
theorem

:: Theorem 2.14, p. 63, see WAYBEL15:17
  for L being continuous complete LATTICE, k being
  directed-sups-preserving kernel Function of L, L holds k = kernel_op
  kernel_congruence k
proof
  let L be continuous complete LATTICE, k be directed-sups-preserving kernel
  Function of L, L;
  set kc = kernel_congruence k, cL = the carrier of L, km = kernel_op kc;
A1: dom k = cL by FUNCT_2:def 1;
A2: km <= id L by WAYBEL_1:def 15;
A3: k <= id L by WAYBEL_1:def 15;
A4: kc is CLCongruence by Th40;
  then
A5: kc = [:km, km:]"(id cL) by Th36;
  reconsider kc9 = kc as Equivalence_Relation of cL by A4;
  field kc9 = cL by ORDERS_1:12;
  then
A6: kc9 is_transitive_in cL by RELAT_2:def 16;
A7: dom [:km, km:] = [:dom km, dom km:] by FUNCT_3:def 8;
A8: dom km = cL by FUNCT_2:def 1;
A9: dom [:k, k:] = [:dom k, dom k:] by FUNCT_3:def 8;
  now
    let x be object;
    assume x in cL;
    then reconsider x9 = x as Element of L;
A10: k.(k.x9) = (k*k).x9 by A1,FUNCT_1:13
      .= k.x9 by QUANTAL1:def 9;
A11: [k.x9, k.x9] in id cL & [:k, k:].(k.x9, x9) = [k.(k.x9), k.x9] by A1,
FUNCT_3:def 8,RELAT_1:def 10;
    [k.x9, x9] in dom [:k, k:] by A1,A9,ZFMISC_1:def 2;
    then
A12: [k.x9, x9] in kc by A10,A11,FUNCT_1:def 7;
A13: km.(km.x9) = (km*km).x9 by A8,FUNCT_1:13
      .= km.x9 by QUANTAL1:def 9;
    km.(k.x9) <= (id L).(k.x9) by A2,YELLOW_2:9;
    then
A14: km.(k.x9) <= k.x9 by FUNCT_1:18;
A15: [km.x9, km.x9] in id cL & [:km, km:].(x9, km.x9) = [km.x9, km.(km.x9)
    ] by A8,FUNCT_3:def 8,RELAT_1:def 10;
    [x9, km.x9] in dom [:km, km:] by A8,A7,ZFMISC_1:def 2;
    then [x9, km.x9] in kc by A5,A13,A15,FUNCT_1:def 7;
    then
A16: [k.x9, km.x9] in kc by A6,A12;
    then [:k, k:].(k.x9, km.x9) in id cL by FUNCT_1:def 7;
    then [k.(k.x9), k.(km.x9)] in id cL by A1,FUNCT_3:def 8;
    then
A17: k.(km.x9) = k.(k.x9) by RELAT_1:def 10
      .= (k*k).x9 by A1,FUNCT_1:13
      .= k.x9 by QUANTAL1:def 9;
    [:km, km:].(k.x9, km.x9) in id cL by A5,A16,FUNCT_1:def 7;
    then [km.(k.x9), km.(km.x9)] in id cL by A8,FUNCT_3:def 8;
    then
A18: km.(k.x9) = km.(km.x9) by RELAT_1:def 10
      .= (km*km).x9 by A8,FUNCT_1:13
      .= km.x9 by QUANTAL1:def 9;
    k.(km.x9) <= (id L).(km.x9) by A3,YELLOW_2:9;
    then k.(km.x9) <= km.x9 by FUNCT_1:18;
    hence k.x = km.x by A17,A18,A14,YELLOW_0:def 3;
  end;
  hence thesis by FUNCT_2:12;
end;
