reserve AS for AffinSpace;
reserve A,K,M,X,Y,Z,X9,Y9 for Subset of AS;
reserve zz for Element of AS;
reserve x,y for set;
reserve x,y,z,t,u,w for Element of AS;
reserve K,X,Y,Z,X9,Y9 for Subset of AS;
reserve a,b,c,d,p,q,r,p9 for POINT of IncProjSp_of(AS);
reserve A for LINE of IncProjSp_of(AS);
reserve A,K,M,N,P,Q for LINE of IncProjSp_of(AS);

theorem
  IncProjSp_of(AS) is Pappian implies AS is Pappian
proof
  set XX = IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Pappian;
  for M,N being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st M is being_line & N is being_line & M<>N & o in M & o in N & o
<>a & o<>a9 & o<>b & o<>b9 & o<>c & o<>c9 & a in M & b in M & c in M & a9 in N
  & b9 in N & c9 in N & a,b9 // b,a9 & b,c9 // c,b9 holds a,c9 // c,a9
  proof
    let M,N be Subset of AS, o,a,b,c,a9,b9,c9 be Element of AS
    such that
A2: M is being_line and
A3: N is being_line and
A4: M<>N and
A5: o in M and
A6: o in N and
A7: o<>a and
A8: o<>a9 and
A9: o<>b and
A10: o<>b9 and
A11: o<>c and
A12: o<>c9 and
A13: a in M and
A14: b in M and
A15: c in M and
A16: a9 in N and
A17: b9 in N and
A18: c9 in N and
A19: a,b9 // b,a9 and
A20: b,c9 // c,b9;
A21: b<>c9 by A2,A3,A4,A5,A6,A9,A14,A18,AFF_1:18;
    then
A22: Line(b,c9) is being_line by AFF_1:def 3;
    c <>a9 by A2,A3,A4,A5,A6,A8,A15,A16,AFF_1:18;
    then
A23: Line(c,a9) is being_line by AFF_1:def 3;
A24: b<>a9 by A2,A3,A4,A5,A6,A8,A14,A16,AFF_1:18;
    then
A25: Line(b,a9) is being_line by AFF_1:def 3;
A26: c <>b9 by A2,A3,A4,A5,A6,A10,A15,A17,AFF_1:18;
    then
A27: Line(c,b9) is being_line by AFF_1:def 3;
    reconsider A3=[M,1],B3=[N,1] as Element of the Lines of XX by A2,A3,Th23;
    reconsider p=o,a1=a9,a2=c9,a3=b9,b1=a,b2=c,b3=b as Element of the Points
    of XX by Th20;
A28: p on A3 by A2,A5,Th26;
A29: a<>b9 by A2,A3,A4,A5,A6,A7,A13,A17,AFF_1:18;
    then
A30: Line(a,b9) is being_line by AFF_1:def 3;
    then reconsider c1=LDir(Line(b,c9)),c2=LDir(Line(a,b9)) as Element of the
    Points of XX by A22,Th20;
A31: b1 on A3 by A2,A13,Th26;
    a<>c9 by A2,A3,A4,A5,A6,A7,A13,A18,AFF_1:18;
    then
A32: Line(a,c9) is being_line by AFF_1:def 3;
    then reconsider
    A1=[Line(b,c9),1],A2=[Line(b,a9),1],B1=[Line(a,b9),1], B2=[Line
(c,b9),1],C1=[Line(c,a9),1],C2=[Line(a,c9),1] as Element of the Lines of XX by
A30,A25,A22,A27,A23,Th23;
A33: c2 on B1 by A30,Th30;
A34: b3 on A3 by A2,A14,Th26;
A35: b2 on A3 by A2,A15,Th26;
    consider Y such that
A36: M c= Y and
A37: N c= Y and
A38: Y is being_plane by A2,A3,A5,A6,AFF_4:38;
    reconsider C39=[PDir(Y),2] as Element of the Lines of XX by A38,Th23;
A39: c1 on C39 by A14,A18,A36,A37,A38,A21,A22,Th31,AFF_4:19;
A40: c2 on C39 by A13,A17,A36,A37,A38,A29,A30,Th31,AFF_4:19;
A41: a1 on B3 by A3,A16,Th26;
A42: a3 on B3 by A3,A17,Th26;
A43: p on B3 by A3,A6,Th26;
    b9 in Line(a,b9) by AFF_1:15;
    then
A44: a3 on B1 by A30,Th26;
    a in Line(a,b9) by AFF_1:15;
    then
A45: b1 on B1 by A30,Th26;
A46: c in Line(c,a9) by AFF_1:15;
    then
A47: b2 on C1 by A23,Th26;
    Line(b,c9) // Line(c,b9) by A20,A21,A26,AFF_1:37;
    then Line(b,c9) '||' Line(c,b9) by A22,A27,AFF_4:40;
    then
