 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 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;
