
theorem Th18:
  for m,n be non zero Element of NAT,
      s be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
      K be Real st
      for i be Element of NAT, si be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1)
      st si=Proj(i,n)*s & 1 <=i & i <= n holds ||. si .|| <= K
  holds ||. s .|| <= n*K
proof
deffunc RealNormSpaceOfBLO(non zero Nat, non zero Nat)
  = R_NormSpace_of_BoundedLinearOperators(REAL-NS $1,REAL-NS $2);
let m,n be non zero Element of NAT,
    s be Point of RealNormSpaceOfBLO(m,n), K be Real;
  assume
A1: for i be Element of NAT,
         si be Point of RealNormSpaceOfBLO(m,1)
       st si = Proj(i,n)*s & 1 <=i & i <= n holds ||. si .|| <= K;
  reconsider s0 = s as Lipschitzian LinearOperator of REAL-NS m,REAL-NS n
    by LOPBAN_1:def 9;
A2: now
    let x be Point of REAL-NS m;
    assume
A3: ||.x.|| <= 1;
    now let i be Element of NAT;
      assume A4: 1 <=i & i <= n;
      set si = Proj(i,n)*s;
      reconsider si as Point of RealNormSpaceOfBLO(m,1) by Th7,A4;
      reconsider sii = si as Lipschitzian LinearOperator of REAL-NS m,REAL-NS 1
        by LOPBAN_1:def 9;
A5:   ||. sii.x .|| <= ||. si .|| * ||.x.|| by LOPBAN_1:32;
A6: the carrier of REAL-NS m = REAL m by REAL_NS1:def 4;
A7: Proj(i,n) is Lipschitzian LinearOperator of REAL-NS n,REAL-NS 1
    by A4,Th6;
s is Lipschitzian LinearOperator of REAL-NS m,REAL-NS n
  by LOPBAN_1:def 9; then
Proj(i,n)*s is LinearOperator of REAL-NS m,REAL-NS 1
  by A7,LOPBAN_2:1; then
  dom (Proj(i,n)*s) = REAL m by A6,FUNCT_2:def 1; then
A8:   sii.x = Proj(i,n).(s.x) by A6,FUNCT_1:12;
A9:   0 <= ||. x .|| by NORMSP_1:4;
      ||. si .|| * ||.x.|| <= K * ||.x.|| by A4,A1,A9,XREAL_1:64;
      hence ||. Proj(i,n).(s.x) .|| <= K * ||.x.|| by A5,A8,XXREAL_0:2;
   end; then
A10: ||. (s.x) .|| <= n*(K * ||.x.||) by Th16;
A11: 1 <=1 & 1 <= n by NAT_1:14; then
 reconsider s1 =Proj(1,n)*s  as Point of RealNormSpaceOfBLO(m,1) by Th7;
 ||. s1 .|| <= K by A11,A1; then
A12: 0 <= K by NORMSP_1:4;
 (n*K)*||.x.|| <= (n*K)*1 by A12,A3,XREAL_1:64;
 hence ||. (s0.x) .|| <= (n*K) by A10,XXREAL_0:2;
 end;
     set PreNormS = PreNorms(modetrans(s,REAL-NS m,REAL-NS n));
A13: for y be ExtReal st
       y in PreNorms(modetrans(s,REAL-NS m,REAL-NS n)) holds y <= (n*K)
     proof
     let y be ExtReal;
     assume A14: y in PreNormS;
     consider x be VECTOR of REAL-NS m such that
A15: y = ||.modetrans(s,REAL-NS m,REAL-NS n).x.|| & ||. x .|| <= 1 by A14;
     y = ||. s0.x .|| by A15,LOPBAN_1:29;
     hence thesis by A2,A15;
     end;
A16: PreNormS is bounded_above
     proof
     n*K is UpperBound of PreNormS by A13,XXREAL_2:def 1;
     hence thesis by XXREAL_2:def 10;
     end;
  set UBPreNormS = upper_bound PreNormS;
  not UBPreNormS > n*K
  proof
  assume
A17: UBPreNormS > n*K;
  set dif = UBPreNormS - n*K;
A18: dif > 0 by A17,XREAL_1:50;
  consider w being Real such that
A19: w in PreNormS & UBPreNormS - dif < w by A18,A16,SEQ_4:def 1;
  thus contradiction by A19,A13;
  end; then
upper_bound PreNorms(s0) <= (n*K) by LOPBAN_1:def 11;
hence thesis by LOPBAN_1:30;
end;
