
theorem Th15:
  for K be Real, n be Nat,
      s be Element of REAL n st
      for i be Element of NAT st 1 <= i & i <= n holds |.s.i.| <= K
  holds |.s.| <= n*K
proof
let K be Real;
defpred P[Nat] means for s be Element of REAL $1 st
   for i be Element of NAT st 1 <= i & i <= $1
      holds |.s.i.| <= K holds |.s.| <= $1*K;
A1: P[0]
proof
 let s be Element of REAL 0;
 s = 0*0;
 hence thesis by EUCLID:7;
end;
A2: for n be Nat st P[n] holds P[n+1]
proof
  let n be Nat;
  assume A3: P[n];
  let s be Element of REAL (n+1);
  assume A4:for i be Element of NAT st 1 <=i & i <= n+1 holds |.s.i.| <= K;
  set sn = s | n;
  len s = n+1 by CARD_1:def 7; then
  len sn = n by FINSEQ_3:53;
  then reconsider sn as Element of REAL n by FINSEQ_2:92;
A5:now let i be Element of NAT;
      assume A6:1 <=i & i <= n;
      n <= n+1 by NAT_1:11; then
      1 <=i & i <= n+1 by A6,XXREAL_0:2; then
      |.s.i.| <= K by A4;
      hence |.sn.i.| <= K by A6,FINSEQ_3:112;
   end;
A7:  n+1 in NAT by ORDINAL1:def 12;
   1 <= (n+1) & (n+1) <= (n+1) by NAT_1:11; then
A8:  |.s.(n+1).| <= K by A4,A7;
A9:  |.s.|^2 = |.sn.|^2 + (s. (n+1))^2 by REAL_NS1:10;
A10:   K >= 0 by A8,COMPLEX1:46;
A11: |.sn.|^2 <= (n*K)^2 by A3,A5,SQUARE_1:15;
A12: (s.(n+1))^2 <= K^2 by A8,SERIES_3:24;
A13:  ((n*K )^2 + K^2) + 2*(n*K)*K >= ((n*K )^2 + K^2) by A10,XREAL_1:38;
  |.sn.|^2 + (s.(n+1))^2 <= (n*K )^2 + K^2 by A11,A12,XREAL_1:7; then
A14:  |.s.|^2 <= ((n+1)*K )^2 by A9,A13,XXREAL_0:2;
A15:  sqrt(((n+1)*K )^2) = |.(n+1)*K .| by COMPLEX1:72;
  sqrt(|.s.|^2 ) <= sqrt(((n+1)*K )^2) by A14,SQUARE_1:26; then
  |.|.|.s.|.|.| <= sqrt(((n+1)*K )^2) by COMPLEX1:72; then
  |.s.| <= sqrt(((n+1)*K )^2) by ABSVALUE:def 1;
hence thesis by A10,A15,ABSVALUE:def 1;
end;
 thus for n being Nat holds P[n] from NAT_1:sch 2(A1,A2);
end;
