 reserve X,Y for set,
         n,m,k,i for Nat,
         r for Real,
         R for Element of F_Real,
         K for Field,
         f,f1,f2,g1,g2 for FinSequence,
         rf,rf1,rf2 for real-valued FinSequence,
         cf,cf1,cf2 for complex-valued FinSequence,
         F for Function;
reserve f,f1,f2 for n-element real-valued FinSequence,
        p,p1,p2 for Point of TOP-REAL n,
        M,M1,M2 for Matrix of n,m,F_Real,
        A,B for Matrix of n,F_Real;

theorem Th45:
  ex L be Real st L>0 & for f holds |.(Mx2Tran M).f.| <= L*|.f.|
proof
  consider L be m-element FinSequence of REAL such that
    A1: for i st i in dom L holds L.i=|.@Col(M,i).| and
    A2: for f be n-element real-valued FinSequence
    holds|.(Mx2Tran M).f.|<=Sum L*|.f.| by Th44;
   reconsider S1 = 1+Sum L as Real;
  take S1;
  now let i;
   assume i in dom L;
   then L.i=|.@Col(M,i).| by A1;
   hence 0<=L.i;
  end;
  then Sum L>=0 by RVSUM_1:84;
  hence S1>0;
  let f;
  Sum L<=S1 by XREAL_1:29;
  then A3: Sum L*|.f.|<=S1*|.f.| by XREAL_1:64;
  |.(Mx2Tran M).f.|<=Sum L*|.f.| by A2;
  hence thesis by A3,XXREAL_0:2;
end;
