reserve V for non empty RealLinearSpace;

theorem Th17:
for X be RealLinearSpace holds LinearFunctionals X is linearly-closed
proof
  let X be RealLinearSpace;
  set W = LinearFunctionals X;
A1: for v,u be VECTOR of RealVectSpace(the carrier of X) st
     v in LinearFunctionals X & u in LinearFunctionals X
      holds v + u in LinearFunctionals X
  proof
    let v,u be VECTOR of RealVectSpace(the carrier of X) such that
A2: v in W & u in W;
    reconsider f=v+u as Functional of X by FUNCT_2:66;
A3: f is additive
    proof
     let x,y be VECTOR of X;
     reconsider vZ1=v, uZ1=u as Element of Funcs(the carrier of X,REAL);
A4:  uZ1 is linear-Functional of X by A2,Def7;
     reconsider uZ11=uZ1 as additive homogeneous Functional of X by A2,Def7;
     reconsider x1=x, y1=y as Element of X;
A5:  vZ1 is linear-Functional of X by A2,Def7;
     f.(x+y) = uZ1.(x+y)+vZ1.(x+y) by FUNCSDOM:1
            .= (uZ1.x+uZ1.y)+vZ1.(x+y) by A4,HAHNBAN:def 2
            .= uZ1.x+uZ1.y+(vZ1.x+vZ1.y) by A5,HAHNBAN:def 2
            .= uZ1.x+vZ1.x+uZ1.y+vZ1.y
            .= f.x+ uZ1.y+vZ1.y by FUNCSDOM:1
            .= f.x+ (uZ1.y+vZ1.y);
     hence f.(x+y) = f.x+f.y by FUNCSDOM:1;
    end;
    f is homogeneous
    proof
     let x be VECTOR of X, s be Real;
     reconsider v1 = v, u1 = u as Element of Funcs(the carrier of X,REAL);
A6:  u1 is linear-Functional of X & v1 is linear-Functional of X by A2,Def7;
     f.(s*x) = u1.(s*x)+v1.(s*x) by FUNCSDOM:1
            .= (s*(u1.x))+v1.(s*x) by A6,HAHNBAN:def 3
            .= s*u1.x+s*v1.x by A6,HAHNBAN:def 3
            .= s*(u1.x+v1.x);
     hence f.(s*x) = s*f.x by FUNCSDOM:1;
    end;
    hence v+u in W by A3,Def7;
  end;
  for a be Real, v be VECTOR of RealVectSpace(the carrier of X)
    st v in W holds a * v in W
  proof
    let a be Real;
    let v be VECTOR of RealVectSpace(the carrier of X) such that
A8: v in W;
    reconsider f=a*v as Functional of X by FUNCT_2:66;
A9: f is additive
    proof
     let x,y be VECTOR of X;
     reconsider vZ1=v as Element of Funcs(the carrier of X,REAL);
A10: vZ1 is linear-Functional of X by Def7,A8;
     f.(x+y) = a*(vZ1.(x+y)) by FUNCSDOM:4
            .= a*(vZ1.x+vZ1.y) by A10,HAHNBAN:def 2
            .= a*vZ1.x+a*vZ1.y
            .= f.x+a*vZ1.y by FUNCSDOM:4;
     hence f.(x+y) = f.x+ f.y by FUNCSDOM:4;
    end;
    f is homogeneous
    proof
     let x be VECTOR of X, s be Real;
     reconsider vZ1=v as Element of Funcs(the carrier of X,REAL);
A11: vZ1 is linear-Functional of X by Def7,A8;
     f.(s*x) = a*vZ1.(s*x) by FUNCSDOM:4
            .= a*(s*(vZ1.x)) by A11,HAHNBAN:def 3
            .= s*(a*vZ1.x);
     hence f.(s*x) = s*f.x by FUNCSDOM:4;
    end;
    hence thesis by A9,Def7;
  end;
  hence thesis by A1;
end;
