
theorem Th15:
  for X,Y be RealNormSpace, x be Point of X, y be Point of Y holds
    ||.x.|| <= ||. [x,y] .|| & ||.y.|| <= ||. [x,y] .||
proof
   let X,Y be RealNormSpace, x be Point of X, y be Point of Y;
   consider w be Element of REAL 2 such that
   A1: w=<* ||.x.||,||.y.|| *> & ||.[x,y].|| = |.w.| by PRVECT_3:18;
   (proj (1,2)).w = w.1 by PDIFF_1:def 1
                 .= ||.x.|| by A1,FINSEQ_1:44;
   then |. ||.x.|| .| <= |.w.| by PDIFF_8:5;
   hence ||.x.|| <= ||.[x,y].|| by A1, ABSVALUE:def 1;
   (proj (2,2)).w = w.2 by PDIFF_1:def 1
                 .= ||.y.|| by A1,FINSEQ_1:44;
   then |. ||.y.|| .| <= |.w.| by PDIFF_8:5;
   hence ||.y.|| <= ||.[x,y].|| by A1, ABSVALUE:def 1;
end;
