reconsider f1=f, g1=g, h1=h as VECTOR of
  R_VectorSpace_of_BoundedLinearOperators(X,Y);
  h=f+g iff h1=f1+g1;
  hence thesis by Th24;
end;
