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;
