reserve X, Y for non empty set;

theorem Th6:
  for X,Y,Z being non empty set, R,S being RMembership_Func of X,Y,
T,U being RMembership_Func of Y,Z holds R c= S & T c= U implies R(#)T c= S(#)U
proof
  let X,Y,Z be non empty set;
  let R,S be RMembership_Func of X,Y;
  let T,U be RMembership_Func of Y,Z;
  assume
A1: R c= S & T c= U;
  for c being Element of [:X,Z:] holds (R(#)T).c <= (S(#)U).c
  proof
    let c be Element of [:X,Z:];
    consider x,z being object such that
A2: [x,z] = c by RELAT_1:def 1;
A3: x in X & z in Z by A2,ZFMISC_1:87;
    for y be set st y in Y holds R. [x,y] <= S. [x,y] & T. [y,z] <= U. [y, z]
    proof
      let y be set;
      assume y in Y;
      then [x,y] in [:X,Y:] & [y,z] in [:Y,Z:] by A3,ZFMISC_1:87;
      hence thesis by A1;
    end;
    hence thesis by A2,A3,FUZZY_4:18;
  end;
  hence thesis;
end;
