reserve x,y,z for object,
  i,j,n,m for Nat,
  D for non empty set,
  s,t for FinSequence,
  a,a1,a2,b1,b2,d for Element of D,
  p, p1,p2,q,r for FinSequence of D;
reserve M,M1,M2 for Matrix of D;
reserve f for FinSequence of D;
reserve i,j,i1,j1 for Nat;
reserve k for Nat, G for Matrix of D;
reserve x,y,x1,x2,y1,y2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  s,s2 for FinSequence,
  a,b,c,d for Element of D,
  q,r for FinSequence of D,
  a9,b9 for Element of D;

theorem Th58:
  for M being Matrix of D holds for i st i in dom M holds Line(M,i )=Col(M@,i)
proof
  let M be Matrix of D;
  let i;
A1: len (M@)=width M by Def6;
  len Col(M@,i) = len (M@) by Def8;
  then
A2: len Col(M@,i)=width M by Def6;
  then
A3: dom Col(M@,i) = Seg width M by FINSEQ_1:def 3;
  assume
A4: i in dom M;
A5: now
    let j be Nat;
A6: dom (M@) = Seg len (M@) by FINSEQ_1:def 3;
    assume
A7: j in dom Col(M@,i);
    then Line(M,i).j=M*(i,j) & [i,j] in Indices M by A4,A3,Def7,ZFMISC_1:87;
    hence Line(M,i).j=M@*(j,i) by Def6
      .=Col(M@,i).j by A1,A3,A7,A6,Def8;
  end;
  len Line(M,i) = width M by Def7;
  hence thesis by A2,A5,FINSEQ_2:9;
end;
