
theorem Th37: :: Theorem 2.11, p. 61-62, (3) implies (2)
  for L being continuous complete LATTICE, R be Subset of [:L, L:]
, k being kernel Function of L, L st k is directed-sups-preserving & R = [:k, k
:]"(id the carrier of L) ex LR being continuous complete strict LATTICE st the
  carrier of LR = Class EqRel R & the InternalRel of LR = {[Class(EqRel R, x),
  Class(EqRel R, y)] where x, y is Element of L : k.x <= k.y } & for g being
  Function of L, LR st for x being Element of L holds g.x = Class(EqRel R, x)
  holds g is CLHomomorphism of L, LR
proof
  let L be continuous complete LATTICE, R be Subset of [:L, L:], k be kernel
  Function of L, L such that
A1: k is directed-sups-preserving and
A2: R = [:k, k:]"(id the carrier of L);
  set ER = EqRel R;
  R is Equivalence_Relation of the carrier of L by A2,Th2;
  then
A3: ER = R by Def1;
  reconsider rngk = rng k as non empty set;
  defpred P[set,set] means ex x, y being Element of L st $1 = Class(ER, x) &
  $2 = Class(ER, y) & k.x <= k.y;
  set xx = the Element of L;
  set cL = the carrier of L;
  Class(ER, xx) in Class ER by EQREL_1:def 3;
  then reconsider CER = Class ER as non empty Subset-Family of cL;
  consider LR being non empty strict RelStr such that
A4: the carrier of LR = CER and
A5: for a, b being Element of LR holds a <= b iff P[a,b] from YELLOW_0:
  sch 1;
  defpred P[set, set] means ex a being Element of cL st $1 = Class(ER, a) & $2
  = k.a;
A6: dom k = cL by FUNCT_2:def 1;
A7: for x being Element of CER ex y being Element of rngk st P[x, y]
  proof
    let x be Element of CER;
    consider y being object such that
A8: y in cL and
A9: x = Class(ER, y) by EQREL_1:def 3;
    reconsider y as Element of L by A8;
    reconsider ky = k.y as Element of rngk by A6,FUNCT_1:def 3;
    take ky;
    thus thesis by A9;
  end;
  consider f being Function of CER, rngk such that
A10: for x being Element of CER holds P[x, f.x] from FUNCT_2:sch 3(A7);
A11: dom [:k, k:] = [:cL, cL:] by A6,FUNCT_3:def 8;
A12: for a, b being Element of cL holds Class(ER, a) = Class(ER, b) iff k.a
  = k.b
  proof
    let a, b be Element of cL;
    hereby
      assume Class(ER, a) = Class(ER, b);
      then a in Class(ER, b) by EQREL_1:23;
      then [a, b] in R by A3,EQREL_1:19;
      then [:k, k:].(a, b) in id cL by A2,FUNCT_1:def 7;
      then [k.a, k.b] in id cL by A6,FUNCT_3:def 8;
      hence k.a = k.b by RELAT_1:def 10;
    end;
    assume k.a = k.b;
    then [k.a, k.b] in id cL by RELAT_1:def 10;
    then [a, b] in [:cL, cL:] & [:k, k:].(a, b) in id cL by A6,FUNCT_3:def 8
,ZFMISC_1:def 2;
    then [a, b] in ER by A2,A3,A11,FUNCT_1:def 7;
    then a in Class(ER, b) by EQREL_1:19;
    hence thesis by EQREL_1:23;
  end;
A13: for x being Element of L holds f.Class(ER, x) = k.x
  proof
    let x be Element of L;
    reconsider CERx = Class(ER, x) as Element of CER by EQREL_1:def 3;
    ex a being Element of cL st CERx = Class(ER, a) & f.CERx = k.a by A10;
    hence thesis by A12;
  end;
A14: for x being Element of LR ex a being Element of L st x = Class(ER, a)
  proof
    let x be Element of LR;
    x in CER by A4;
    then consider a being object such that
A15: a in cL and
A16: x = Class(ER, a) by EQREL_1:def 3;
    reconsider a as Element of L by A15;
    take a;
    thus thesis by A16;
  end;
  now
    let x1, x2 be object;
    assume that
A17: x1 in CER and
A18: x2 in CER and
A19: f.x1 = f.x2;
    reconsider x19 = x1 as Element of LR by A4,A17;
    consider a being Element of L such that
A20: x19 = Class(ER, a) by A14;
    reconsider x29 = x2 as Element of LR by A4,A18;
    consider b being Element of L such that
A21: x29 = Class(ER, b) by A14;
A22: f.x29 = k.b by A13,A21;
    f.x19 = k.a by A13,A20;
    hence x1 = x2 by A12,A19,A20,A21,A22;
  end;
  then
A23: f is one-to-one by FUNCT_2:19;
  set tIR = the InternalRel of LR;
A24: dom f = CER by FUNCT_2:def 1;
  reconsider f as Function of LR, Image k by A4,YELLOW_0:def 15;
  now
    let y be object;
    hereby
      assume y in rng f;
      then consider x being object such that
