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 Th103:
  x in lines M iff ex i st i in Seg m & x = Line(M,i)
proof
  thus x in lines M implies ex i st i in Seg m & x = Line(M,i)
  proof
    assume x in lines M;
    then consider i be object such that
A1: i in dom M and
A2: M.i=x by FUNCT_1:def 3;
A3: len M=m by MATRIX_0:def 2;
    reconsider i as Element of NAT by A1;
A4: dom M=Seg len M by FINSEQ_1:def 3;
    then M.i=Line(M,i) by A1,A3,MATRIX_0:52;
    hence thesis by A1,A2,A4,A3;
  end;
  given i such that
A5: i in Seg m and
A6: x = Line(M,i);
A7: len M=m by MATRIX_0:def 2;
  dom M=Seg len M by FINSEQ_1:def 3;
  then M.i in rng M by A5,A7,FUNCT_1:def 3;
  hence thesis by A5,A6,MATRIX_0:52;
end;
