reserve x, y, z, r, s, t for Real;

theorem
  |.r.| = |.s.| implies r = s or r = -s
proof
  assume
A1: |.r.| = |.s.|;
  assume
A2: r <> s;
  per cases by Def1;
  suppose
    |.r.| = r & |.s.| = s;
    hence thesis by A1,A2;
  end;
  suppose
    |.r.| = r & |.s.| = -s;
    hence thesis by A1;
  end;
  suppose
    |.r.| = -r & |.s.| = s;
    hence thesis by A1;
  end;
  suppose
    |.r.| = -r & |.s.| = -s;
    hence thesis by A1,A2;
  end;
end;
