theorem Th44:
  ex L be m-element FinSequence of REAL st
     (for i st i in dom L holds L.i=|.@Col(M,i).|) &
      for f holds|.(Mx2Tran M).f.|<=Sum L*|.f.|
proof
  set F=the n-element real-valued FinSequence;
  consider L be m-element FinSequence of REAL such that
   |.(Mx2Tran M).F.|<=Sum L*|.F.| and
   A1: for i be Nat st i in dom L holds L.i=|.@Col(M,i).| by Lm4;
  take L;
  thus for i st i in dom L holds L.i=|.@Col(M,i).| by A1;
  let f be n-element real-valued FinSequence;
  consider L1 be m-element FinSequence of REAL such that
   A2: |.(Mx2Tran M).f.|<=Sum L1*|.f.| and
   A3: for i be Nat st i in dom L1 holds L1.i=|.@Col(M,i).| by Lm4;
  len L1=m & len L=m by CARD_1:def 7;
  then A4: dom L=dom L1 by FINSEQ_3:29;
  now let i be Nat;
   assume A5: i in dom L;
   hence L.i=|.@Col(M,i).| by A1
    .=L1.i by A3,A4,A5;
  end;
  hence thesis by A2,A4,FINSEQ_1:13;
end;
