 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;

theorem Th06:
  for a11,a12,a13,a21,a22,a23,a31,a32,a33 being Element of F_Real
  for A being Matrix of 3,F_Real st
  A = <* <* a11,a12,a13 *>, <* a21,a22,a23 *>, <* a31,a32,a33 *> *>
  holds
  Col(A,1) = <* a11,a21,a31 *> & Col(A,2) = <* a12,a22,a32 *> &
  Col(A,3) = <* a13,a23,a33 *>
  proof
    let a11,a12,a13,a21,a22,a23,a31,a32,a33 be Element of F_Real;
    let A be Matrix of 3,F_Real;
    assume
A1: A = <* <* a11,a12,a13 *>, <* a21,a22,a23 *>, <* a31,a32,a33 *> *>;
    |[ a11,a12,a13 ]| is Point of TOP-REAL 3 &
      |[ a21,a22,a23 ]| is Point of TOP-REAL 3 &
      |[ a31,a32,a33 ]| is Point of TOP-REAL 3;
    then reconsider p = <* a11,a12,a13 *>, q = <* a21,a22,a23 *>,
                    r = <* a31,a32,a33 *> as Point of TOP-REAL 3;
    p.1 = a11 & p.2 = a12 & p.3 = a13 & q.1 = a21 & q.2 = a22 & q.3 = a23 &
      r.1 = a31 & r.2 = a32 & r.3 = a33;
    then
A2: p`1 = a11 & p`2 = a12 & p`3 = a13 & q`1 = a21 & q`2 = a22 & q`3 = a23 &
      r`1 = a31 & r`2 = a32 & r`3 = a33
      by EUCLID_5:def 1,EUCLID_5:def 2,EUCLID_5:def 3;
    A@ = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*>
      by A1,ANPROJ_8:23; then
A3: Line(A@,1) = <* a11,a21,a31 *> & Line(A@,2) = <* a12,a22,a32 *> &
      Line(A@,3) = <* a13,a23,a33 *> by A2,Th05;
A4: width A = 3 by MATRIX_0:24;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    hence thesis by A3,A4,MATRIX_0:59;
  end;
