reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem Th105:
  the_rank_of M = m implies M is without_repeated_line
proof
  assume
A1: the_rank_of M = m;
A2: len M = m by MATRIX_0:def 2;
  assume not M is without_repeated_line;
  then consider x1,x2 be object such that
A3: x1 in dom M and
A4: x2 in dom M and
A5: M.x1=M.x2 and
A6: x1<>x2;
  reconsider x1,x2 as Element of NAT by A3,A4;
  consider k such that
A7: len M = k + 1 and
A8: len Del(M,x1) = k by A3,FINSEQ_3:104;
A9: dom M=Seg len M by FINSEQ_1:def 3;
  then
A10: M.x2=Line(M,x2) by A4,A2,MATRIX_0:52;
  M.x1=Line(M,x1) by A3,A9,A2,MATRIX_0:52;
  then M = RLine(M,x1,Line(M,x2)) by A5,A10,MATRIX11:30
    .= RLine(M,x1,1_K*Line(M,x2)) by FVSUM_1:57;
  then m = the_rank_of DelLine(M,x1) by A1,A4,A6,A9,Th93;
  then m <= k by A8,Th74;
  hence thesis by A2,A7,NAT_1:13;
end;
