
theorem Th49:
  for l1,l2,l3 being Element of ProjectiveLines real_projective_plane holds
  l1,l2,l3 are_concurrent iff #l1,#l2,#l3 are_concurrent
  proof
    let l1,l2,l3 be Element of ProjectiveLines real_projective_plane;
    now
      l1 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l1 = B &
        B is LINE of real_projective_plane;
      l2 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l2 = B &
        B is LINE of real_projective_plane;
      l3 in {B where B is Subset of real_projective_plane:
        B is LINE of real_projective_plane};
      hence ex B be Subset of real_projective_plane st l3 = B &
        B is LINE of real_projective_plane;
    end;
    then reconsider m1 = l1,m2 = l2,m3 = l3 as LINE of real_projective_plane;
    reconsider l91 = #l1, l92 = #l2, l93 = #l3 as
      LINE of IncProjSp_of real_projective_plane;
    hereby
      assume l1,l2,l3 are_concurrent;
      then consider P be Point of real_projective_plane such that
A1:   P in l1 and
A2:   P in l2 and
A3:   P in l3;
      reconsider P as Element of the Points of IncProjSp_of
        real_projective_plane;
      reconsider P9 = P as POINT of IncProjSp_of real_projective_plane;
      P in m1 & P in m2 & P in m3 by A1,A2,A3;
      then P9 on l91 & P9 on l92 & P9 on l93 by INCPROJ:5;
      hence #l1,#l2,#l3 are_concurrent;
    end;
    assume #l1,#l2,#l3 are_concurrent;
    then consider o be Element of the Points of IncProjSp_of
      real_projective_plane
    such that
A4: o on #l1 and
A5: o on #l2 and
A6: o on #l3;
    reconsider o9 = o as Point of real_projective_plane;
    o9 in m1 & o9 in m2 & o9 in m3 by A4,A5,A6,INCPROJ:5;
    hence l1,l2,l3 are_concurrent;
  end;
