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;
reserve X for OrtAfPl;
reserve o9,a9,a19,a29,a39,a49,b9,b19,b29,b39,b49,c9,c19 for Element of X;
reserve o,a,a1,a2,a3,a4,b,b1,b2,b3,b4,c,c1 for Element of the AffinStruct of X;
reserve M9,N9 for Subset of X;
reserve A,M,N for Subset of the AffinStruct of X;

theorem
  X is satisfying_PAP iff the AffinStruct of X is Pappian
proof
A1: X is satisfying_PAP implies the AffinStruct of X is Pappian
  proof
    assume
A2: X is satisfying_PAP;
    now
      let M,N,o,a,b,c,a1,b1,c1;
      assume that
A3:   M is being_line and
A4:   N is being_line and
A5:   M<>N and
A6:   o in M and
A7:   o in N and
A8:   o<>a and
A9:   o<>a1 and
A10:  o<>b and
A11:  o<>b1 and
A12:  o<>c and
A13:  o<>c1 and
A14:  a in M and
A15:  b in M and
A16:  c in M and
A17:  a1 in N and
A18:  b1 in N and
A19:  c1 in N and
A20:  a,b1 // b,a1 and
A21:  b,c1 // c,b1;
      reconsider a9=a,b9=b,c9=c,a19=a1,b19=b1,c19=c1 as Element of X;
      reconsider M9=M,N9=N as Subset of X;
A22:  N9 is being_line by A4,ANALMETR:43;
A23:  not c19 in M9 & not b9 in N9
      proof
        assume c19 in M9 or b9 in N9;
        then
A24:    o,c1 // M or o,b // N by A3,A4,A6,A7,AFF_1:52;
A25:    o,b // M by A3,A6,A15,AFF_1:52;
        o,c1 // N by A4,A7,A19,AFF_1:52;
        hence contradiction by A5,A6,A7,A10,A13,A24,A25,AFF_1:45,53;
      end;
      b,a1 // a,b1 by A20,AFF_1:4;
      then
A26:  b9,a19 // a9,b19 by ANALMETR:36;
A27:  b9,c19 // c9,b19 by A21,ANALMETR:36;
      M9 is being_line by A3,ANALMETR:43;
      then
      a9,c19 // c9,a19 by A2,A6,A7,A8,A9,A11,A12,A14,A15,A16,A17,A18,A19,A22
,A26,A27,A23,CONMETR:def 2;
      hence a,c1 // c,a1 by ANALMETR:36;
    end;
    hence thesis by AFF_2:def 2;
  end;
  the AffinStruct of X is Pappian implies X is satisfying_PAP
  proof
    assume
A28: the AffinStruct of X is Pappian;
    now
      let o9,a19,a29,a39,b19,b29,b39,M9,N9;
      assume that
A29:  M9 is being_line and
A30:  N9 is being_line and
A31:  o9 in M9 and
A32:  a19 in M9 and
A33:  a29 in M9 and
A34:  a39 in M9 and
A35:  o9 in N9 and
A36:  b19 in N9 and
A37:  b29 in N9 and
A38:  b39 in N9 and
A39:  not b29 in M9 and
A40:  not a39 in N9 and
A41:  o9<>a19 and
A42:  o9<>a29 and
      o9<>a39 and
A43:  o9<>b19 and
      o9<>b29 and
A44:  o9<>b39 and
A45:  a39,b29 // a29,b19 and
A46:  a39,b39 // a19,b19;
      reconsider a1=a19,a2=a29,a3=a39,b1=b19,b2=b29,b3=b39
                 as Element of the AffinStruct of X;
      reconsider M=M9,N=N9 as Subset of the AffinStruct of X;
A47:  N is being_line by A30,ANALMETR:43;
A48:  M is being_line by A29,ANALMETR:43;
      now
        assume M<>N;
        a3,b3 // a1,b1 by A46,ANALMETR:36;
        then
A49:    a1,b1 // a3,b3 by AFF_1:4;
        a3,b2 // a2,b1 by A45,ANALMETR:36;
        then
        a1,b2 // a2,b3 by A28,A31,A32,A33,A34,A35,A36,A37,A38,A39,A40,A41,A42
,A43,A44,A48,A47,A49,AFF_2:def 2;
        hence a19,b29 // a29,b39 by ANALMETR:36;
      end;
      hence a19,b29 // a29,b39 by A37,A39;
    end;
    hence thesis by CONMETR:def 2;
  end;
  hence thesis by A1;
end;
