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 Th9:
  G is IncProjSp & A<>B implies ex b st b on A & b|'B & a<>b
proof
  assume that
A1: G is IncProjSp and
A2: A<>B;
  consider b,c such that
A3: {b,c} on A and
A4: a,b,c are_mutually_distinct by A1,Th8;
A5: a<>c by A4,ZFMISC_1:def 5;
A6: c on A by A3,INCSP_1:1;
A7: b<>c by A4,ZFMISC_1:def 5;
A8: b on A by A3,INCSP_1:1;
A9: a<>b by A4,ZFMISC_1:def 5;
  now
    per cases by A1,A2,A7,A8,A6,INCPROJ:def 4;
    case
      b|'B;
      hence thesis by A9,A8;
    end;
    case
      c|'B;
      hence thesis by A5,A6;
    end;
  end;
  hence thesis;
end;
