
theorem
for H being RealUnitarySpace,
    x be FinSequence of H
  st x is one-to-one
       & rng x is linearly-independent
       & 1 <= len x holds
rng GramSchmidt (x) is linearly-independent
proof
  let H be RealUnitarySpace,
      x be FinSequence of H;
  assume x is one-to-one
       & rng x is linearly-independent
       & 1 <= len x; then
  len x = len GramSchmidt (x) &
    rng GramSchmidt (x) is OrthonormalFamily of H &
    GramSchmidt (x) is one-to-one by Def2;
  hence thesis by Th8;
end;
