
theorem Th46:
  for P,Q,R being non point_at_infty Point of ProjectiveSpace TOP-REAL 3
  holds P,Q,R are_collinear iff Collinear RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R
  proof
    let P,Q,R be non point_at_infty Point of ProjectiveSpace TOP-REAL 3;
    reconsider p = RP3_to_T2 P,q = RP3_to_T2 Q,
    r = RP3_to_T2 R as POINT of TarskiEuclid2Space;
    consider u be non zero Element of TOP-REAL 3 such that
A1: P = Dir u & u`3 = 1 & RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
    consider v be non zero Element of TOP-REAL 3 such that
A2: Q = Dir v & v`3 = 1 & RP3_to_REAL2 Q = |[v`1,v`2]| by Def05;
    consider w be non zero Element of TOP-REAL 3 such that
A3: R = Dir w & w`3 = 1 & RP3_to_REAL2 R = |[w`1,w`2]| by Def05;
    hereby
      assume
A4:   P,Q,R are_collinear;
      u,v,w are_collinear by A4,A1,A2,A3,Th43;
      then per cases by TOPREAL9:67;
      suppose u in LSeg(v,w);
        then Tn2TR p in LSeg ( Tn2TR q, Tn2TR r ) by A1,A2,A3,Th44;
        hence Collinear RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R by GTARSKI2:20;
      end;
      suppose v in LSeg(w,u);
        then Tn2TR q in LSeg ( Tn2TR r, Tn2TR p ) by A1,A2,A3,Th44;
        hence Collinear RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R by GTARSKI2:20;
      end;
      suppose w in LSeg(u,v);
        then Tn2TR r in LSeg ( Tn2TR p, Tn2TR q ) by A1,A2,A3,Th44;
        hence Collinear RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R by GTARSKI2:20;
      end;
    end;
    assume Collinear RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R;
    then
A5: Tn2TR q in LSeg( Tn2TR p, Tn2TR r) or
    Tn2TR r in LSeg( Tn2TR q, Tn2TR p) or
    Tn2TR p in LSeg( Tn2TR r, Tn2TR q) by GTARSKI2:20;
    reconsider u1 = Tn2TR p, v1 = Tn2TR q,
               w1 = Tn2TR r as Element of TOP-REAL 2;
    reconsider u9 = |[u1`1,u1`2,1]|,v9 = |[v1`1,v1`2,1]|,
               w9 = |[w1`1,w1`2,1]| as Element of TOP-REAL 3;
    now
      u1`1 = u`1 & u1`2 = u`2 & v1`1 = v`1 &
        v1`2 = v`2 & w1`1 = w`1 & w1`2 = w`2 by A1,A2,A3,EUCLID:52;
      hence Dir u9 = P & Dir v9 = Q & Dir w9 = R by A1,A2,A3,EUCLID_5:3;
      thus u9`3 = 1 & v9`3 = 1 & w9`3 = 1 by EUCLID_5:2;
      v9 in LSeg(u9,w9) or w9 in LSeg(v9,u9) or u9 in LSeg(w9,v9) by A5,Th45;
      hence u9,v9,w9 are_collinear by TOPREAL9:67;
    end;
    hence P,Q,R are_collinear by Th43;
  end;
