reserve X, Y for non empty set;

theorem
  for r being Relation of X st r is transitive holds chi(r,[:X,X:]) is
  transitive
proof
  let r be Relation of X;
  assume
A1: r is transitive;
  let x,y,z be Element of X;
  r is_transitive_in field r by A1,RELAT_2:def 16;
  then
A2: x in field r & y in field r & z in field r & [x,y] in r & [y,z] in r
  implies [x,z] in r by RELAT_2:def 8;
  per cases;
  suppose
A3: [x,y] in r;
    thus thesis
    proof
      per cases;
      suppose
A4:     [y,z] in r;
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] = min(1,chi(r,[:X
        ,X:]). [y,z]) by A3,FUNCT_3:def 3
          .= min(1,1) by A4,FUNCT_3:def 3
          .= chi(r,[:X,X:]). [x,z] by A2,A3,A4,FUNCT_3:def 3,RELAT_1:15;
        hence thesis by LFUZZY_0:3;
      end;
      suppose
A5:     not [y,z] in r;
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] = min(1,chi(r,[:X
        ,X:]). [y,z]) by A3,FUNCT_3:def 3
          .= min(1,0) by A5,FUNCT_3:def 3
          .= 0 by XXREAL_0:def 9;
        then
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] <= chi(r,[:X,X:])
        . [x,z] by FUZZY_2:1;
        hence thesis by LFUZZY_0:3;
      end;
    end;
  end;
  suppose
A6: not [x,y] in r;
    thus thesis
    proof
      per cases;
      suppose
A7:     [y,z] in r;
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] = min(0,chi(r,[:
        X,X:]). [y,z]) by A6,FUNCT_3:def 3
          .= min(0,1) by A7,FUNCT_3:def 3
          .= 0 by XXREAL_0:def 9;
        then
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] <= chi(r,[:X,X:]
        ). [x,z] by FUZZY_2:1;
        hence thesis by LFUZZY_0:3;
      end;
      suppose
A8:     not [y,z] in r;
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] = min(0,chi(r,[:
        X,X:]). [y,z]) by A6,FUNCT_3:def 3
          .= min(0,0) by A8,FUNCT_3:def 3
          .= 0;
        then
        chi(r,[:X,X:]). [x,y] "/\" chi(r,[:X,X:]). [y,z] <= chi(r,[:X,X:]
        ). [x,z] by FUZZY_2:1;
        hence thesis by LFUZZY_0:3;
      end;
    end;
  end;
end;
