
theorem Th1:
for X,Y,Z be RealNormSpace, u be Point of [:X,Y,Z:], x be Point of X,
  y be Point of Y, z be Point of Z st u = [x,y,z] holds
    ||.u.|| <= ||.x.|| + ||.y.||+ ||.z.||
  & ||.x.|| <= ||.u.|| & ||.y.|| <= ||.u.|| & ||.z.|| <= ||.u.||
proof
    let X,Y,Z be RealNormSpace, u be Point of [:X,Y,Z:], x be Point of X,
    y be Point of Y, z be Point of Z;
    assume u = [x,y,z]; then
A1: ||.u.|| = sqrt(||.x.||^2 + ||.y.||^2+ ||.z.||^2) by PRVECT_4:9;
    ||.x.||^2 + ||.y.||^2 + ||.z.||^2 + 0
      <= ( ||.x.||^2 + ||.y.||^2 + ||.z.||^2 ) + ( 2*||.x.||) * ||.y.||
      by XREAL_1:7; then
    ||.x.||^2 + ||.y.||^2 + ||.z.||^2 + 0
      <= ( ||.x.||^2 + ||.y.||^2 + ||.z.||^2) + ( 2 * ||.x.||) * ||.y.||
     + ( 2 * ||.y.||) * ||.z.|| by XREAL_1:7; then
    ||.x.||^2 + ||.y.||^2 + ||.z.||^2 + 0
      <= ( ||.x.||^2 + ||.y.||^2 + ||.z.||^2) + ( 2 * ||.x.||) * ||.y.||
     + ( 2 * ||.y.||) * ||.z.||+ ( 2 * ||.x.||) * ||.z.||
      by XREAL_1:7; then
    sqrt(||.x.||^2 + ||.y.||^2 + ||.z.||^2)
     <= sqrt(( ||.x.|| + ||.y.|| + ||.z.|| ) ^2 ) by SQUARE_1:26;
    hence ||.u.|| <= ||.x.|| + ||.y.||+ ||.z.|| by A1,SQUARE_1:22;

    ||.x.||^2 <= ||.x.||^2 + ||.y.||^2
  & ||.x.||^2 + ||.y.||^2 <= ||.x.||^2 + ||.y.||^2 + ||.z.||^2
       by XREAL_1:31; then
    ||.x.||^2 <= ||.x.||^2 + ||.y.||^2 + ||.z.||^2 by XXREAL_0:2; then
    sqrt(||.x.||^2) <= sqrt(||.x.||^2 + ||.y.||^2 + ||.z.||^2) by SQUARE_1:26;
    hence ||.x.|| <= ||.u.|| by A1,SQUARE_1:22;

    ||.y.||^2 <= ||.x.||^2 + ||.y.||^2
  & ||.x.||^2 + ||.y.||^2 <= ||.x.||^2 + ||.y.||^2 + ||.z.||^2
       by XREAL_1:31; then
    ||.y.||^2 <= ||.x.||^2 + ||.y.||^2 + ||.z.||^2 by XXREAL_0:2; then
    sqrt(||.y.||^2) <= sqrt(||.x.||^2 + ||.y.||^2 + ||.z.||^2) by SQUARE_1:26;
    hence ||.y.|| <= ||.u.|| by A1,SQUARE_1:22;

    sqrt(||.z.||^2) <= sqrt(||.x.||^2 + ||.y.||^2 + ||.z.||^2)
      by SQUARE_1:26,XREAL_1:31;
    hence ||.z.|| <= ||.u.|| by A1,SQUARE_1:22;
end;
