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

theorem Th9:
  |.||.x.|| - ||.y.||.| <= ||.x - y.||
proof
  (y - x) + x = y - (x - x) by RLVECT_1:29
    .= y - 09(RNS) by RLVECT_1:15
    .= y;
  then ||.y.|| <= ||.y - x.|| + ||.x.|| by Def1;
  then ||.y.|| - ||.x.|| <= ||.y - x.|| by XREAL_1:20;
  then ||.y.|| - ||.x.|| <= ||.x - y.|| by Th7;
  then
A1: -(||.y.|| - ||.x.||) >= -||.x - y.|| by XREAL_1:24;
  ||.x.|| - ||.y.|| <= ||.x - y.|| by Th8;
  hence thesis by A1,ABSVALUE:5;
end;
