
theorem Th35:
  for X be RealBanachSpace, Y be Subset of X holds
  ex Z be Subset of X
  st Z = the carrier of Lin(Y) & ClNLin(Y) = NLin(Cl(Z))
   & Cl(Z) is linearly-closed & Cl(Z) <> {}
  proof
    let X be RealBanachSpace, Y be Subset of X;
    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 NORMSP_3:def 20;
    A2: the carrier of Lin(Cl(Z)) = Cl(Z) by A1,NORMSP_3:13,31;
    A3: NLin(Cl(Z))= NORMSTR(# the carrier of Lin(Cl(Z)), 0.Lin(Cl(Z)),
                     the addF of Lin(Cl(Z)), the Mult of Lin(Cl(Z)),
                     Norm_(the carrier of Lin(Cl(Z)),X) #);
    A4: Zero_(Cl(Z),X) = 0.X by A1,NORMSP_3:13,RSSPACE:def 10
                      .= 0.Lin(Cl(Z)) by RLSUB_1:def 2;
    A5: Add_(Cl(Z),X) = (the addF of X) || (Cl(Z))
                         by A1,NORMSP_3:13,RSSPACE:def 8
                     .= the addF of Lin(Cl(Z)) by A2,RLSUB_1:def 2;
    A6: Mult_(Cl(Z),X) = (the Mult of X) | [:REAL,Cl(Z):]
                          by A1,NORMSP_3:13,RSSPACE:def 9
                      .= the Mult of Lin(Cl(Z)) by A2,RLSUB_1:def 2;
    Norm_(Cl(Z), X) = Norm_(the carrier of Lin(Cl(Z)),X) by A1,NORMSP_3:13,31;
    hence thesis by A1,A3,A4,A5,A6,NORMSP_3:13;
  end;
