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 Desarguesian implies AS is Desarguesian
proof
  set XX= IncProjSp_of(AS);
  assume
A1: IncProjSp_of(AS) is Desarguesian;
  for A,P,C being Subset of AS, o,a,b,c,a9,b9,c9 being Element
 of AS st o in A & o in P & o in C & o<>a & o<>b & o<>c & a in A & a9 in
A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C
  is being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9 holds b,c // b9,c9
  proof
    let A,P,C be Subset of AS, o,a,b,c,a9,b9,c9 be Element of
    AS such that
A2: o in A and
A3: o in P and
A4: o in C and
A5: o<>a and
A6: o<>b and
A7: o<>c and
A8: a in A and
A9: a9 in A and
A10: b in P and
A11: b9 in P and
A12: c in C and
A13: c9 in C and
A14: A is being_line and
A15: P is being_line and
A16: C is being_line and
A17: A<>P and
A18: A<>C and
A19: a,b // a9,b9 and
A20: a,c // a9,c9;
    now
      assume
A21:  P<>C;
      now
        reconsider p=o,a1=a,b1=a9,a2=b,b2=b9,a3=c,b3=c9 as Element of the
        Points of XX by Th20;
        reconsider C1=[A,1],C2=[P,1],C39=[C,1] as Element of the Lines of XX
        by A14,A15,A16,Th23;
        assume that
A22:    a<>a9 and
A23:    o<>a9;
A24:    o<>c9 by A2,A4,A5,A7,A8,A9,A12,A14,A16,A18,A20,A23,AFF_4:8;
A25:    a9<>c9 by A2,A4,A9,A13,A14,A16,A18,A23,AFF_1:18;
        then
A26:    Line(a9,c9) is being_line by AFF_1:def 3;
A27:    o<>b9 by A2,A3,A5,A6,A8,A9,A10,A14,A15,A17,A19,A23,AFF_4:8;
        then b9<>c9 by A3,A4,A11,A13,A15,A16,A21,AFF_1:18;
        then
A28:    Line(b9,c9) is being_line by AFF_1:def 3;
        b<>c by A3,A4,A6,A10,A12,A15,A16,A21,AFF_1:18;
        then
A29:    Line(b,c) is being_line by AFF_1:def 3;
A30:    a<>c by A2,A4,A5,A8,A12,A14,A16,A18,AFF_1:18;
        then
A31:    Line(a,c) is being_line by AFF_1:def 3;
A32:    a<>b by A2,A3,A5,A8,A10,A14,A15,A17,AFF_1:18;
        then
A33:    Line(a,b) is being_line by AFF_1:def 3;
        then reconsider s=LDir(Line(a,b)),r=LDir(Line(a,c)) as Element of the
        Points of XX by A31,Th20;
A34:    p on C2 by A3,A15,Th26;
A35:    a9<>b9 by A2,A3,A9,A11,A14,A15,A17,A23,AFF_1:18;
        then
A36:    Line(a9,b9) is being_line by AFF_1:def 3;
        then reconsider
        A1=[Line(b,c),1],A2=[Line(a,c),1],A3=[Line(a,b),1], B1=[
Line(b9,c9),1],B2=[Line(a9,c9),1],B3=[Line(a9,b9),1] as Element of the Lines of
        XX by A33,A29,A31,A28,A26,Th23;
A37:    r on A2 by A31,Th30;
A38:    c in Line(b,c) by AFF_1:15;
        then
A39:    a3 on A1 by A29,Th26;
A40:    a3 on A1 by A29,A38,Th26;
A41:    c9 in Line(a9,c9) by AFF_1:15;
        then
A42:    b3 on B2 by A26,Th26;
A43:    a9 in Line(a9,c9) by AFF_1:15;
        then
A44:    b1 on B2 by A26,Th26;
A45:    Line(a,c) // Line(a9,c9) by A20,A30,A25,AFF_1:37;
        then Line(a,c) '||' Line(a9,c9) by A31,A26,AFF_4:40;
        then r on B2 by A31,A26,Th28;
        then
A46:    {b1,r,b3} on B2 by A44,A42,INCSP_1:2;
A47:    c <>c9 by A2,A4,A5,A7,A8,A9,A12,A14,A16,A18,A20,A22,AFF_4:9;
A48:    b1 on C1 by A9,A14,Th26;
A49:    a3 on C39 by A12,A16,Th26;
A50:    b9 in Line(a9,b9) by AFF_1:15;
        then
A51:    b2 on B3 by A36,Th26;
A52:    a9 in Line(a9,b9) by AFF_1:15;
        then
A53:    b1 on B3 by A36,Th26;
A54:    Line(a,b) // Line(a9,b9) by A19,A32,A35,AFF_1:37;
        then Line(a,b) '||' Line(a9,b9) by A33,A36,AFF_4:40;
        then s on B3 by A33,A36,Th28;
        then
A55:    {b1,s,b2} on B3 by A53,A51,INCSP_1:2;
A56:    now
          assume C2=C39;
          then P=[C,1]`1
            .=C;
          hence contradiction by A21;
        end;
A57:    now
          assume C1=C39;
          then A=[C,1]`1
            .=C;
          hence contradiction by A18;
        end;
        now
          assume C1=C2;
          then A=[P,1]`1
            .=P;
          hence contradiction by A17;
        end;
        then
A58:    C1,C2,C39 are_mutually_distinct by A56,A57,ZFMISC_1:def 5;
