
theorem
  for u,v,w being Element of TOP-REAL 2 st u,v,w are_collinear holds
  |[u`1,u`2,1]|,|[v`1,v`2,1]|,|[w`1,w`2,1]| are_collinear
  proof
    let u,v,w be Element of TOP-REAL 2;
    assume
A1: u,v,w are_collinear;
    reconsider u1 = |[u`1,u`2,1]|, v1 = |[v`1,v`2,1]|,
               w1 = |[w`1,w`2,1]| as non zero Point of TOP-REAL 3;
    u in LSeg(v,w) or v in LSeg(w,u) or w in LSeg(u,v) by A1,TOPREAL9:67;
    then u1 in LSeg(v1,w1) or v1 in LSeg(w1,u1) or w1 in LSeg(u1,v1) by Th45;
    hence thesis by TOPREAL9:67;
  end;
