reserve AP for AffinPlane,
  a,a9,b,b9,c,c9,x,y,o,p,q,r,s for Element of AP,
  A,C,C9,D,K,M,N,P,T for Subset of AP;

theorem Th10:
  AP is satisfying_PPAP iff AP is Pappian & AP is satisfying_pap
proof
A1: AP is Pappian & AP is satisfying_pap implies AP is satisfying_PPAP
  proof
    assume that
A2: AP is Pappian and
A3: AP is satisfying_pap;
    thus AP is satisfying_PPAP
    proof
      let M,N,a,b,c,a9,b9,c9;
      assume that
A4:   M is being_line and
A5:   N is being_line and
A6:   a in M and
A7:   b in M and
A8:   c in M and
A9:   a9 in N and
A10:  b9 in N and
A11:  c9 in N and
A12:  a,b9 // b,a9 and
A13:  b,c9 // c,b9;
      now
        assume
A14:    M<>N;
        assume
A15:    not thesis;
        now
          assume not M // N;
          then consider o such that
A16:      o in M and
A17:      o in N by A4,A5,AFF_1:58;
A18:      o<>a
          proof
            assume
A19:        o=a;
            then o,b9 // a9,b by A12,AFF_1:4;
            then o=b9 or b in N by A5,A9,A10,A17,AFF_1:48;
            then o=b9 or o=b by A4,A5,A7,A14,A16,A17,AFF_1:18;
            then c,o // b,c9 or o,c9 // b9,c by A13,AFF_1:4;
            then c9 in M or c =o or c in N or o=c9 by A4,A5,A7,A8,A10,A11,A16
,A17,AFF_1:48;
            then o=c or o=c9 by A4,A5,A8,A11,A14,A16,A17,AFF_1:18;
            hence contradiction by A5,A9,A11,A15,A17,A19,AFF_1:3,51;
          end;
A20:      o<>b
          proof
            assume
A21:        o=b;
            then o,c9 // b9,c by A13,AFF_1:4;
            then o=c9 or c in N by A5,A10,A11,A17,AFF_1:48;
            then
A22:        o=c9 or o=c by A4,A5,A8,A14,A16,A17,AFF_1:18;
            o,a9 // b9,a by A12,A21,AFF_1:4;
            then o=a9 or a in N by A5,A9,A10,A17,AFF_1:48;
            hence contradiction by A4,A5,A6,A8,A14,A15,A16,A17,A18,A22,AFF_1:3
,18,51;
          end;
A23:      o<>c9
          proof
            assume
A24:        o=c9;
            then b9 in M by A4,A7,A8,A13,A16,A20,AFF_1:48;
            then a,o // b,a9 by A4,A5,A10,A12,A14,A16,A17,AFF_1:18;
            then a9 in M by A4,A6,A7,A16,A18,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A16,A24,AFF_1:51;
          end;
A25:      o<>c
          proof
            assume
A26:        o=c;
            then o,b9 // c9,b by A13,AFF_1:4;
            then o=b9 or b in N by A5,A10,A11,A17,AFF_1:48;
            then a9 in M by A4,A5,A6,A7,A9,A12,A16,A17,A18,A20,AFF_1:18,48;
            then a9=o by A4,A5,A9,A14,A16,A17,AFF_1:18;
            hence contradiction by A15,A26,AFF_1:3;
          end;
A27:      o<>a9
          proof
            assume
A28:        o=a9;
            then o,b // a,b9 by A12,AFF_1:4;
            then b9 in M by A4,A6,A7,A16,A20,AFF_1:48;
            then o=b9 by A4,A5,A10,A14,A16,A17,AFF_1:18;
            then c,o // b,c9 by A13,AFF_1:4;
            then c9 in M by A4,A7,A8,A16,A25,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A16,A28,AFF_1:51;
          end;
          o<>b9
          proof
            assume
A29:        o=b9;
            then o,c // b,c9 by A13,AFF_1:4;
            then
A30:        c9 in M by A4,A7,A8,A16,A25,AFF_1:48;
            a9 in M by A4,A6,A7,A12,A16,A18,A29,AFF_1:48;
            hence contradiction by A4,A6,A8,A15,A30,AFF_1:51;
          end;
          hence thesis by A2,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14,A16,A17,A18
,A20,A25,A27,A23;
        end;
        hence thesis by A3,A4,A5,A6,A7,A8,A9,A10,A11,A12,A13,A14;
      end;
      hence thesis by A4,A6,A8,A9,A11,AFF_1:51;
    end;
  end;
  thus thesis by A1;
end;
