
theorem Th7:
  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 st 1 <= i <= n holds
  Proj(i,n)*s is Point of
    R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1)
  & BoundedLinearOperatorsNorm(REAL-NS m,REAL-NS 1).(Proj(i,n)*s)
    <= ( BoundedLinearOperatorsNorm(REAL-NS n,REAL-NS 1).(Proj(i,n)))
         *(BoundedLinearOperatorsNorm(REAL-NS m,REAL-NS n).s)
proof
let m,n be non zero Element of NAT;
let s be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n);
let i be Nat;
deffunc BLONorm(Nat,Nat)
          = BoundedLinearOperatorsNorm(REAL-NS $1,REAL-NS $2);
assume 1 <=i & i <= n; then
A1: Proj(i,n) is Lipschitzian LinearOperator of REAL-NS n,REAL-NS 1 by Th6;
    s is Lipschitzian LinearOperator of REAL-NS m,REAL-NS n
      by LOPBAN_1:def 9; then
    Proj(i,n)*s is Lipschitzian LinearOperator of REAL-NS m,REAL-NS 1 &
    BLONorm(m,1).(Proj(i,n)*s) <= ( BLONorm(n,1).(Proj(i,n)))*(BLONorm(m,n).s)
      by A1,LOPBAN_2:2;
  hence thesis by LOPBAN_1:def 9;
end;
