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 Fanoian implies AS is Fanoian
proof
  set XX=IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Fanoian;
  for a,b,c,d being Element of AS st a,b // c,d & a,c // b,d & a,d // b,c
  holds a,b // a,c
  proof
    let a,b,c,d be Element of AS such that
A2: a,b // c,d and
A3: a,c // b,d and
A4: a,d // b,c;
    assume
A5: not a,b // a,c;
    then
A6: a<>d by A2,AFF_1:4;
    then
A7: Line(a,d) is being_line by AFF_1:def 3;
A8: now
      assume b=d;
      then b,a // b,c by A2,AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
    then
A9: Line(b,d) is being_line by AFF_1:def 3;
A10: now
      assume c =d;
      then c,a // c,b by A3,AFF_1:4;
      then LIN c,a,b by AFF_1:def 1;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
    then
A11: Line(c,d) is being_line by AFF_1:def 3;
A12: a<>c by A5,AFF_1:3;
    then
A13: Line(a,c) is being_line by AFF_1:def 3;
A14: a<>b by A5,AFF_1:3;
    then
A15: Line(a,b) is being_line by AFF_1:def 3;
    then reconsider
    a9=LDir(Line(a,b)),b9=LDir(Line(a,c)),c9=LDir(Line(a,d)) as
    Element of the Points of XX by A13,A7,Th20;
A16: b<>c by A5,AFF_1:2;
    then
A17: Line(b,c) is being_line by AFF_1:def 3;
    then reconsider
    L1=[Line(a,b),1],Q1=[Line(c,d),1],R1=[Line(b,d),1],S1=[Line(a,c
),1], A1=[Line(a,d),1],B1=[Line(b,c),1] as Element of the Lines of XX by A15
,A11,A9,A13,A7,Th23;
    reconsider p=a,q=d,r=c,s=b as Element of the Points of XX by Th20;
A18: a9 on L1 by A15,Th30;
    c in Line(b,c) by AFF_1:15;
    then
A19: r on B1 by A17,Th26;
    b in Line(b,c) by AFF_1:15;
    then
A20: s on B1 by A17,Th26;
    Line(a,d) // Line(b,c) by A4,A16,A6,AFF_1:37;
    then Line(a,d) '||' Line(b,c) by A7,A17,AFF_4:40;
    then c9 on B1 by A7,A17,Th28;
    then
A21: {c9,r,s} on B1 by A19,A20,INCSP_1:2;
A22: d in Line(b,d) by AFF_1:15;
    then
A23: q on R1 by A9,Th26;
A24: c in Line(a,c) by AFF_1:15;
    then
A25: r on S1 by A13,Th26;
A26: b9 on S1 by A13,Th30;
A27: a in Line(a,c) by AFF_1:15;
    then p on S1 by A13,Th26;
    then
A28: {b9,p,r} on S1 by A25,A26,INCSP_1:2;
A29: Line(a,c) // Line(b,d) by A3,A12,A8,AFF_1:37;
    then Line(a,c) '||' Line(b,d) by A9,A13,AFF_4:40;
    then
A30: b9 on R1 by A9,A13,Th28;
A31: b in Line(b,d) by AFF_1:15;
    then s on R1 by A9,Th26;
    then
A32: {b9,q,s} on R1 by A23,A30,INCSP_1:2;
A33: now
      assume Line(a,c)=Line(b,d);
      then LIN a,c,b by A31,AFF_1:def 2;
      then LIN a,b,c by AFF_1:6;
      hence contradiction by A5,AFF_1:def 1;
    end;
A34: now
      assume q on S1 or s on S1;
      then d in Line(a,c) or b in Line(a,c) by Th26;
      hence contradiction by A31,A22,A33,A29,AFF_1:45;
    end;
A35: now
      assume p on R1 or r on R1;
      then a in Line(b,d) or c in Line(b,d) by Th26;
      hence contradiction by A27,A24,A33,A29,AFF_1:45;
    end;
A36: a in Line(a,b) by AFF_1:15;
    then consider Y such that
A37: Line(a,b) c= Y and
A38: Line(a,c) c= Y and
A39: Y is being_plane by A27,A15,A13,AFF_4:38;
    reconsider C1=[PDir(Y),2] as Element of the Lines of XX by A39,Th23;
A40: b9 on C1 by A13,A38,A39,Th31;
A41: Line(a,b) // Line(c,d) by A2,A14,A10,AFF_1:37;
    then Line(a,b) '||' Line(c,d) by A15,A11,AFF_4:40;
    then
A42: a9 on Q1 by A15,A11,Th28;
    d in Line(a,d) by AFF_1:15;
    then
A43: q on A1 by A7,Th26;
    a in Line(a,d) by AFF_1:15;
    then
A44: p on A1 by A7,Th26;
    c9 on A1 by A7,Th30;
    then
A45: {c9,p,q} on A1 by A44,A43,INCSP_1:2;
A46: b in Line(a,b) by AFF_1:15;
    then
A47: s on L1 by A15,Th26;
    a9 on C1 by A15,A37,A39,Th31;
    then
A48: {a9,b9} on C1 by A40,INCSP_1:1;
A49: d in Line(c,d) by AFF_1:15;
    then
A50: q on Q1 by A11,Th26;
A51: c in Line(c,d) by AFF_1:15;
    then r on Q1 by A11,Th26;
    then
A52: {a9,q,r} on Q1 by A50,A42,INCSP_1:2;
A53: now
      assume Line(a,b)=Line(c,d);
      then LIN a,b,c by A51,AFF_1:def 2;
      hence contradiction by A5,AFF_1:def 1;
    end;
A54: now
      assume q on L1 or r on L1;
      then d in Line(a,b) or c in Line(a,b) by Th26;
      hence contradiction by A51,A49,A53,A41,AFF_1:45;
    end;
A55: now
      assume p on Q1 or s on Q1;
      then a in Line(c,d) or b in Line(c,d) by Th26;
      hence contradiction by A36,A46,A53,A41,AFF_1:45;
    end;
    Line(b,d)=b*Line(a,c) by A31,A13,A29,AFF_4:def 3;
    then Line(b,d) c= Y by A46,A13,A37,A38,A39,AFF_4:28;
    then
A56: c9 on C1 by A36,A22,A6,A7,A37,A39,Th31,AFF_4:19;
    p on L1 by A36,A15,Th26;
    then {a9,p,s} on L1 by A47,A18,INCSP_1:2;
    hence contradiction by A1,A56,A54,A34,A55,A35,A52,A32,A28,A45,A21,A48,
INCPROJ:def 12;
  end;
  hence thesis by PAPDESAF:def 1;
end;
