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;
reserve PQR for Matrix of 3,F_Real;

theorem
  PQR = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*> implies
  Col(PQR,1) = <*p`1,q`1,r`1*> & Col(PQR,2) = <*p`2,q`2,r`2*> &
  Col(PQR,3) = <*p`3,q`3,r`3*>
  proof
    assume
A1: PQR = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*>;
    len PQR = 3 by MATRIX_0:24; then
A2: dom PQR = Seg 3 by FINSEQ_1:def 3;
A3: PQR = <* <* PQR*(1,1), PQR*(1,2), PQR*(1,3) *>,
             <* PQR*(2,1), PQR*(2,2), PQR*(2,3) *>,
             <* PQR*(3,1), PQR*(3,2), PQR*(3,3) *> *> by MATRIXR2:37;
    then PQR.1 = <* PQR*(1,1), PQR*(1,2), PQR*(1,3) *>; then
A4: <* PQR*(1,1), PQR*(1,2), PQR*(1,3) *> = <*p`1,p`2,p`3*> by A1;
    PQR.2 = <* PQR*(2,1), PQR*(2,2), PQR*(2,3) *> by A3; then
A5: <* PQR*(2,1), PQR*(2,2), PQR*(2,3) *> = <*q`1,q`2,q`3*>
      by A1;
    PQR.3 = <* PQR*(3,1), PQR*(3,2), PQR*(3,3) *> by A3; then
A6: <* PQR*(3,1), PQR*(3,2), PQR*(3,3) *> = <*r`1,r`2,r`3*>
      by A1;
    now
      thus len Col(PQR,1) = len PQR by MATRIX_0:def 8
                         .= 3 by MATRIX_0:24;
      thus len Col(PQR,2) = len PQR by MATRIX_0:def 8
                         .= 3 by MATRIX_0:24;
      thus len Col(PQR,3) = len PQR by MATRIX_0:def 8
                         .= 3 by MATRIX_0:24;
      thus Col(PQR,1).1 = PQR*(1,1) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= p`1 by A4,FINSEQ_1:78;
      thus Col(PQR,1).2 = PQR*(2,1) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= q`1 by A5,FINSEQ_1:78;
      thus Col(PQR,1).3 = PQR*(3,1) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= r`1 by A6,FINSEQ_1:78;
      thus Col(PQR,2).1 = PQR*(1,2) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= p`2 by A4,FINSEQ_1:78;
      thus Col(PQR,2).2 = PQR*(2,2) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= q`2 by A5,FINSEQ_1:78;
      thus Col(PQR,2).3 = PQR*(3,2) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= r`2 by A6,FINSEQ_1:78;
      thus Col(PQR,3).1 = PQR*(1,3) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= p`3 by A4,FINSEQ_1:78;
      thus Col(PQR,3).2 = PQR*(2,3) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= q`3 by A5,FINSEQ_1:78;
      thus Col(PQR,3).3 = PQR*(3,3) by A2,MATRIX_0:def 8,FINSEQ_1:1
                       .= r`3 by A6,FINSEQ_1:78;
    end;
    hence thesis by FINSEQ_1:45;
  end;
