reserve X, Y for non empty set;

theorem
  for r being Relation of X st r is symmetric holds chi(r,[:X,X:]) is symmetric
proof
  let r be Relation of X;
  assume r is symmetric;
  then
A1: r is_symmetric_in field r by RELAT_2:def 11;
  let x,y be Element of X;
A2: x in field r & y in field r & [x,y] in r implies [y,x] in r by A1,
RELAT_2:def 3;
A3: x in field r & y in field r & [y,x] in r implies [x,y] in r by A1,
RELAT_2:def 3;
  per cases;
  suppose
A4: [x,y] in r;
    then chi(r,[:X,X:]). [x,y] = 1 by FUNCT_3:def 3;
    hence thesis by A2,A4,FUNCT_3:def 3,RELAT_1:15;
  end;
  suppose
    not [x,y] in r;
    then ( not [y,x] in r)& chi(r,[:X,X:]). [x,y] = 0 by A3,FUNCT_3:def 3
,RELAT_1:15;
    hence thesis by FUNCT_3:def 3;
  end;
end;
