
theorem CLTh37:
  for X be RealNormSpace, Y be Subset of X,
  f,g be Point of ClNLin(Y), a be Real holds
    (||.f.|| = 0 iff f = 0.ClNLin(Y))
  & ||.a*f.|| = |.a.| * ||.f.||
  & ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    let X be RealNormSpace, Y be Subset of X;
    let f,g be Point of ClNLin(Y);
    let a be Real;
    consider Z be Subset of X such that
    A1: Z = the carrier of Lin(Y)
      & ClNLin(Y) = NORMSTR(# Cl(Z),
                              Zero_(Cl(Z), X),
                              Add_(Cl(Z), X),
                              Mult_(Cl(Z), X),
                              Norm_(Cl(Z),X) #) by defClN;
    reconsider CL = Cl(Z) as Subset of X;
    reconsider f1 = f, g1 = g as Point of X by A1,TARSKI:def 3;
    A3: f1+g1 = f+g by SUBTHCL;
    A4: a*f1 = a*f by SUBTHCL;
    A5: ||.f+g.|| = ||.f1+g1.|| by A3,SUBTHCL;
    A6: ||.a*f.|| = ||.a*f1.|| by A4,SUBTHCL;
    A7: ||.f.|| = ||.f1.|| by SUBTHCL;
    A8: ||.g.|| = ||.g1.|| by SUBTHCL;
    0.ClNLin(Y) = 0.X by A1,Cl01,RSSPACE:def 10;
    hence thesis by A5,A6,A7,A8,NORMSP_0:def 5,NORMSP_1:def 1;
  end;
