reserve X, Y for non empty set;

theorem
  for R being RMembership_Func of X,X holds min(R,converse R) is symmetric
proof
  let R be RMembership_Func of X,X;
  set S = min(R,converse R);
  let x,y be Element of X;
  thus S.(x,y) = S. [x,y] .= min(R.(x,y), (converse R).(x,y)) by FUZZY_1:def 3
    .= min(R.(x,y), R.(y,x)) by FUZZY_4:26
    .= min((converse R).(y,x), R.(y,x)) by FUZZY_4:26
    .= S. [y,x] by FUZZY_1:def 3
    .= S.(y,x);
end;
