
theorem Th14:
  for K be add-associative right_zeroed right_complementable non
empty addLoopStr for V be non empty ModuleStr over K for f be Functional of V
  holds f-f = 0Functional(V)
proof
  let K be add-associative right_zeroed right_complementable non empty
  addLoopStr;
  let V be non empty ModuleStr over K;
  let f be Functional of V;
  now
    let x be Element of V;
    thus (f-f).x = f.x+(-f).x by Def3
      .= f.x+-f.x by Def4
      .= 0.K by RLVECT_1:5
      .= (0Functional(V)).x by FUNCOP_1:7;
  end;
  hence thesis by FUNCT_2:63;
end;
