
theorem Th19:
  for m,n be non zero Element of NAT,
      s,t be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS n),
      K be Real st
      for i be Element of NAT, si,ti be Point of
      R_NormSpace_of_BoundedLinearOperators(REAL-NS m,REAL-NS 1)
      st si = Proj(i,n)*s & ti = Proj(i,n)*t &  1 <=i & i <= n
      holds ||. si - ti .|| <= K
  holds ||. s-t .|| <= 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,t be Point of RealNormSpaceOfBLO(m,n), K be Real;
  assume
A1: for i be Element of NAT,
        si,ti be Point of RealNormSpaceOfBLO(m,1)
      st si = Proj(i,n)*s & ti = Proj(i,n)*t &  1 <=i & i <= n
    holds ||. si - ti .|| <= K;
now
  let i be Element of NAT, sti be Point of RealNormSpaceOfBLO(m,1);
  assume
A2: sti = Proj(i,n)*(s-t) & 1 <=i & i <= n;
  reconsider si = Proj(i,n)*s,
    ti = Proj(i,n)*t as Point of RealNormSpaceOfBLO(m,1) by Th7,A2;
   si - ti = sti by A2,Lm1;
   hence ||. sti .|| <= K by A2,A1;
end;
hence thesis by Th18;
end;
