
theorem
  for n be non zero Element of NAT,
      x be Element of REAL n,
      i be Element of NAT st 1 <= i & i <= n holds
   |. proj(i,n).x .| <= |. x .|
proof
let n be non zero Element of NAT;
let x be Element of REAL n;
let i be Element of NAT;
assume
A1:  1 <=i & i <= n;
reconsider y = x as Element of REAL-NS n by REAL_NS1:def 4;
A2: ||. Proj(i,n).y .|| = |. proj(i,n).y .| by Th4;
||. Proj(i,n).y .|| <= ||.y.|| by A1,Th3;
hence thesis by A2,REAL_NS1:1;
end;
