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
  AP is satisfying_pap iff AP is satisfying_pap_1
proof
    hereby assume
A1: AP is satisfying_pap;
    thus AP is satisfying_pap_1
    proof
      let M,N,a,b,c,a9,b9,c9;
      assume that
A2:   M is being_line and
A3:   N is being_line and
A4:   a in M and
A5:   b in M and
A6:   c in M and
A7:   M // N and
A8:   M<>N and
A9:  a9 in N and
A10:  b9 in N and
A11:  a,b9 // b,a9 and
A12:  b,c9 // c,b9 and
A13:  a,c9 // c,a9 and
A14:  a9<>b9;
A15:  c <>a9 by A6,A7,A8,A9,AFF_1:45;
      set C=Line(c,b9);
A16:  c <>b9 by A6,A7,A8,A10,AFF_1:45;
      then C is being_line by AFF_1:24;
      then consider K such that
A17:  b in K and
A18:  C // K by AFF_1:49;
A19:  c in C & b9 in C by A16,AFF_1:24;
A20:  not K // N
      proof
        assume K // N;
        then C // N by A18,AFF_1:44;
        then C // M by A7,AFF_1:44;
        then b9 in M by A6,A19,AFF_1:45;
        hence contradiction by A7,A8,A10,AFF_1:45;
      end;
      K is being_line by A18,AFF_1:36;
      then consider x such that
A21:  x in K and
A22:  x in N by A3,A20,AFF_1:58;
A23:  b,x // c,b9 by A19,A17,A18,A21,AFF_1:39;
      then a,x // c,a9 by A1,A2,A3,A4,A5,A6,A7,A8,A9,A10,A11,A22;
      then a,x // a,c9 by A13,A15,AFF_1:5;
      then LIN a,x,c9 by AFF_1:def 1;
      then
A24:  LIN c9,x,a by AFF_1:6;
A25:  a<>b
      proof
        assume a=b;
        then LIN a,b9,a9 by A11,AFF_1:def 1;
        then LIN a9,b9,a by AFF_1:6;
        then a9=b9 or a in N by A3,A9,A10,AFF_1:25;
        hence contradiction by A4,A7,A8,A14,AFF_1:45;
      end;
A26:  c9<>b
      proof
        assume c9=b;
        then a9 in M by A2,A4,A5,A6,A13,A25,AFF_1:48;
        hence contradiction by A7,A8,A9,AFF_1:45;
      end;
      b,x // b,c9 by A12,A16,A23,AFF_1:5;
      then LIN b,x,c9 by AFF_1:def 1;
      then
A27:  LIN c9,x,b by AFF_1:6;
      assume
A28:  not c9 in N;
      LIN c9,x,c9 by AFF_1:7;
      then c9 in M by A2,A4,A5,A28,A25,A22,A24,A27,AFF_1:8,25;
      then b9 in M by A2,A5,A6,A12,A26,AFF_1:48;
      hence contradiction by A7,A8,A10,AFF_1:45;
    end;
  end;
    assume
A29: AP is satisfying_pap_1;
      let M,N,a,b,c,a9,b9,c9;
      assume that
A30:  M is being_line and
A31:  N is being_line and
A32:  a in M and
A33:  b in M and
A34:  c in M and
A35:  M // N & M<>N and
A36:  a9 in N and
A37:  b9 in N and
A38:  c9 in N and
A39:  a,b9 // b,a9 and
A40:  b,c9 // c,b9;
      set A=Line(c,a9), D=Line(b,c9);
A41:  b<>c9 by A33,A35,A38,AFF_1:45;
      then
A42:  b in D & c9 in D by AFF_1:24;
      assume
A43:  not a,c9 // c,a9;
      then
A44:  c <>a9 by AFF_1:3;
      then
A45:  c in A by AFF_1:24;
A46:  a9 in A by A44,AFF_1:24;
A47:  A is being_line by A44,AFF_1:24;
      then consider P such that
A48:  a in P and
A49:  A // P by AFF_1:49;
A50:  a9<>b9
      proof
        assume
A51:    a9=b9;
        then a9,a // a9,b by A39,AFF_1:4;
        then LIN a9,a,b by AFF_1:def 1;
        then LIN a,b,a9 by AFF_1:6;
        then a=b or a9 in M by A30,A32,A33,AFF_1:25;
        hence contradiction by A35,A36,A40,A43,A51,AFF_1:45;
      end;
A52:  not D // P
      proof
        assume D // P;
        then c,b9 // P by A40,A41,A42,AFF_1:32,40;
        then c,b9 // A by A49,AFF_1:43;
        then b9 in A by A47,A45,AFF_1:23;
        then c in N by A31,A36,A37,A50,A47,A45,A46,AFF_1:18;
        hence contradiction by A34,A35,AFF_1:45;
      end;
A53:  D is being_line by A41,AFF_1:24;
      P is being_line by A49,AFF_1:36;
      then consider x such that
A54:  x in D and
A55:  x in P by A53,A52,AFF_1:58;
      LIN b,x,c9 by A53,A42,A54,AFF_1:21;
      then b,x // b,c9 by AFF_1:def 1;
      then
A56:  b,x // c,b9 by A40,A41,AFF_1:5;
      a,x // c,a9 by A45,A46,A48,A49,A55,AFF_1:39;
      then x in N by A29,A30,A31,A32,A33,A34,A35,A36,A37,A39,A50,A56;
      then x=c9 or b in N by A31,A38,A53,A42,A54,AFF_1:18;
      hence contradiction by A33,A35,A43,A45,A46,A48,A49,A55,AFF_1:39,45;
end;
