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 Th22:
  M = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*> implies
  Line(M,1) = p & Line(M,2) = q & Line(M,3) = r
  proof
    assume
A1: M = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*>;
    1 in Seg 3 & 2 in Seg 3 & 3 in Seg 3 by FINSEQ_1:1;
    then Line(M,1) = M.1 & Line(M,2) = M.2 & Line(M,3) = M.3 by MATRIX_0:52;
    then Line(M,1) = <*p`1,p`2,p`3*> & Line(M,2) = <*q`1,q`2,q`3*> &
      Line(M,3) = <*r`1,r`2,r`3*> by A1;
    hence thesis by EUCLID_5:3;
  end;
