reserve G for IncProjStr;
reserve a,a1,a2,b,b1,b2,c,d,p,q,r for POINT of G;
reserve A,B,C,D,M,N,P,Q,R for LINE of G;

theorem Th4:
  G is IncProjSp iff G is configuration & (for p,q ex P st {p,q} on
P) & (ex p,P st p|'P) & (for P ex a,b,c st a,b,c are_mutually_distinct & {a,b,
c} on P) & for a,b,c,d,p,M,N,P,Q st {a,b,p} on M & {c,d,p} on N & {a,c} on P &
  {b,d} on Q & p|'P & p|'Q & M<>N holds ex q st q on P,Q
proof
  hereby
    assume
A1: G is IncProjSp;
    then
    for p,q,P,Q st p on P & q on P & p on Q & q on Q holds p = q or P = Q
    by INCPROJ:def 4;
    hence G is configuration;
    thus for p,q ex P st {p,q} on P
    proof
      let p,q;
      consider P such that
A2:   p on P & q on P by A1,INCPROJ:def 5;
      take P;
      thus thesis by A2,INCSP_1:1;
    end;
    thus ex p,P st p|'P by A1,INCPROJ:def 6;
    thus for P ex a,b,c st a,b,c are_mutually_distinct & {a,b,c} on P
    proof
      let P;
      consider a,b,c such that
A3:   a<>b & b<>c & c <>a and
A4:   a on P & b on P & c on P by A1,INCPROJ:def 7;
      take a,b,c;
      thus a,b,c are_mutually_distinct by A3,ZFMISC_1:def 5;
      thus thesis by A4,INCSP_1:2;
    end;
    thus for a,b,c,d,p,M,N,P,Q st {a,b,p} on M & {c,d,p} on N & {a,c} on P & {
    b,d} on Q & p|'P & p|'Q & M<>N holds ex q st q on P,Q
    proof
      let a,b,c,d,p,M,N,P,Q such that
A5:   {a,b,p} on M and
A6:   {c,d,p} on N and
A7:   {a,c} on P and
A8:   {b,d} on Q and
A9:   p|'P & p|'Q & M<>N;
A10:  a on M & b on M by A5,INCSP_1:2;
A11:  a on P & c on P by A7,INCSP_1:1;
A12:  d on N & p on N by A6,INCSP_1:2;
A13:  b on Q & d on Q by A8,INCSP_1:1;
      p on M & c on N by A5,A6,INCSP_1:2;
      then consider q such that
A14:  q on P & q on Q by A1,A9,A10,A12,A11,A13,INCPROJ:def 8;
      take q;
      thus thesis by A14;
    end;
  end;
  hereby
    assume that
A15: G is configuration and
A16: for p,q ex P st {p,q} on P and
A17: ex p,P st p|'P and
A18: for P ex a,b,c st a,b,c are_mutually_distinct & {a,b,c} on P and
A19: for a,b,c,d,p,M,N,P,Q st {a,b,p} on M & {c,d,p} on N & {a,c} on P
    & {b,d} on Q & p|'P & p|'Q & M<>N holds ex q st q on P,Q;
A20: now
      let p,q;
      consider P such that
A21:  {p,q} on P by A16;
      take P;
      thus p on P & q on P by A21,INCSP_1:1;
    end;
A22: now
      let P;
      consider a,b,c such that
A23:  a,b,c are_mutually_distinct and
A24:  {a,b,c} on P by A18;
A25:  a<>b & b<>c by A23,ZFMISC_1:def 5;
A26:  b on P & c on P by A24,INCSP_1:2;
      c <>a & a on P by A23,A24,INCSP_1:2,ZFMISC_1:def 5;
      hence
      ex a,b,c st a<>b & b<>c & c <>a & a on P & b on P & c on P by A25,A26;
    end;
A27: for a,b,c,d,p,q,M,N,P,Q st a on M & b on M & c on N & d on N & p on M
& p on N & a on P & c on P & b on Q & d on Q & not p on P & not p on Q & M<>N
    holds ex q st q on P & q on Q
    proof
      let a,b,c,d,p,q,M,N,P,Q such that
A28:  a on M & b on M & c on N & d on N & p on M & p on N and
A29:  a on P & c on P & b on Q & d on Q and
A30:  ( not p on P)& not p on Q & M<>N;
A31:  {a,c} on P & {b,d} on Q by A29,INCSP_1:1;
      {a,b,p} on M & {c,d,p} on N by A28,INCSP_1:2;
      then consider q1 being POINT of G such that
A32:  q1 on P,Q by A19,A30,A31;
      take q1;
      thus thesis by A32;
    end;
    for p,q,P,Q st p on P & q on P & p on Q & q on Q holds p = q or P = Q
    by A15;
    hence G is IncProjSp by A17,A20,A22,A27,INCPROJ:def 4,def 5,def 6,def 7
,def 8;
  end;
end;
