
theorem
  BK-model-Plane is satisfying_Pasch
  proof
    let A,B,P,Q,Z be POINT of BK-model-Plane;
    assume that
A1: between A,P,Z and
A2: between B,Q,Z;
    reconsider a = A,b = B,p = P,q = Q,z = Z as Element of BK_model;
    reconsider A2 = BK_to_T2 A, B2 = BK_to_T2 B, P2 = BK_to_T2 P,
    Q2 = BK_to_T2 Q, Z2 = BK_to_T2 Z as POINT of TarskiEuclid2Space;
    between A2,P2,Z2 & between B2,Q2,Z2 by A1,A2,Th05;
    then consider X2 be POINT of TarskiEuclid2Space such that
A3: between P2,X2,B2 and
A4: between Q2,X2,A2 by GTARSKI1:def 11;
    reconsider X = T2_to_BK X2 as POINT of BK-model-Plane;
A5: Tn2TR X2 in LSeg(Tn2TR P2,Tn2TR B2) by A3,GTARSKI2:20;
    set P9 = Tn2TR P2, B9 = Tn2TR B2;
    P9 in inside_of_circle(0,0,1) & B9 in inside_of_circle(0,0,1) by Th02;
    then Tn2TR X2 is Element of inside_of_circle(0,0,1) by A5,Th15;
    then X2 = BK_to_T2 X by Th03;
    then between P,X,B & between Q,X,A by A3,A4,Th05;
    hence thesis;
  end;
