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 Th16:
  for X, Y be RealLinearSpace for f,g,h be VECTOR of
  R_VectorSpace_of_LinearOperators(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 RealLinearSpace;
  let f,g,h be VECTOR of R_VectorSpace_of_LinearOperators(X,Y);
  reconsider f9=f,g9=g,h9=h as LinearOperator of X,Y by Def6;
A1: R_VectorSpace_of_LinearOperators(X,Y) is Subspace of RealVectSpace(the
  carrier of X,Y) by Th14,RSSPACE:11;
  then reconsider f1=f as VECTOR of RealVectSpace(the carrier of X,Y) by
RLSUB_1:10;
  reconsider h1=h as VECTOR of RealVectSpace(the carrier of X,Y) by A1,
RLSUB_1:10;
  reconsider g1=g as VECTOR of RealVectSpace(the carrier of X,Y) by A1,
RLSUB_1:10;
A2: now
    assume
A3: h = f+g;
    let x be Element of X;
    h1=f1+g1 by A1,A3,RLSUB_1:13;
    hence h9.x=f9.x+g9.x by Th1;
  end;
  now
    assume for x be Element of X holds h9.x=f9.x+g9.x;
    then h1=f1+g1 by Th1;
    hence h =f+g by A1,RLSUB_1:13;
  end;
  hence thesis by A2;
end;
