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 & R c= R9 holds max(R, converse R) c= R9
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: R c= T;
  let x,y be Element of X;
  R. [y,x] <= T. [y,x] by A2;
  then R.(y,x) <= T.(y,x);
  then
A3: R.(y,x) <= T.(x,y) by A1;
  R. [x,y] <= T. [x,y] by A2;
  then max(R.(x,y), R.(y,x)) <= T.(x,y) by A3,XXREAL_0:28;
  then max(R.(x,y), (converse R).(x,y)) <= T.(x,y) by FUZZY_4:26;
  then max(R. [x,y], (converse R). [x,y]) <= T. [x,y];
  hence thesis by FUZZY_1:def 4;
end;