A48: c1 on B2 by A22,A27,Th28;
A49: c9 in Line(a,c9) by AFF_1:15;
    then
A50: a2 on C2 by A32,Th26;
    b9 in Line(c,b9) by AFF_1:15;
    then
A51: a3 on B2 by A27,Th26;
    c in Line(c,b9) by AFF_1:15;
    then
A52: b2 on B2 by A27,Th26;
    c9 in Line(b,c9) by AFF_1:15;
    then
A53: a2 on A1 by A22,Th26;
    b in Line(b,c9) by AFF_1:15;
    then
A54: b3 on A1 by A22,Th26;
A55: a2 on B3 by A3,A18,Th26;
    Line(a,b9) // Line(b,a9) by A19,A29,A24,AFF_1:37;
    then Line(a,b9) '||' Line(b,a9) by A30,A25,AFF_4:40;
    then
A56: c2 on A2 by A30,A25,Th28;
A57: a in Line(a,c9) by AFF_1:15;
    then
A58: b1 on C2 by A32,Th26;
    a9 in Line(b,a9) by AFF_1:15;
    then
A59: a1 on A2 by A25,Th26;
    b in Line(b,a9) by AFF_1:15;
    then
A60: b3 on A2 by A25,Th26;
A61: a9 in Line(c,a9) by AFF_1:15;
    then
A62: a1 on C1 by A23,Th26;
A63: c1 on A1 by A22,Th30;
    now
A64:  A3<>B3
      proof
        assume A3=B3;
        then M=[N,1]`1
          .= N;
        hence contradiction by A4;
      end;
      not p on C1 & not p on C2
      proof
        assume p on C1 or p on C2;
        then a1 on A3 or a2 on A3 by A7,A11,A28,A31,A35,A58,A50,A47,A62,Lm2;
        hence contradiction by A8,A12,A28,A43,A41,A55,A64,INCPROJ:def 4;
      end;
      then consider c3 being Element of the Points of XX such that
A65:  c3 on C1 and
A66:  c3 on C2 by A28,A31,A35,A43,A41,A55,A58,A50,A47,A62,A64,INCPROJ:def 8;
A67:  {a2,b1,c3} on C2 by A58,A50,A66,INCSP_1:2;
A68:  {a1,b3,c2} on A2 by A60,A59,A56,INCSP_1:2;
A69:  {a3,b1,c2} on B1 by A45,A44,A33,INCSP_1:2;
      assume that
A70:  b1<>b2 and
A71:  b2<>b3 and
A72:  b3<>b1;
A73:  p,b1,b2,b3 are_mutually_distinct by A7,A9,A11,A70,A71,A72,ZFMISC_1:def 6
;
      a1<>a2 & a2<>a3 & a1<>a3
      proof
A74:    now
          assume a9=c9;
          then a,b9 // c,b9 by A19,A20,A24,AFF_1:5;
          hence contradiction by A2,A3,A4,A5,A6,A7,A10,A13,A15,A17,A70,AFF_4:9;
        end;
        assume not thesis;
        hence contradiction by A2,A3,A4,A5,A6,A7,A9,A10,A13,A14,A15,A17,A19,A20
,A71,A72,A74,AFF_4:9;
      end;
      then
A75:  p,a1,a2,a3 are_mutually_distinct by A8,A10,A12,ZFMISC_1:def 6;
A76:  {a1,a2,a3} on B3 by A41,A55,A42,INCSP_1:2;
A77:  {b1,b2,b3} on A3 by A31,A35,A34,INCSP_1:2;
A78:  {a3,b2,c1} on B2 by A51,A52,A48,INCSP_1:2;
A79:  {a2,b3,c1} on A1 by A53,A54,A63,INCSP_1:2;
A80:  p on B3 by A3,A6,Th26;
A81:  p on A3 by A2,A5,Th26;
A82:  {c1,c2} on C39 by A39,A40,INCSP_1:1;
      {a1,b2,c3} on C1 by A47,A62,A65,INCSP_1:2;
      then c3 on C39 by A1,A75,A73,A64,A81,A80,A79,A69,A68,A78,A67,A77,A76,A82,
INCPROJ:def 14;
      then not c3 is Element of AS by Th27;
      then consider Y such that
A83:  c3=LDir(Y) and
A84:  Y is being_line by Th20;
      Y '||' Line(c,a9) by A23,A65,A83,A84,Th28;
      then
A85:  Y // Line(c,a9) by A23,A84,AFF_4:40;
      Y '||' Line(a,c9) by A32,A66,A83,A84,Th28;
      then Y // Line(a,c9) by A32,A84,AFF_4:40;
      then Line(a,c9) // Line(c,a9) by A85,AFF_1:44;
      hence thesis by A57,A49,A46,A61,AFF_1:39;
    end;
    hence thesis by A2,A3,A4,A5,A6,A9,A10,A12,A14,A15,A16,A17,A18,A19,A20,Th48;
  end;
  hence thesis by AFF_2:def 2;
end;
