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;
reserve pf for FinSequence of D;

theorem
  for pr being Element of REAL 3 st
  pf = pr holds MXR2MXF ColVec2Mx pr = <*pf*>@
  proof
    let pr being Element of REAL 3;
    assume
A1: pf = pr;
    set M1 = MXR2MXF ColVec2Mx pr, M2 = ColVec2Mx pr;
A2: M1 = ColVec2Mx pr by MATRIXR1:def 1;
    pr in REAL 3; then
A3: pr in 3-tuples_on REAL by EUCLID:def 1; then
A4: len pr = 3 by FINSEQ_2:133;
A5: len M2 = len pr & width M2 = 1 & for j be Nat st j in dom pr holds
      M2.j = <*pr.j*> by A4,MATRIXR1:def 9;
    now
A6:   width <*pf*> = len pr by A1,MATRIX_0:23;
      hence len M2 = len (<*pf*>@) by A5,MATRIX_0:def 6;
      thus for k be Nat st 1 <= k & k <= len M2 holds M2.k = (<*pf*>@).k
      proof
        let k be Nat;
        assume that
A7:     1 <= k and
A8:     k <= len M2;
A9:     k in Seg len pr by A5,A7,A8,FINSEQ_1:1; then
A10:    k in dom pr by FINSEQ_1:def 3;
A11:    len <*pf*> = 1 by MATRIX_0:23;
A12:    width <*pf*> = 3 by A6,A3,FINSEQ_2:133;
        Seg len (<*pf*>@) = Seg len pr by A6,MATRIX_0:def 6;
        then dom (<*pf*>@) = Seg len pr by FINSEQ_1:def 3
                          .= dom pr
          by FINSEQ_1:def 3; then
A13:    k in dom (<*pf*>@) by A9,FINSEQ_1:def 3; then
A14:    (<*pf*>@).k = Line(<*pf*>@,k) by MATRIX_0:60
                   .= Col(<*pf*>@@,k) by A13,MATRIX_0:58
                   .= Col(<*pf*>,k) by A12,A11,MATRIX_0:57
                   .= <*pf.k*> by A10,A1,Th59;
        k in dom pr by A9,FINSEQ_1:def 3;
        hence thesis by A4,MATRIXR1:def 9,A1,A14;
      end;
    end;
    hence thesis by A2,FINSEQ_1:def 18;
  end;
