
theorem
  for P,Q being non point_at_infty Element of ProjectiveSpace TOP-REAL 3 st
  P in absolute & Q in absolute
  holds RP3_to_T2 Dir001, RP3_to_T2 P equiv RP3_to_T2 Dir001,RP3_to_T2 Q
  proof
    let P,Q be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: P in absolute and
A2: Q in absolute;
    reconsider p = RP3_to_T2 P, q = RP3_to_T2 Q,
               r = RP3_to_T2 Dir001 as Element of TarskiEuclid2Space;
    consider u be non zero Element of TOP-REAL 3 such that
A3: 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
A4: 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
A5: Dir001 = Dir w & w`3 = 1 & RP3_to_REAL2 Dir001 = |[w`1,w`2]| by Def05;
    are_Prop |[0,0,1]|,w by A5,ANPROJ_1:22;
    then consider a be Real such that a <> 0 and
A6: w = a * |[0,0,1]| by ANPROJ_1:1;
    w = |[ a * 0, a * 0, a * 1]| by A6,EUCLID_5:8;
    then
A8: w`1 = 0 & w`2 = 0 by EUCLID_5:2;
    reconsider u1=u`1,u2=u`2,v1=v`1,v2=v`2,w1=w`1,w2=w`2 as Real;
    now
A9:   (Tn2TR RP3_to_T2 P)`1 = u`1 & (Tn2TR RP3_to_T2 P)`2 = u`2 &
        (Tn2TR RP3_to_T2 Q)`1 = v`1 & (Tn2TR RP3_to_T2 Q)`2 = v`2 &
        (Tn2TR RP3_to_T2 Dir001)`1 = w`1 & (Tn2TR RP3_to_T2 Dir001)`2 = w`2
        by A3,A4,A5,EUCLID:52;
      reconsider uP = |[u`1,u`2,1]| as non zero Element of TOP-REAL 3;
      now
        thus P in absolute by A1;
        thus P = Dir uP by A3,EUCLID_5:3;
        uP`3 = 1 by EUCLID_5:2;
        hence uP.3 = 1 by EUCLID_5:def 3;
      end;
      then |[uP.1,uP.2]| in circle(0,0,1) by BKMODEL1:84;
      then (uP.1)^2+(uP.2)^2 = 1 by BKMODEL1:13;
      then (uP`1)^2+(uP.2)^2 = 1 by EUCLID_5:def 1;
      then (uP`1)^2+(uP`2)^2 = 1 by EUCLID_5:def 2;
      then (u`1)^2+(uP`2)^2 = 1 by EUCLID_5:2;
      then
A10:  (u`1)^2+(u`2)^2 = 1 by EUCLID_5:2;
      reconsider vQ = |[v`1,v`2,1]| as non zero Element of TOP-REAL 3;
      now
        thus Q in absolute by A2;
        thus Q = Dir vQ by A4,EUCLID_5:3;
        vQ`3 = 1 by EUCLID_5:2;
        hence vQ.3 = 1 by EUCLID_5:def 3;
      end;
      then |[vQ.1,vQ.2]| in circle(0,0,1) by BKMODEL1:84;
      then (vQ.1)^2+(vQ.2)^2 = 1 by BKMODEL1:13;
      then (vQ`1)^2+(vQ.2)^2 = 1 by EUCLID_5:def 1;
      then (vQ`1)^2+(vQ`2)^2 = 1 by EUCLID_5:def 2;
      then (v`1)^2+(vQ`2)^2 = 1 by EUCLID_5:2;
      then
A11:  (v`1)^2+(v`2)^2 = 1 by EUCLID_5:2;
      now
        thus dist(RP3_to_T2 Dir001,RP3_to_T2 P)
          = sqrt((( 0 )-(u`1))^2+(( 0 )-(u`2))^2) by A9,A8,GTARSKI2:16
         .= sqrt ((u`1)^2+(-u`2)^2) by SQUARE_1:3
         .= 1 by A10,SQUARE_1:3,18;
        thus dist(RP3_to_T2 Dir001,RP3_to_T2 Q)
          = sqrt(((w`1)-(v`1))^2+((w`2)-(v`2))^2) by A9,GTARSKI2:16
         .= sqrt ((v`1)^2+(-v`2)^2) by A8,SQUARE_1:3
         .= 1 by A11,SQUARE_1:3,18;
      end;
      hence dist(RP3_to_T2 Dir001,RP3_to_T2 P)
        = dist(RP3_to_T2 Dir001,RP3_to_T2 Q);
      thus dist(RP3_to_T2 Dir001,RP3_to_T2 P)
        = |. Tn2TR RP3_to_T2 Dir001 - Tn2TR RP3_to_T2 P .| by GTARSKI2:17;
      thus dist(RP3_to_T2 Dir001,RP3_to_T2 Q)
        = |. Tn2TR RP3_to_T2 Dir001 - Tn2TR RP3_to_T2 Q .| by GTARSKI2:17;
    end;
    hence thesis by GTARSKI2:18;
  end;
