
theorem
  for X be RealNormSpace, Y be SubRealNormSpace of X,
      L be Lipschitzian linear-Functional of X
  holds L | (the carrier of Y) is Lipschitzian linear-Functional of Y
  proof
    let X be RealNormSpace, Y be SubRealNormSpace of X,
        L be Lipschitzian linear-Functional of X;
    set Y1 = the carrier of Y;
    A1: the carrier of Y c= the carrier of X by DUALSP01:def 16; then
    reconsider L1 = L|Y1 as Functional of Y by FUNCT_2:32;
    A2: L1 is additive
    proof
      let x,y be Point of Y;
      reconsider x1 = x, y1 = y as Point of X by A1;
      thus L1.(x + y) = L.(x + y) by FUNCT_1:49
                     .= L.(x1 + y1) by NORMSP_3:28
                     .= L.x1 + L.y1 by HAHNBAN:def 2
                     .= L1.x + L.y by FUNCT_1:49
                     .= L1.x + L1.y by FUNCT_1:49;
    end;
    A3: L1 is homogeneous
    proof
      let x be Point of Y, r be Real;
      reconsider x1 = x as Point of X by A1;
      thus L1.(r * x) = L.(r * x) by FUNCT_1:49
                     .= L.(r * x1) by NORMSP_3:28
                     .= r * L.x1 by HAHNBAN:def 3
                     .= r * L1.x by FUNCT_1:49;
    end;
    consider K be Real such that
    A4: 0 <= K
      & for x be Point of X holds |. L.x .| <= K * ||.x.|| by DUALSP01:def 9;
    for x be Point of Y holds |. L1.x .| <= K * ||.x.||
    proof
      let x be Point of Y;
      reconsider x1 = x as Point of X by A1;
      |. L.x1 .| <= K * ||.x1.|| by A4; then
      |. L.x1 .| <= K * ||.x.|| by NORMSP_3:28;
      hence thesis by FUNCT_1:49;
    end; then
    L1 is Lipschitzian by A4;
    hence thesis by A2,A3;
  end;
