reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;

theorem Th61:
  R = r & L = Line(p,q) implies (R on L iff p,q,r are_collinear)
  proof
    assume
A1: R = r & L = Line(p,q);
    hereby
      assume R on L;
      then [R,L] in the Inc of IncProjSp_of real_projective_plane
        by INCSP_1:def 1;
      then [R,L] in Proj_Inc real_projective_plane by INCPROJ:2;
      then R in the carrier of real_projective_plane &
        L in ProjectiveLines real_projective_plane & ex Y be set
        st L = Y & R in Y by INCPROJ:def 2;
      hence p,q,r are_collinear by A1,COLLSP:11;
    end;
    assume
A2: p,q,r are_collinear;
    now
      R is Point of real_projective_plane by INCPROJ:3;
      hence R in the carrier of real_projective_plane;
      L is Element of ProjectiveLines real_projective_plane
        by INCPROJ:1,INCPROJ:4;
      hence L in ProjectiveLines real_projective_plane;
      thus ex Y be set st L = Y & R in Y by A2,A1,COLLSP:11;
    end;
    then [R,L] in Proj_Inc real_projective_plane by INCPROJ:def 2;
    then [R,L] in the Inc of IncProjSp_of real_projective_plane by INCPROJ:2;
    hence R on L by INCSP_1:def 1;
  end;
