reserve X, Y for non empty set;

theorem
  for R being RMembership_Func of X,X for R9 being RMembership_Func of X
  ,X st R9 is symmetric & R9 c= R holds R9 c= min(R, converse R)
proof
  let R be RMembership_Func of X,X;
  let T be RMembership_Func of X,X;
  assume that
A1: T is symmetric and
A2: T c= R;
  let x,y be Element of X;
  T. [y,x] <= R. [y,x] by A2;
  then T.(y,x) <= R.(y,x);
  then
A3: T.(x,y) <= R.(y,x) by A1;
  T. [x,y] <= R. [x,y] by A2;
  then T.(x,y) <= min(R.(x,y), R.(y,x)) by A3,XXREAL_0:20;
  then T.(x,y) <= min(R.(x,y), (converse R).(x,y)) by FUZZY_4:26;
  then T. [x,y] <= min(R. [x,y], (converse R). [x,y]);
  hence thesis by FUZZY_1:def 3;
end;
