reserve V for non empty RealLinearSpace;

theorem Th1:
  for x be object holds
    x in the carrier of V*' iff x is linear-Functional of V
proof
  let x be object;
  consider X be non empty VectSp of F_Real such that
AS1:X = RLSp2RVSp V & V*'= RVSp2RLSp X*' by def2;
  x is linear-Functional of X iff x is linear-Functional of V
  proof
   hereby assume A21: x is linear-Functional of X;
    then
    reconsider f=x as Functional of V by AS1;
    reconsider g=x as linear-Functional of X by A21;
A1: f is additive
    proof
     let v,w be Element of V;
     reconsider v1=v,w1=w as Element of X by AS1;
     f.(v+w) = g.(v1+w1) by AS1
            .= g.v1 + g.w1 by VECTSP_1:def 20;
     hence f.(v+w) = f.v+f.w;
    end;
    f is homogeneous
    proof
     let v be VECTOR of V, r be Real;
     reconsider v1=v as Element of X by AS1;
     reconsider r1 =r as Scalar of X by XREAL_0:def 1;
     f.(r*v) = g.(r1*v1) by AS1
            .= r1*g.v1 by HAHNBAN1:def 8;
     hence f.(r*v) = r*f.v;
    end;
    hence x is linear-Functional of V by A1;
   end;
   assume
A21: x is linear-Functional of V; then
   reconsider f=x as Functional of X by AS1;
   reconsider g = x as linear-Functional of V by A21;
A1:f is additive
   proof
    let v,w be Element of X;
    reconsider v1=v,w1=w as VECTOR of V by AS1;
    f.(v+w) = g.(v1+w1) by AS1;
    hence f.(v+w) = f.v+f.w by HAHNBAN:def 2;
   end;
   f is homogeneous
   proof
    let v be Element of  X, r be Element of F_Real;
    reconsider v1=v as Element of V by AS1;
    reconsider r1=r as Element of REAL;
    f.(r*v) = g.(r1*v1) by AS1;
    hence f.(r*v) = r*f.v by HAHNBAN:def 3;
   end;
   hence x is linear-Functional of X by A1;
  end;
  hence thesis by AS1,HAHNBAN1:def 10;
end;