A59:    a1 on C1 by A8,A14,Th26;
A60:    b3 on C39 by A13,A16,Th26;
A61:    p on C39 by A4,A16,Th26;
        then
A62:    {p,a3,b3} on C39 by A49,A60,INCSP_1:2;
        p on C1 by A2,A14,Th26;
        then
A63:    {p,b1,a1} on C1 by A48,A59,INCSP_1:2;
A64:    b2 on C2 by A11,A15,Th26;
A65:    a in Line(a,c) by AFF_1:15;
        then
A66:    a1 on A2 by A31,Th26;
A67:    c in Line(a,c) by AFF_1:15;
        then a3 on A2 by A31,Th26;
        then
A68:    {a3,r,a1} on A2 by A37,A66,INCSP_1:2;
A69:    b9 in Line(b9,c9) by AFF_1:15;
        then
A70:    b2 on B1 by A28,Th26;
A71:    c9 in Line(b9,c9) by AFF_1:15;
        then
A72:    b3 on B1 by A28,Th26;
A73:    b3 on B1 by A28,A71,Th26;
A74:    a2 on C2 by A10,A15,Th26;
        then
A75:    {p,a2,b2} on C2 by A34,A64,INCSP_1:2;
A76:    b in Line(b,c) by AFF_1:15;
        then
A77:    a2 on A1 by A29,Th26;
        not p on A1 & not p on B1
        proof
          assume p on A1 or p on B1;
          then a3 on C2 or b3 on C2 by A6,A27,A34,A74,A64,A77,A40,A70,A73,
INCPROJ:def 4;
          hence contradiction by A7,A24,A34,A61,A49,A60,A56,INCPROJ:def 4;
        end;
        then consider t being Element of the Points of XX such that
A78:    t on A1 and
A79:    t on B1 by A34,A61,A74,A64,A49,A60,A77,A40,A70,A73,A56,INCPROJ:def 8;
        a2 on A1 by A29,A76,Th26;
        then
A80:    {a3,a2,t} on A1 by A78,A39,INCSP_1:2;
        b2 on B1 by A28,A69,Th26;
        then
A81:    {t,b2,b3} on B1 by A79,A72,INCSP_1:2;
A82:    a in Line(a,b) by AFF_1:15;
        then
A83:    a1 on A3 by A33,Th26;
A84:    s on A3 by A33,Th30;
A85:    b in Line(a,b) by AFF_1:15;
        then a2 on A3 by A33,Th26;
        then
A86:    {a2,s,a1} on A3 by A84,A83,INCSP_1:2;
        b<>b9 by A2,A3,A5,A6,A8,A9,A10,A14,A15,A17,A19,A22,AFF_4:9;
        then consider O being Element of the Lines of XX such that
A87:    {r,s,t} on O by A1,A5,A6,A7,A22,A23,A27,A24,A47,A63,A75,A62,A80,A68,A86
,A81,A46,A55,A58,INCPROJ:def 13;
A88:    t on O by A87,INCSP_1:2;
A89:    s on O by A87,INCSP_1:2;
A90:    r on O by A87,INCSP_1:2;
A91:    now
          assume
A92:      r<>s;
          ex X st O=[PDir(X),2] & X is being_plane
          proof
            reconsider x=LDir(Line(a,b)),y=LDir(Line(a,c)) as Element of the
            Points of ProjHorizon(AS) by A33,A31,Th14;
A93:        [y,O] in the Inc of IncProjSp_of(AS) by A90,INCSP_1:def 1;
            [x,O] in the Inc of IncProjSp_of(AS) by A89,INCSP_1:def 1;
            then consider
            Z9 being Element of the Lines of ProjHorizon(AS) such
            that
A94:        O=[Z9,2] by A92,A93,Th41;
            consider X such that
A95:        Z9=PDir(X) and
A96:        X is being_plane by Th15;
            take X;
            thus thesis by A94,A95,A96;
          end;
          then not t is Element of AS by A88,Th27;
          then consider Y such that
A97:      t=LDir(Y) and
A98:      Y is being_line by Th20;
          Y '||' Line(b9,c9) by A28,A79,A97,A98,Th28;
          then
A99:      Y // Line(b9,c9) by A28,A98,AFF_4:40;
          Y '||' Line(b,c) by A29,A78,A97,A98,Th28;
          then Y // Line(b,c) by A29,A98,AFF_4:40;
          then Line(b,c) // Line(b9,c9) by A99,AFF_1:44;
          hence thesis by A76,A38,A69,A71,AFF_1:39;
        end;
        now
          assume r=s;
          then
A100:     Line(a,b) // Line(a,c) by A33,A31,Th11;
          then Line(a,b) // Line(a9,c9) by A45,AFF_1:44;
          then Line(a9,b9) // Line(a9, c9) by A54,AFF_1:44;
          then
A101:     c9 in Line(a9,b9) by A52,A43,A41,AFF_1:45;
          c in Line(a,b) by A82,A65,A67,A100,AFF_1:45;
          hence thesis by A85,A50,A54,A101,AFF_1:39;
        end;
        hence thesis by A91;
      end;
      hence
      thesis by A2,A3,A4,A5,A6,A7,A8,A10,A11,A12,A13,A14,A15,A16,A17,A18,A19
,A20,Th50;
    end;
    hence thesis by A10,A11,A12,A13,A15,AFF_1:51;
  end;
  hence thesis by AFF_2:def 4;
end;
