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;
reserve G for IncProjectivePlane;
reserve a,q for POINT of G;
reserve A,B for LINE of G;
reserve G for IncProjSp;
reserve a,b,c,d for POINT of G;
reserve P for LINE of G;

theorem
  a,b,c,d is_a_quadrangle implies a,b,c,d are_mutually_distinct
proof
  assume that
A1: a,b,c,d is_a_quadrangle and
A2: not a,b,c,d are_mutually_distinct;
  now
    per cases by A2,ZFMISC_1:def 6;
    case
      a=b;
      then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
      then not a,b,c is_a_triangle by Th26;
      hence contradiction by A1;
    end;
    case
      b=c;
      then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
      then not a,b,c is_a_triangle by Th26;
      hence contradiction by A1;
    end;
    case
      c =a;
      then not a,b,c are_mutually_distinct by ZFMISC_1:def 5;
      then not a,b,c is_a_triangle by Th26;
      hence contradiction by A1;
    end;
    case
      d=a;
      then not d,a,b are_mutually_distinct by ZFMISC_1:def 5;
      then not d,a,b is_a_triangle by Th26;
      hence contradiction by A1;
    end;
    case
      d=b;
      then not d,a,b are_mutually_distinct by ZFMISC_1:def 5;
      then not d,a,b is_a_triangle by Th26;
      hence contradiction by A1;
    end;
    case
      d=c;
      then not b,c,d are_mutually_distinct by ZFMISC_1:def 5;
      then not b,c,d is_a_triangle by Th26;
      hence contradiction by A1;
    end;
  end;
  hence thesis;
end;
