
theorem LM4:
  for X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL
  st f=g holds
   f is_Lipschitzian_on the carrier of X
 iff
   g is_Lipschitzian_on the carrier of RUSp2RNSp X
proof
  let X be RealUnitarySpace,
      f be Function of X,REAL, g be Function of RUSp2RNSp X,REAL;
  assume AS: f=g;
  set Y=RUSp2RNSp X;
  thus f is_Lipschitzian_on the carrier of X implies
    g is_Lipschitzian_on the carrier of RUSp2RNSp X
  proof
    assume A11: f is_Lipschitzian_on the carrier of X; then
    consider r be Real such that
A2:   0 < r &
      for x1, x2 be Point of X
        st x1 in the carrier of X & x2 in the carrier of X
       holds |.f/.x1 - f/.x2.| <= r * ||.x1 - x2.||;
    for y1, y2 be Point of Y
      st y1 in the carrier of Y & y2 in the carrier of Y
     holds |.g/.y1 - g/.y2.| <= r * ||.y1 - y2.||
    proof
      let y1, y2 be Point of Y;
      assume y1 in the carrier of Y & y2 in the carrier of Y;
      reconsider x1=y1,x2=y2 as Point of X;
      ||.y1 - y2.|| = ||.x1 - x2.|| by RHS4,RHS6;
      hence |.g/.y1 - g/.y2.| <= r * ||.y1 - y2.|| by A2,AS;
    end;
    hence g is_Lipschitzian_on the carrier of Y by A2,A11,AS;
  end;
    assume A11: g is_Lipschitzian_on the carrier of Y; then
    consider r be Real such that
A2:   0 < r &
      for y1, y2 be Point of Y
        st y1 in the carrier of Y & y2 in the carrier of Y
       holds |.g/.y1 - g/.y2.| <= r * ||.y1 - y2.||;
    for x1, x2 be Point of X
      st x1 in the carrier of X & x2 in the carrier of X
     holds |.f/.x1 - f/.x2.| <= r * ||.x1 - x2.||
    proof
      let x1, x2 be Point of X;
      assume x1 in the carrier of X & x2 in the carrier of X;
      reconsider y1=x1,y2=x2 as Point of Y;
      ||.x1 - x2.|| = ||.y1 - y2.|| by RHS4,RHS6;
      hence |.f/.x1 - f/.x2.| <= r * ||.x1 - x2.|| by A2,AS;
    end;
    hence thesis by A2,A11,AS;
end;
