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
  for M being Matrix of D holds for j st j in Seg width M holds Line(M@,
  j)=Col(M,j)
proof
  let M be Matrix of D;
  let j;
  assume
A1: j in Seg width M;
  then j in Seg len (M@) by Def6;
  then
A2: j in dom (M@) by FINSEQ_1:def 3;
  per cases;
  suppose
A3:  len M=0;
        then Seg len M = {};
    hence thesis by A1,A3,Def3;
  end;
  suppose
A4: len M>0;
    per cases;
    suppose
      width M=0;
      hence thesis by A1;
    end;
    suppose
      width M>0;
      then (M@)@=M by A4,Th57;
      hence thesis by A2,Th58;
    end;
  end;
end;
