
theorem RSSPACE11:
  for X be RealNormSpace, V1 be Subset of X, CL be Subset of X
  st CL = the carrier of ClNLin(V1)
  holds RLSStruct(# CL,Zero_(CL,X), Add_(CL,X),Mult_(CL,X) #) is Subspace of X
  proof
    let X be RealNormSpace, V1 be Subset of X, CL be Subset of X;
    assume
    A1: CL = the carrier of ClNLin(V1);
    consider Z be Subset of X such that
    A2: Z = the carrier of Lin(V1)
      & ClNLin(V1) = NORMSTR(# Cl(Z),
                               Zero_(Cl(Z), X),
                               Add_(Cl(Z), X),
                               Mult_(Cl(Z), X),
                               Norm_(Cl(Z),X) #) by defClN;
    thus thesis by A1,RSSPACE:11,A2,Cl01;
  end;
