reserve a, b for Real;
reserve RNS for RealNormSpace;
reserve x, y, z, g, g1, g2 for Point of RNS;

theorem
  ||.a * x + b * y.|| <= |.a.| * ||.x.|| + |.b.| * ||.y.||
proof
  ||.a * x + b * y.|| <= ||.a * x.|| + ||.b * y.|| by Def1;
  then ||.a * x + b * y.|| <= |.a.| * ||.x.|| + ||.b * y.|| by Def1;
  hence thesis by Def1;
end;
