reserve X, Y for non empty set;
reserve X for non empty set;
reserve R for RMembership_Func of X,X;

theorem Th22:
  Imf(X,X) (#) R = R
proof
A1: for x,y being object st [x,y] in dom (Imf(X,X) (#) R)
holds (Imf(X,X) (#) R).(x,y) = R.(x,y)
  proof
    let x,y be object;
    assume [x,y] in dom (Imf(X,X) (#) R);
    then reconsider x,y as Element of X by ZFMISC_1:87;
    set S = the set of all Imf(X,X).(x,z) "/\" R.(z,y) where z is Element of X;
    for c being Element of RealPoset [. 0,1 .] st c in S holds c <<= R. [x ,y]
    proof
      let c be Element of RealPoset [. 0,1 .];
      assume c in S;
      then consider z being Element of X such that
A2:   c = Imf(X,X).(x,z) "/\" R.(z,y);
      per cases;
      suppose
A3:     x = z;
A4:     R.(z,y) <= 1 by FUZZY_4:4;
        c = min(R. [z,y],1) by A2,A3,FUZZY_4:25
          .= R. [x,y] by A3,A4,XXREAL_0:def 9;
        hence thesis by LFUZZY_0:3;
      end;
      suppose
A5:     x <> z;
A6:     0 <= R.(z,y) by FUZZY_4:4;
        c = min(R. [z,y],0) by A2,A5,FUZZY_4:25
          .= 0 by A6,XXREAL_0:def 9;
        then c <= R.(x,y) by FUZZY_4:4;
        hence thesis by LFUZZY_0:3;
      end;
    end;
    then
A7: (Imf(X,X) (#) R).(x,y) = "\/"((the set of all
Imf(X,X).(x,z) "/\" R.(z,y) where z is
Element of X), RealPoset [. 0,1 .]) & R.(x,y) is_>=_than S
    by LATTICE3:def 9,LFUZZY_0:22;
    for b being Element of RealPoset [. 0,1 .] st b is_>=_than S holds R.
    (x,y) <<= b
    proof
      let b be Element of RealPoset [. 0,1 .];
A8:   R.(x,y) <= 1 by FUZZY_4:4;
      Imf(X,X).(x,x) "/\" R. [x,y] = min(1, R.(x,y)) by FUZZY_4:25
        .= R. [x,y] by A8,XXREAL_0:def 9;
      then
A9:   R.(x,y) in S;
      assume b is_>=_than S;
      hence thesis by A9,LATTICE3:def 9;
    end;
    hence thesis by A7,YELLOW_0:32;
  end;
  dom (Imf(X,X) (#) R) = [:X,X:] by FUNCT_2:def 1
    .= dom R by FUNCT_2:def 1;
  hence thesis by A1,BINOP_1:20;
end;
