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;
