
theorem Th22:
  for V be non empty VectSp of F_Complex for p be Semi-Norm of V
  holds p is Banach-Functional of RealVS(V)
proof
  let V be non empty VectSp of F_Complex;
  let p be Semi-Norm of V;
  reconsider p1=p as Functional of RealVS(V) by Th21;
A1: p1 is positively_homogeneous
  proof
    let x be VECTOR of RealVS(V);
    let r be Real;
    assume
A2: r > 0;
    the addLoopStr of V = the addLoopStr of RealVS(V) by Def17;
    then reconsider x1=x as Vector of V;
    r*x = [**r,0**]*x1 by Def17;
    hence p1.(r*x) = |.r.|*p1.x by Def14
      .= r*p1.x by A2,ABSVALUE:def 1;
  end;
  p1 is subadditive
  proof
    let x,y be VECTOR of RealVS(V);
A3: the addLoopStr of V = the addLoopStr of RealVS(V) by Def17;
    then reconsider x1=x, y1=y as Vector of V;
    x+y = x1+y1 by A3;
    hence thesis by Def11;
  end;
  hence thesis by A1;
end;
