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 Th3:
  (FuncAdd(X,Y)).(f,g) = (FuncAdd(X,Y)).(g,f)
proof
  now
    let x be Element of X;
    thus ((FuncAdd(X,Y)).(f,g)).x = g.x + f.x by Th1
      .= ((FuncAdd(X,Y)).(g,f)).x by Th1;
  end;
  hence thesis by FUNCT_2:63;
end;
