
theorem Th26:
  for A being partial non-empty UAStr, P being a_partition of A holds
  P is_finer_than Class LimDomRel A
proof
  let A be partial non-empty UAStr;
  let P be a_partition of A;
  consider EP being Equivalence_Relation of the carrier of A such that
A1: P = Class EP by EQREL_1:34;
  let a be set;
  assume a in P;
  then reconsider aa = a as Element of P;
  set x = the Element of aa;
  take Class(LimDomRel A, x);
  thus Class(LimDomRel A, x) in Class LimDomRel A by EQREL_1:def 3;
  let y be object;
  assume y in a;
  then reconsider y as Element of aa;
  reconsider x,y as Element of A;
  defpred P[Nat] means EP c= (DomRel A)|^(A,$1);
A2: P[ 0 ]
  proof
    let x,y be object;
    assume
A3: [x,y] in EP;
    then reconsider x,y as Element of A by ZFMISC_1:87;
    reconsider a = Class(EP, y) as Element of P by A1,EQREL_1:def 3;
A4: x in a by A3,EQREL_1:19;
A5: y in a by EQREL_1:20;
    for f being operation of A, p,q being FinSequence holds
    p^<*x*>^q in dom f iff p^<*y*>^q in dom f
    proof
      let f be operation of A, p,q be FinSequence;
A6:   f is_exactly_partitable_wrt P by Def10;
      now
        let p,q be FinSequence, x,y be Element of a;
        assume
A7:     p^<*x*>^q in dom f;
        then p^<*x*>^q is FinSequence of the carrier of A by FINSEQ_1:def 11;
        then consider pp being FinSequence of P such that
A8:     p^<*x*>^q in product pp by Th6;
        product pp meets dom f by A7,A8,XBOOLE_0:3;
        then
A9:     product pp c= dom f by A6;
        p^<*y*>^q in product pp by A8,Th25;
        hence p^<*y*>^q in dom f by A9;
      end;
      hence thesis by A4,A5;
    end;
    then [x,y] in DomRel A by Def4;
    hence thesis by Th15;
  end;
A10: for i being Nat st P[i] holds P[i+1]
  proof
    let i be Nat;
    assume
A11: EP c= (DomRel A)|^(A,i);
    let x,y be object;
    assume
A12: [x,y] in EP;
    then reconsider x,y as Element of A by ZFMISC_1:87;
    reconsider a = Class(EP, y) as Element of P by A1,EQREL_1:def 3;
    now
      let f be operation of A, p,q be FinSequence;
      assume that
A13:  p^<*x*>^q in dom f and
A14:  p^<*y*>^q in dom f;
      p^<*x*>^q is FinSequence of the carrier of A by A13,FINSEQ_1:def 11;
      then consider pp being FinSequence of P such that
A15:  p^<*x*>^q in product pp by Th6;
      f is_exactly_partitable_wrt P by Def10;
      then f is_partitable_wrt P;
      then consider c being Element of P such that
A16:  f.:product pp c= c;
A17:  x in a by A12,EQREL_1:19;
      y in a by EQREL_1:20;
      then
A18:  p^<*y*>^q in product pp by A15,A17,Th25;
A19:  f.(p^<*x*>^q) in f.:product pp by A13,A15,FUNCT_1:def 6;
A20:  f.(p^<*y*>^q) in f.:product pp by A14,A18,FUNCT_1:def 6;
      ex x being object st x in the carrier of A & c = Class(EP,x) by A1,
EQREL_1:def 3;
      then [f.(p^<*x*>^q), f.(p^<*y*>^q)] in EP by A16,A19,A20,EQREL_1:22;
      hence [f.(p^<*x*>^q), f.(p^<*y*>^q)] in (DomRel A)|^(A,i) by A11;
    end;
    then [x,y] in (DomRel A)|^(A,i)|^A by A11,A12,Def5;
    hence thesis by Th16;
  end;
A21: for i being Nat holds P[i] from NAT_1:sch 2(A2,A10);
  now
    let i be Element of NAT;
    ex x being object st x in the carrier of A & aa = Class(EP, x)
    by A1,EQREL_1:def 3;
    then
A22: [x,y] in EP by EQREL_1:22;
    EP c= (DomRel A)|^(A,i) by A21;
    hence [x,y] in (DomRel A)|^(A,i) by A22;
  end;
  then [x,y] in LimDomRel A by Def7;
  then [y,x] in LimDomRel A by EQREL_1:6;
  hence thesis by EQREL_1:19;
end;
