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 Th26:
  for A be linearly-independent Subset of TOP-REAL m st the_rank_of M = n
  holds (Mx2Tran M)"A is linearly-independent
proof
  let A be linearly-independent Subset of TOP-REAL m such that
   A1: the_rank_of M=n;
  set nV=n-VectSp_over F_Real,mV=m-VectSp_over F_Real;
  reconsider Bm=MX2FinS 1.(F_Real,m) as OrdBasis of mV by MATRLIN2:45;
  reconsider A1=A as Subset of mV by Lm1;
  reconsider Bn=MX2FinS 1.(F_Real,n) as OrdBasis of nV 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;
  A4: MT is one-to-one by A1,A3,MATRTOP1:39;
  reconsider R=A/\rng MT as Subset of mV;
  A5:R c= A by XBOOLE_1:17;
  A1 is linearly-independent by Th7;
  then A6: dom MT=[#]nV & R is linearly-independent
    by A5,FUNCT_2:def 1,VECTSP_7:1;
  MT"R is linearly-independent
  proof
   assume MT"R is non linearly-independent;
   then consider L be Linear_Combination of MT"R such that
    A7: Carrier L<>{} and
    A8: Sum L=0.nV by RANKNULL:41;
   set C=Carrier L;
   A9: C c=MT"R by VECTSP_6:def 4;
   MT.:(MT"R)=R & MT@L is Linear_Combination of MT.:C
     by FUNCT_1:77,RANKNULL:29,XBOOLE_1:17;
   then A10: MT@L is Linear_Combination of R by A9,RELAT_1:123,VECTSP_6:4;
   MT|C is one-to-one by A4,FUNCT_1:52;
   then A11: Carrier(MT@L)=MT.:C by RANKNULL:39;
   MT| (MT"R) is one-to-one by A4,FUNCT_1:52;
   then Sum(MT@L)=MT.(Sum L) by Th14
    .=0.mV by A8,RANKNULL:9;
   hence contradiction by A6,A7,A11,A10,VECTSP_7:def 1;
  end;
  then MT"A is linearly-independent by RELAT_1:133;
  hence thesis by A3,Th7;
end;
