reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem
  for B be OrdBasis of Lin lines M st B = M
  for Mf be Element of Lin lines M st Mf = (Mx2Tran M).f holds Mf|--B = f
proof
  set LM=lines M;
  let B be OrdBasis of Lin LM such that
   A1: B=M;
  A2: B is one-to-one by MATRLIN:def 2;
  let Mf be Element of Lin LM such that
   A3: Mf=(Mx2Tran M).f;
  consider L be Linear_Combination of LM such that
   A4: Sum L=Mf and
   A5: for i be Nat st i in dom f holds L.Line(M,i)=f.i by A1,A3,A2,Th16;
  reconsider L1=L|the carrier of Lin LM as Linear_Combination of Lin LM by Th8;
  A6: len M=n by MATRIX_0:def 2;
  A7: len f=n by CARD_1:def 7;
  A8: LM c=[#]Lin LM by Lm4;
  A9: now let k;
   assume A10: 1<=k & k<=n;
   then k in Seg n;
   then A11: M.k=Line(M,k) by MATRIX_0:52;
   A12: k in dom M by A6,A10,FINSEQ_3:25;
   then A13: B/.k=M.k by A1,PARTFUN1:def 6;
   M.k in LM by A12,FUNCT_1:def 3;
   then A14: L.(M.k)=L1.(M.k) by A8,FUNCT_1:49;
   A15: k in dom f by A7,A10,FINSEQ_3:25;
   then f.k=@f/.k by PARTFUN1:def 6;
   hence @f/.k=L1.(B/.k) by A5,A15,A13,A11,A14;
  end;
  A16: Carrier L c=LM by VECTSP_6:def 4;
  then Carrier L c=[#]Lin LM by A8;
  then Carrier L=Carrier L1 & Sum L1=Sum L by VECTSP_9:7;
  hence thesis by A1,A4,A6,A7,A16,A9,MATRLIN:def 7;
end;
