
theorem LM6B:
  for X be RealUnitarySpace,
      f be linear-Functional of X,
      g be linear-Functional of RUSp2RNSp X
    st f=g holds
   f is Lipschitzian iff g is Lipschitzian
proof
  let X be RealUnitarySpace,
      f be linear-Functional of X,
      g be linear-Functional of RUSp2RNSp X;
  assume AS: f=g;
  set Y=RUSp2RNSp X;
  hereby assume f is Lipschitzian; then
    consider K be Real such that
A1:   0 < K &
      for x be Point of X holds |.f.x.| <= K * ||.x.||
        by BHSP_6:def 4;
    for y be Point of Y holds |.g.y.| <= K * ||.y.||
    proof
      let y be Point of Y;
      reconsider x=y as Point of X;
      ||.y.|| = ||.x.|| by Def1;
      hence |.g.y.| <= K * ||.y.|| by A1,AS;
    end;
    hence g is Lipschitzian by A1;
  end;
  assume g is Lipschitzian; then
  consider K be Real such that
A2: 0 <= K &
    for y be Point of Y holds |.g.y.| <= K * ||.y.||;
A4: K + 0 < K + 1 by XREAL_1:8;
  for x be Point of X holds |.f.x.| <= (K+1) * ||.x.||
  proof
    let x be Point of X;
    reconsider y=x as Point of Y;
    ||.x.|| = ||.y.|| by Def1; then
B3: |.f.x.| <= K * ||.x.|| by A2,AS;
    0 <= ||.x.|| by BHSP_1:28; then
    K * ||.x.|| <= (K+1) * ||.x.|| by A4,XREAL_1:64;
    hence |.f.x.| <= (K+1) * ||.x.|| by B3,XXREAL_0:2;
  end;
  hence f is Lipschitzian by A2,BHSP_6:def 4;
end;
