
theorem LM8:
  for X be RealUnitarySpace,
      u be linear-Functional of X,
      v be linear-Functional of (RUSp2RNSp X)
  st u=v holds
  PreNorms u = PreNorms v
proof
  let X be RealUnitarySpace,
      u be linear-Functional of X,
      v be linear-Functional of (RUSp2RNSp X);
  assume AS: u=v;
  set Y = RUSp2RNSp X;
A11: now let x be object;
    assume AS1: x in PreNorms u; then
    reconsider y=x as Real;
    consider t be VECTOR of X such that
B1:   y = |.u.t.| & ||.t.|| <= 1 by AS1;
    reconsider t1=t as VECTOR of Y;
    ||.t1.|| <= 1 by B1,Def1;
    hence x in PreNorms v by AS,B1;
  end;
  now let x be object;
    assume AS2: x in PreNorms v; then
    reconsider y=x as Real;
    consider t be VECTOR of Y such that
B1:   y = |.v.t.| & ||.t.|| <= 1 by AS2;
    reconsider t1=t as VECTOR of X;
    ||.t1.|| <= 1 by B1,Def1;
    hence x in PreNorms u by AS,B1;
  end;
  hence PreNorms u = PreNorms v by A11;
end;
