reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  |.||.x.|| - ||.y.||.| <= ||.x - y.||
proof
  (y - x) + x = y - (x - x) by RLVECT_1:29
    .= y - 09(X) by RLVECT_1:15
    .= y by RLVECT_1:13;
  then ||.y.|| <= ||.y - x.|| + ||.x.|| by Th41;
  then ||.y.|| - ||.x.|| <= ||.y - x.|| by XREAL_1:20;
  then ||.y.|| - ||.x.|| <= ||.-(x - y).|| by RLVECT_1:33;
  then ||.y.|| - ||.x.|| <= ||.x - y.|| by Th42;
  then
A1: -(||.x - y.||) <= -(||.y.|| - ||.x.||) by XREAL_1:24;
  ||.x.|| - ||.y.|| <= ||.x - y.|| by Th43;
  hence thesis by A1,ABSVALUE:5;
end;
