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

theorem Th36:
  for R being RMembership_Func of X,X holds (TrCl R)(#)(TrCl R) =
"\/"({(i iter R) (#) (j iter R) where i is Element of NAT, j is Element of NAT:
  i > 0 & j > 0},FuzzyLattice [:X,X:])
proof
  let R be RMembership_Func of X,X;
  set Q = {n iter R where n is Element of NAT : n > 0}, FL = FuzzyLattice [:X,
  X:];
A1: Q c= the carrier of FL
  proof
    let q be object;
    assume q in Q;
    then consider i being Element of NAT such that
A2: q = (i iter R) and
    i > 0;
    ([:X,X:],(i iter R) )@ = i iter R by LFUZZY_0:def 6;
    hence thesis by A2;
  end;
A3: {"\/"({@r (#) @s where s is Element of FL : s in Q},FL) where r is
Element of FL : r in Q} = {"\/"({([:X,X:],@r9 (#) @s9)@ where s9 is Element of
FL: s9 in Q},FL) where r9 is Element of FL : r9 in Q}
  proof
    deffunc G(Element of FL) = "\/"({([:X,X:],@$1 (#) @s)@ where s is Element
    of FL : s in Q},FL);
    deffunc F(Element of FL) = "\/"({@$1 (#) @s where s is Element of FL : s
    in Q},FL);
    defpred P[Element of FL] means $1 in Q;
    for r being Element of FL holds "\/"({@r (#) @s where s is Element of
FL: s in Q},FL) = "\/"({([:X,X:],@r (#) @s)@ where s is Element of FL : s in Q
    },FL)
    proof
      let r be Element of FL;
      {@r (#) @s where s is Element of FL : s in Q} = {([:X,X:],@r (#) @s
      )@ where s is Element of FL : s in Q}
      proof
        deffunc g(Element of FL) = ([:X,X:],@r (#) @$1)@;
        deffunc f(Element of FL) = @r (#) @$1;
        defpred P[Element of FL] means $1 in Q;
A4:     for s being Element of FL holds f(s) = g(s) by LFUZZY_0:def 6;
        {f(s) where s is Element of FL : P[s]} = {g(s) where s is Element
        of FL : P[s]} from FRAENKEL:sch 5 (A4);
        hence thesis;
      end;
      hence thesis;
    end;
    then
A5: for r being Element of FL holds F(r) = G(r);
    {F(r) where r is Element of FL : P[r]} = {G(r) where r is Element of
FL: P[r]} from FRAENKEL:sch 5 (A5);
    hence thesis;
  end;
  defpred R[Element of FL] means $1 in Q;
  defpred P[Element of FL] means $1 in Q;
  deffunc f(Element of FL,Element of FL) = ([:X,X:],@$1 (#) @$2)@;
A6: {@r (#) @s where r is Element of FL,s is Element of FL:r in Q & s in Q}
= {(i iter R) (#) (j iter R) where i is Element of NAT, j is Element of NAT:i >
  0 & j > 0}
  proof
    set A = {@r (#) @s where r is Element of FL,s is Element of FL:r in Q & s
in Q}, B = {(i iter R) (#) (j iter R) where i is Element of NAT, j is Element
    of NAT:i > 0 & j > 0};
    thus A c= B
    proof
      let a be object;
      assume a in A;
      then consider r,s being Element of FL such that
A7:   a = @r (#) @s and
A8:   r in Q & s in Q;
A9:   r = @r & s = @s by LFUZZY_0:def 5;
      (ex i being Element of NAT st r = i iter R & i > 0 )& ex j being
      Element of NAT st s = j iter R & j > 0 by A8;
      hence thesis by A7,A9;
    end;
    thus B c= A
    proof
      let b be object;
      assume b in B;
      then consider i,j being Element of NAT such that
A10:  b = (i iter R) (#) (j iter R) and
A11:  i > 0 & j > 0;
      (j iter R) = ([:X,X:],(j iter R))@ by LFUZZY_0:def 6;
      then reconsider s = j iter R as Element of FL;
      (i iter R) = ([:X,X:],(i iter R))@ by LFUZZY_0:def 6;
      then reconsider r = i iter R as Element of FL;
A12:  @r = r & @s = s by LFUZZY_0:def 5;
      (i iter R) in Q & (j iter R) in Q by A11;
      hence thesis by A10,A12;
    end;
  end;
A13: {([:X,X:],@r (#) @s)@ where r is Element of FL, s is Element of FL:r in
Q & s in Q} = {@r (#) @s where r is Element of FL, s is Element of FL:r in Q &
  s in Q}
  proof
    deffunc G(Element of FL,Element of FL) = @$1 (#) @$2;
    deffunc F(Element of FL,Element of FL) = ([:X,X:],@$1 (#) @$2)@;
    defpred P[Element of FL,Element of FL] means $1 in Q & $2 in Q;
A14: for r being Element of FL, s being Element of FL holds F(r,s) = G(r,s
    ) by LFUZZY_0:def 6;
    {F(r,s) where r is Element of FL,s is Element of FL:P[r,s]} ={G(r,s)
where r is Element of FL,s is Element of FL:P[r,s]} from FRAENKEL:sch 7(A14);
    hence thesis;
  end;
A15: "\/"(Q,FL) = @"\/"(Q,FL) by LFUZZY_0:def 5;
  {@r (#) (TrCl R) where r is Element of FL:r in Q} = {"\/"({@r9 (#) @s
  where s is Element of FL:s in Q},FL) where r9 is Element of FL : r9 in Q}
  proof
    set A = {@r (#) (TrCl R) where r is Element of FL:r in Q}, B = {"\/"({@r9
(#) @s where s is Element of FL:s in Q},FL) where r9 is Element of FL : r9 in Q
    };
    thus A c= B
    proof
      let a be object;
      assume a in A;
      then consider r being Element of FL such that
A16:  a = @r (#) (TrCl R) and
A17:  r in Q;
      a = "\/"({@r (#) @s where s is Element of FL:s in Q},FL) by A15,A1,A16
,Th34;
      hence thesis by A17;
    end;
    thus B c= A
    proof
      let a be object;
      assume a in B;
      then consider r being Element of FL such that
A18:  a = "\/"({@r (#) @s where s is Element of FL:s in Q},FL) and
A19:  r in Q;
      a = @r (#) (TrCl R) by A15,A1,A18,Th34;
      hence thesis by A19;
    end;
  end;
  hence
  (TrCl R)(#)(TrCl R) = "\/"({"\/"({f(r,s) where s is Element of FL:R[s]}
  ,FL) where r is Element of FL :P[r]},FL) by A15,A1,A3,Th35
    .= "\/"({f(r,s) where r is Element of FL, s is Element of FL: P[r] & R[s
  ]},FL) from LFUZZY_0:sch 1
    .= "\/"({(i iter R) (#) (j iter R) where i is Element of NAT, j is
  Element of NAT: i > 0 & j > 0},FL) by A6,A13;
end;
