let X be RealUnitarySpace; :: thesis: for x, y being Point of X holds |.(||.x.|| - ||.y.||).| <= ||.(x - y).||
let x, y be Point of X; :: thesis: |.(||.x.|| - ||.y.||).| <= ||.(x - y).||
(y - x) + x = y - (x - x) by RLVECT_1:29
.= y - H₁(X) by RLVECT_1:15
.= y by RLVECT_1:13 ;
then ||.y.|| <= ||.(y - x).|| + ||.x.|| by Th30;
then ||.y.|| - ||.x.|| <= ||.(y - x).|| by XREAL_1:20;
then ||.y.|| - ||.x.|| <= ||.(- (x - y)).|| by RLVECT_1:33;
then ||.y.|| - ||.x.|| <= ||.(x - y).|| by Th31;
then A1: - ||.(x - y).|| <= - (||.y.|| - ||.x.||) by XREAL_1:24;
||.x.|| - ||.y.|| <= ||.(x - y).|| by Th32;
hence |.(||.x.|| - ||.y.||).| <= ||.(x - y).|| by A1, ABSVALUE:5; :: thesis: verum