
theorem Th14:
  for m,n be non zero Element of NAT,
      s,t be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
      si,ti be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1),
      i be Nat st   si = Proj(i,n)*s & ti = Proj(i,n)*t & 1 <= i <= n
  holds ||. si - ti .|| <= ||. s - t .||
proof
let m,n be non zero Element of NAT,
    s,t be Point of R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
    si,ti be Point of
       R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1),
    i be Nat;
deffunc BLONorm(Element of NAT,Element of NAT)
    = BoundedLinearOperatorsNorm(REAL-NS $1,REAL-NS $2);
  assume
A1: si = Proj(i,n)*s & ti = Proj(i,n)*t & 1 <=i & i <= n;
  si - ti = Proj(i,n)*(s-t) by Lm1,A1;
  hence thesis by A1,Th13;
end;
