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;

theorem Th42:
  for D being non empty set,M being Matrix of D holds for i,j st j
  in dom M & i in Seg width M holds Col(M,i).j = Line(M,j).i
proof
  let D be non empty set,M be Matrix of D;
  let i,j;
  assume that
A1: j in dom M and
A2: i in Seg width M;
  thus Col(M,i).j = M*(j,i) by A1,Def8
    .= Line(M,j).i by A2,Def7;
end;
