reserve V for non empty RealLinearSpace;

theorem Th23:
for X be RealLinearSpace holds
 (the carrier of X) --> 0 is linear-Functional of X
proof
  let X be RealLinearSpace;
  set f=(the carrier of X) --> 0;
  reconsider f as Functional of X by FUNCOP_1:45,XREAL_0:def 1;
A1: f is additive
  proof
   let x,y be VECTOR of X;
   thus f.(x+y) = f.x+f.y;
  end;
  f is homogeneous
  proof
   let x be VECTOR of X, r be Real;
   thus f.(r*x) = r*f.x;
  end;
  hence thesis by A1;
end;
