let RNS be RealNormSpace; :: thesis: for x, y being Point of RNS holds |.(||.x.|| - ||.y.||).| <= ||.(x - y).||
let x, y be Point of RNS; :: thesis: |.(||.x.|| - ||.y.||).| <= ||.(x - y).||
(y - x) + x = y - (x - x) by RLVECT_1:29
.= y - H₁(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 |.(||.x.|| - ||.y.||).| <= ||.(x - y).|| by A1, ABSVALUE:5; :: thesis: verum