A25:  x in dom f and
A26:  y = f.x by FUNCT_1:def 3;
      reconsider x as Element of LR by A25;
      consider a being Element of L such that
A27:  x = Class(ER, a) by A14;
      f.x = k.a by A13,A27;
      hence y in rngk by A6,A26,FUNCT_1:def 3;
    end;
    assume y in rngk;
    then consider x being object such that
A28: x in dom k and
A29: y = k.x by FUNCT_1:def 3;
    reconsider x as Element of L by A28;
    Class(ER, x) in CER & f.Class(ER, x) = k.x by A13,EQREL_1:def 3;
    hence y in rng f by A24,A29,FUNCT_1:def 3;
  end;
  then
A30: the carrier of Image k = rngk & rng f = rngk by TARSKI:2,YELLOW_0:def 15;
  for x, y being Element of LR holds x <= y iff f.x <= f.y
  proof
    let x, y be Element of LR;
    x in CER by A4;
    then consider a being object such that
A31: a in cL and
A32: x = Class(ER, a) by EQREL_1:def 3;
    hereby
      assume x <= y;
      then consider c, d being Element of L such that
A33:  x = Class(ER, c) & y = Class(ER, d) and
A34:  k.c <= k.d by A5;
      f.x = k.c & f.y = k.d by A13,A33;
      hence f.x <= f.y by A34,YELLOW_0:60;
    end;
    reconsider a as Element of L by A31;
    assume
A35: f.x <= f.y;
    y in CER by A4;
    then consider b being object such that
A36: b in cL and
A37: y = Class(ER, b) by EQREL_1:def 3;
    reconsider b as Element of L by A36;
A38: f.y = k.b by A13,A37;
    f.x = k.a by A13,A32;
    then k.a <= k.b by A38,A35,YELLOW_0:59;
    hence thesis by A5,A32,A37;
  end;
  then
A39: f is isomorphic by A23,A30,WAYBEL_0:66;
  then
A40: LR, Image k are_isomorphic;
  then Image k, LR are_isomorphic by WAYBEL_1:6;
  then reconsider LR as non empty strict Poset by Th15,Th16,Th17;
  Image k is complete by WAYBEL_1:54;
  then reconsider LR as complete non empty strict Poset by A40,Th18,WAYBEL_1:6;
  reconsider LR as complete strict LATTICE;
  Image k is continuous LATTICE by A1,WAYBEL15:14;
  then reconsider LR as continuous complete strict LATTICE by A40,WAYBEL15:9
,WAYBEL_1:6;
  reconsider f9 = ((f qua Function)") as Function of Image k, LR by A23,A30,
FUNCT_2:25;
  set IR = {[Class(ER, x), Class(ER, y)] where x, y is Element of L : k.x <= k
  .y };
A41: f9 is isomorphic by A39,WAYBEL_0:68;
  then
A42: corestr k is infs-preserving & f9 is infs-preserving by WAYBEL13:20
,WAYBEL_1:56;
  take LR;
  thus the carrier of LR = Class ER by A4;
  now
    let z be object;
    hereby
      assume
A43:  z in tIR;
      then consider a, b being object such that
A44:  a in CER & b in CER and
A45:  z = [a, b] by A4,ZFMISC_1:def 2;
      reconsider a, b as Element of LR by A4,A44;
      a <= b by A43,A45,ORDERS_2:def 5;
      then ex x, y being Element of L st a = Class(ER, x) & b = Class(ER, y) &
      k.x <= k.y by A5;
      hence z in IR by A45;
    end;
    assume z in IR;
    then consider x, y being Element of L such that
A46: z = [Class(ER, x), Class(ER, y)] and
A47: k.x <= k.y;
    reconsider b = Class(ER, y) as Element of LR by A4,EQREL_1:def 3;
    reconsider a = Class(ER, x) as Element of LR by A4,EQREL_1:def 3;
    a <= b by A5,A47;
    hence z in tIR by A46,ORDERS_2:def 5;
  end;
  hence the InternalRel of LR = {[Class(ER, x), Class(ER, y)] where x, y is
  Element of L : k.x <= k.y } by TARSKI:2;
  let g be Function of L, LR such that
A48: for x being Element of L holds g.x = Class(ER, x);
  now
    let x be object;
    assume
A49: x in cL;
    then reconsider x9 = x as Element of L;
A50: f.Class(ER, x9) = k.x9 & Class(ER, x9) in CER by A13,EQREL_1:def 3;
    dom corestr k = cL by FUNCT_2:def 1;
    hence (f9*corestr k).x = f9.((corestr k).x) by A49,FUNCT_1:13
      .= f9.(k.x) by WAYBEL_1:30
      .= Class(ER, x9) by A24,A23,A50,FUNCT_1:32
      .= g.x by A48;
  end;
  then
A51: g = f9*corestr k by FUNCT_2:12;
A52: corestr k is directed-sups-preserving by A1,Th22;
  reconsider f9 as sups-preserving Function of Image k, LR by A41,WAYBEL13:20;
  f9 is directed-sups-preserving;
  then
A53: g is directed-sups-preserving by A51,A52,WAYBEL15:11;
  g is infs-preserving by A51,A42,Th25;
  hence thesis by A53,WAYBEL16:def 1;
end;
