reserve X for AffinPlane;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1,c2,d,d1,d2, d3,d4,d5,e1,e2,x,y,z
  for Element of X;
reserve Y,Z,M,N,A,K,C for Subset of X;

theorem Th9:
  X is satisfying_pap iff X is satisfying_minor_indirect_Scherungssatz
proof
A1: X is satisfying_minor_indirect_Scherungssatz implies X is satisfying_pap
  proof
    assume
A2: X is satisfying_minor_indirect_Scherungssatz;
    now
      let M,N,a,b,c,a1,b1,c1;
      assume that
      M is being_line and
      N is being_line and
A3:   a in M and
A4:   b in M and
A5:   c in M and
A6:   M // N and
A7:   M<>N and
A8:   a1 in N and
A9:   b1 in N and
A10:  c1 in N and
A11:  a,b1 // b,a1 and
A12:  b,c1 // c,b1;
A13:  not b in N by A4,A6,A7,AFF_1:45;
A14:  not c1 in M by A6,A7,A10,AFF_1:45;
A15:  not c in N by A5,A6,A7,AFF_1:45;
A16:  b1,b // b,b1 by AFF_1:2;
A17:  not b1 in M by A6,A7,A9,AFF_1:45;
A18:  b,c1 // b1,c by A12,AFF_1:4;
A19:  not a1 in M by A6,A7,A8,AFF_1:45;
A20:  a,b1 // a1,b by A11,AFF_1:4;
      not a in N by A3,A6,A7,AFF_1:45;
      then a,c1 // a1,c by A2,A3,A4,A5,A6,A8,A9,A10,A13,A15,A19,A17,A14,A20,A18
,A16;
      hence a,c1 // c,a1 by AFF_1:4;
    end;
    hence thesis by AFF_2:def 13;
  end;
  X is satisfying_pap implies X is satisfying_minor_indirect_Scherungssatz
  proof
    assume
A21: X is satisfying_pap;
    now
      let a1,a2,a3,a4,b1,b2,b3,b4,M,N;
      assume that
A22:  M // N and
A23:  a1 in M and
A24:  a3 in M and
A25:  b2 in M and
A26:  b4 in M and
A27:  a2 in N and
A28:  a4 in N and
A29:  b1 in N and
A30:  b3 in N and
A31:  not a4 in M and
      not a2 in M and
      not b1 in M and
      not b3 in M and
      not a1 in N and
      not a3 in N and
      not b2 in N and
      not b4 in N and
A32:  a3,a2 // b3,b2 and
A33:  a2,a1 // b2,b1 and
A34:  a1,a4 // b1,b4;
A35:  M is being_line by A22,AFF_1:36;
A36:  b2,b3 // a3,a2 by A32,AFF_1:4;
A37:  N is being_line by A22,AFF_1:36;
A38:  a4,a1 // b1,b4 by A34,AFF_1:4;
      a1,a2 // b2,b1 by A33,AFF_1:4;
      then a1,b3 // a3,b1 by A21,A22,A23,A24,A25,A27,A28,A29,A30,A31,A35,A37
,A36,AFF_2:def 13;
      then b1,a3 // b3,a1 by AFF_1:4;
      then a4,a3 // b3,b4 by A21,A22,A23,A24,A26,A28,A29,A30,A31,A35,A37,A38,
AFF_2:def 13;
      hence a3,a4 // b3,b4 by AFF_1:4;
    end;
    hence thesis;
  end;
  hence thesis by A1;
end;
