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 Th23:
  for A be linearly-independent Subset of TOP-REAL n st the_rank_of M = n
  holds (Mx2Tran M).:A is linearly-independent
proof
  let A be linearly-independent Subset of TOP-REAL n such that
   A1: the_rank_of M=n;
  set nV=n-VectSp_over F_Real,mV=m-VectSp_over F_Real;
  reconsider Bn=MX2FinS 1.(F_Real,n) as OrdBasis of nV by MATRLIN2:45;
  reconsider Bm=MX2FinS 1.(F_Real,m) as OrdBasis of mV by MATRLIN2:45;
A2:  len Bm = m by MATRTOP1:19;
  len Bn=n by MATRTOP1:19;
  then reconsider M1=M as Matrix of len Bn,len Bm,F_Real by A2;
  set MT=Mx2Tran(M1,Bn,Bm);
  A3: Mx2Tran M=MT by MATRTOP1:20;
  reconsider A1=A as Subset of nV by Lm1;
  A4: A1 is linearly-independent by Th7;
  MT.:A1 is linearly-independent
  proof
    assume MT.:A1 is non linearly-independent;
    then consider L be Linear_Combination of MT.:A1 such that
     A5: Carrier L<>{} and
     A6: Sum L=0.mV by RANKNULL:41;
     A7: MT is one-to-one by A1,A3,MATRTOP1:39;
     then A8: ker MT=(0).nV by RANKNULL:15;
     A9: MT|A1 is one-to-one by A7,FUNCT_1:52;
     then A10: MT@(MT#L)=L by RANKNULL:43;
     MT|Carrier(MT#L) is one-to-one by A7,FUNCT_1:52;
     then MT.:Carrier(MT#L)=Carrier L by A10,RANKNULL:39;
     then A11: Carrier(MT#L)<>{} by A5;
     MT.(Sum(MT#L))=0.mV by A6,A9,A10,Th14;
     then Sum(MT#L) in ker MT by RANKNULL:10;
     then Sum(MT#L)=0.nV by A8,VECTSP_4:35;
     hence contradiction by A4,A11,VECTSP_7:def 1;
  end;
  hence thesis by A3,Th7;
end;
