
theorem
  for K be Real,
      n be non zero Element of NAT,
      s be Element of REAL n st
  for i be Element of NAT st 1 <= i & i <= n holds |. proj(i,n).s .| <= K
    holds |. s .| <= n*K
proof
let K be Real,
    n be non zero Element of NAT,
    s be Element of REAL n;
assume
A1: for i be Element of NAT st 1 <= i & i <= n
    holds |. proj(i,n).s .| <= K;
reconsider t = s as Element of REAL-NS n by REAL_NS1:def 4;
for i be Element of NAT st 1 <= i & i <= n holds ||. Proj(i,n).t .|| <= K
  proof
  let i be Element of NAT;
  assume 1 <=i & i <= n;
  then |. proj(i,n).t .| <= K by A1;
  hence ||. Proj(i,n).t .|| <= K by Th4;
  end; then
  ||. t .|| <= n*K by Th16;
  hence thesis by REAL_NS1:1;
end;
