
theorem Th3:
  for X be RealNormSpace, Y,Z be SubRealNormSpace of X
  st ex A be Subset of X
     st A = the carrier of Y & Cl(A) = the carrier of Z
  holds
    for D0 be Subset of Y, D be Subset of Z st D0 is dense & D0 = D
    holds D is dense
  proof
    let X be RealNormSpace, Y,Z be SubRealNormSpace of X;
    given A be Subset of X such that
    A1: A = the carrier of Y & Cl(A) = the carrier of Z;
    let D0 be Subset of Y, D be Subset of Z;
    assume
    A2: D0 is dense & D0 = D;
    A3: the carrier of (LinearTopSpaceNorm Z) = the carrier of Z
        & the carrier of (LinearTopSpaceNorm Y) = the carrier of Y
        & the carrier of (LinearTopSpaceNorm X) = the carrier of X
        by NORMSP_2:def 4;
    A4: LinearTopSpaceNorm Z is SubSpace of LinearTopSpaceNorm X
      & LinearTopSpaceNorm Y is SubSpace of LinearTopSpaceNorm X by Th2;
    for S be Subset of Z st S <> {} & S is open holds D meets S
    proof
      let S be Subset of Z;
      assume
      A5: S <> {} & S is open;
      reconsider SZL = S as Subset of LinearTopSpaceNorm Z by NORMSP_2:def 4;
      reconsider SZT = SZL as Subset of TopSpaceNorm Z;
      SZT is open by A5,NORMSP_2:16; then
      SZL is open by NORMSP_2:20; then
      consider SXL be Subset of LinearTopSpaceNorm X such that
      A6: SXL in the topology of LinearTopSpaceNorm X
        & SZL = SXL /\ [#](LinearTopSpaceNorm Z) by A4,PRE_TOPC:def 4;
      reconsider SYL = SXL /\ [#](LinearTopSpaceNorm Y)
      as Subset of (LinearTopSpaceNorm Y);
      reconsider SX = SXL as Subset of X by NORMSP_2:def 4;
      reconsider SXT = SXL as Subset of TopSpaceNorm X by NORMSP_2:def 4;
      reconsider SY = SYL as Subset of Y by NORMSP_2:def 4;
      reconsider SYT = SYL as Subset of TopSpaceNorm Y by NORMSP_2:def 4;
      SXL is open by A6; then
      SXT is open by NORMSP_2:20; then
      A7: SX is open by NORMSP_2:16;
      SX /\ Cl(A) <> {} by A1,A5,A6,NORMSP_2:def 4; then
      consider v be object such that
      A8: v in SX /\ Cl(A) by XBOOLE_0:def 1;
      v in SX & v in Cl(A) by A8,XBOOLE_0:def 4; then
      SX meets A by A7,NORMSP_3:5; then
      A9: SYL <> {} by A1,NORMSP_2:def 4;
      SYL is open by A4,A6,PRE_TOPC:def 4; then
      SYT is open by NORMSP_2:20; then
      SY is open by NORMSP_2:16; then
      A10: D0 meets SY by A2,A9,NORMSP_3:17;
      SYL c= SZL by A1,A3,A6,NORMSP_3:4,XBOOLE_1:26; then
      D /\ SYL c= D /\ SZL by XBOOLE_1:26;
      hence thesis by A2,A10;
    end;
    hence thesis by NORMSP_3:17;
  end;
