reserve x,y,z for Real,
  R for real non empty RelStr,
  a,b for Element of R;
reserve C for non empty set,
  c for Element of C,
  f,g for Membership_Func of C,
  s,t for Element of FuzzyLattice C;

theorem Th22:
  for X,Y,Z being non empty set for R being RMembership_Func of X,
Y for S being RMembership_Func of Y,Z for x being Element of X, z being Element
of Z holds (R (#) S).(x,z) =
"\/"((the set of all R.(x,y) "/\" S.(y,z) where y is Element of Y),
RealPoset [. 0,1 .])
proof
  let X,Y,Z being non empty set;
  let R being RMembership_Func of X,Y;
  let S being RMembership_Func of Y,Z;
  let x being Element of X, z being Element of Z;
  set L = the set of all R.(x,y) "/\" S.(y,z) where y is Element of Y;
  [x,z] in [:X,Z:];
  then
A1: (R (#) S).(x,z) = upper_bound(rng(min(R,S,x,z))) by FUZZY_4:def 3;
A2: for b being Element of RealPoset [. 0,1 .] st b is_>=_than L holds (R
  (#) S).(x,z) <<= b
  proof
    let b be Element of RealPoset [. 0,1 .];
    assume
A3: b is_>=_than the set of all R.(x,y) "/\" S.(y,z) where y is Element of Y;
A4: rng min(R,S,x,z) c= [. 0,1 .] by RELAT_1:def 19;
A5: L = rng min(R,S,x,z) by Lm4;
A6: for r being Real st r in rng min(R,S,x,z) holds r <= b
    proof
      let r be Real;
      assume
A7:   r in rng min(R,S,x,z);
      then reconsider r as Element of RealPoset [. 0,1 .] by A4,Def3;
      r <<= b by A3,A5,A7;
      hence thesis by Th3;
    end;
    rng min(R,S,x,z) <> {} by Lm4;
    then upper_bound rng min(R,S,x,z) <= b by A6,SEQ_4:144;
    hence thesis by A1,Th3;
  end;
  for b being Element of RealPoset [. 0,1 .] st b in L holds (R (#) S). [x
  ,z] >>= b
  proof
    let b be Element of RealPoset [. 0,1 .];
    assume b in L;
    then consider y being Element of Y such that
A8: b = R.(x,y) "/\" S.(y,z);
    reconsider b as Real;
    dom min(R,S,x,z) = Y & b = min(R,S,x,z).y
    by A8,FUNCT_2:def 1,FUZZY_4:def 2;
    then b <= upper_bound rng min(R,S,x,z) by FUZZY_4:1;
    hence thesis by A1,Th3;
  end;
  then (R (#) S). [x,z] is_>=_than L;
  hence thesis by A2,YELLOW_0:32;
end;
