
theorem Th6:
  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) is Lipschitzian LinearOperator of REAL-NS n,REAL-NS 1 &
  BoundedLinearOperatorsNorm(REAL-NS n,REAL-NS 1).(Proj(i,n)) <= 1
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;
assume
A1: 1 <=i & i <= n;
A2: for y be Point of REAL-NS n, r be Real holds
    Proj(i,n).(r*y) = r*(Proj(i,n).y) by PDIFF_6:27;
for y1,y2 be Point of REAL-NS n holds
    Proj(i,n).(y1 + y2) = Proj(i,n).y1 + Proj(i,n).y2 by PDIFF_6:26;
  hence A3: Proj(i,n) is Lipschitzian LinearOperator of REAL-NS n,REAL-NS 1
    by A2,LOPBAN_1:def 5,VECTSP_1:def 20;
  deffunc BLONorm(Nat,Nat)
          = BoundedLinearOperatorsNorm(REAL-NS $1,REAL-NS $2);
A4: Proj(i,n) in BoundedLinearOperators(REAL-NS n,REAL-NS 1)
      by A3,LOPBAN_1:def 9;
set s0 = modetrans(Proj(i,n),REAL-NS n,REAL-NS 1);
   upper_bound PreNorms(s0) <= 1
proof
A5: for x be Real st x in PreNorms(s0) holds x <= 1
    proof
    let x be Real;
    assume x in PreNorms(s0); then
    consider y being VECTOR of REAL-NS n such that
A6:  x = ||. s0.y .|| & ||. y .|| <= 1;
A7:  ||. Proj(i,n).y .|| <= ||. y .|| by A1,Th3;
    Proj(i,n) = s0 by A3,LOPBAN_1:29;
    hence thesis by A6,A7,XXREAL_0:2;
    end;
A8: PreNorms(s0) is bounded_above
    proof
    for x being ExtReal st x in PreNorms(s0) holds x <= 1 by A5; then
    1 is UpperBound of PreNorms(s0) by XXREAL_2:def 1;
    hence thesis by XXREAL_2:def 10;
    end;
    assume
A9: upper_bound PreNorms(s0) > 1;
    set dif = upper_bound PreNorms(s0) - 1;
A10: dif > 0 by A9,XREAL_1:50;
    ex w being Real st
      w in PreNorms(s0) & upper_bound PreNorms(s0) - dif < w
      by A10,A8,SEQ_4:def 1;
    hence contradiction by A5;
   end;
  hence thesis by A4,LOPBAN_1:def 13;
end;
