
theorem Th13:
  for m,n be non zero Element of NAT,
      s be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
      i be Nat,
      si be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1)
      st si = Proj(i,n)*s & 1 <= i <= n
  holds ||. si .|| <= ||. s .||
proof
let m,n be non zero Element of NAT,
    s be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
    i be Nat,
    si be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1);
assume
A1: si = Proj(i,n)*s & 1 <=i & i <= n;
deffunc BLONorm(Nat,Nat)
    = BoundedLinearOperatorsNorm(REAL-NS $1,REAL-NS $2);
 A2: Proj(i,n) is Lipschitzian LinearOperator of REAL-NS n,REAL-NS 1
     & BLONorm(n,1).(Proj(i,n)) <=1 by Th6,A1;
s is Lipschitzian LinearOperator of REAL-NS m,REAL-NS n
  by LOPBAN_1:def 9; then
A3: ||. si .|| <= (BLONorm(n,1).(Proj(i,n))) * ||. s .|| by A2,A1,LOPBAN_2:2;
0 <= ||. s .|| by NORMSP_1:4; then
(BLONorm(n,1).(Proj(i,n))) * ||. s .|| <= 1* ||. s .|| by Th6,A1,XREAL_1:64;
hence thesis by A3,XXREAL_0:2;
end;
