reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem
  for x be object holds x in lines M@ iff
  ex i be Nat st i in Seg 3 & x = Col(M,i)
  proof
    let x be object;
    hereby
      assume x in lines M@;
      then consider i be Nat such that
A1:   i in Seg 3 and
A2:   x = Line( M@ ,i) by MATRIX13:103;
      i in Seg width M by A1,MATRIX_0:24;
      then x = Col(M,i) by A2,MATRIX_0:59;
      hence ex i be Nat st i in Seg 3 & x = Col(M,i) by A1;
    end;
    given i be Nat such that
A3: i in Seg 3 and
A4: x = Col(M,i);
    i in Seg width M by A3,MATRIX_0:24;
    then x = Line(M@,i) by A4,MATRIX_0:59;
    hence x in lines M@ by A3,MATRIX13:103;
  end;
