
theorem Th16:
  for K be Real,
      n be non zero Element of NAT,
      s be Element of REAL-NS 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-NS n;
assume
A1: for i be Element of NAT st 1 <=i & i <= n
    holds ||. Proj(i,n).s .|| <= K;
consider m be Nat such that
A2: n = m + 1 by NAT_1:6;
reconsider m as Element of NAT by ORDINAL1:def 12;
reconsider s0 = s as Element of REAL n by REAL_NS1:def 4;
now
  let i be Element of NAT;
  assume 1 <=i & i <= m+1; then
A3: ||. Proj(i,n).s .|| <= K by A2,A1;
  Proj(i,n).s = <* proj(i,n).s0 *> by PDIFF_1:def 4
               .= <* s0.i *> by PDIFF_1:def 1;
  hence |.s0.i.| <= K by A3,Th2;
end; then
|. s0 .| <= n*K by A2,Th15;
hence thesis by REAL_NS1:1;
end;
