reserve X, Y for non empty set;

theorem Th11:
  for R,S being RMembership_Func of X,X holds R is transitive & S
  is transitive implies min(R,S) (#) min(R,S) c= min(R,S)
proof
  let R,S be RMembership_Func of X,X;
  assume that
A1: R is transitive and
A2: S is transitive;
  let x be Element of X, y be Element of X;
  min(R(#)S, S(#)S). [x,y] = min((S(#)S). [x,y], (R(#)S). [x,y]) by
FUZZY_1:def 3;
  then
A3: min(R(#)S, S(#)S). [x,y] <= (S(#)S). [x,y] by XXREAL_0:17;
  S(#)S c= S by A2;
  then (S(#)S).(x,y) <= S.(x,y);
  then
A4: min(R(#)S, S(#)S). [x,y] <= S. [x,y] by A3,XXREAL_0:2;
  min(R(#)R, S(#)R). [x,y] = min((R(#)R). [x,y], (S(#)R). [x,y]) by
FUZZY_1:def 3;
  then
A5: min(R(#)R, S(#)R). [x,y] <= (R(#)R). [x,y] by XXREAL_0:17;
  R(#)R c= R by A1;
  then (R(#)R).(x,y) <= R.(x,y);
  then min(R(#)R, S(#)R). [x,y] <= R. [x,y] by A5,XXREAL_0:2;
  then
A6: min(min(R(#)R, S(#)R). [x,y],min(R(#)S, S(#)S). [x,y]) <= min(R. [x,y],
  S. [x,y]) by A4,XXREAL_0:18;
  (min(R,S) (#) min(R,S)) c= min(min(R,S) (#) R, min(R,S) (#) S) by FUZZY_4:15;
  then
A7: (min(R,S) (#) min(R,S)). [x,y] <= min(min(R,S) (#) R, min(R,S) (#) S). [
  x,y];
  min(R,S) (#) S c= min(R(#)S, S(#)S) by FUZZY_4:16;
  then
A8: (min(R,S) (#) S). [x,y] <= min(R(#)S, S(#)S). [x,y];
  min(R,S) (#) R c= min(R(#)R, S(#)R) by FUZZY_4:16;
  then
  min(min(R,S) (#) R, min(R,S) (#) S). [x,y] = min((min(R,S) (#) R). [x,y]
, ( min(R,S) (#) S). [x,y]) & (min(R,S) (#) R). [x,y] <= min(R(#)R, S(#)R). [x,
  y] by FUZZY_1:def 3;
  then
  min(min(R,S) (#) R, min(R,S) (#) S). [x,y] <= min(min(R(#)R, S(#)R). [x
  ,y],min(R(#)S, S(#)S). [x,y]) by A8,XXREAL_0:18;
  then
  (min(R,S) (#) min(R,S)). [x,y] <= min(min(R(#)R, S(#)R). [x,y],min(R(#)
  S, S(#)S). [x,y]) by A7,XXREAL_0:2;
  then (min(R,S) (#) min(R,S)). [x,y] <= min(R. [x,y], S. [x,y]) by A6,
XXREAL_0:2;
  hence thesis by FUZZY_1:def 3;
end;
