
theorem Th34:
  for L, T being continuous complete LATTICE, g being
  CLHomomorphism of L, T, S being Subset of [:L, L:] st S = [:g, g:]"(id the
  carrier of T) holds subrelstr S is CLSubFrame of [:L, L:]
proof
  let L, T be continuous complete LATTICE, g be CLHomomorphism of L, T, SL be
  Subset of [:L, L:] such that
A1: SL = [:g, g:]"(id the carrier of T);
  set x = the Element of L;
A2: dom g = the carrier of L by FUNCT_2:def 1;
  then
A3: [x, x] in [:dom g, dom g:] by ZFMISC_1:87;
  [g.x, g.x] in id the carrier of T by RELAT_1:def 10;
  then
  dom [:g, g:] = [: dom g, dom g :] & [:g, g:].(x, x) in id the carrier of
  T by A2,FUNCT_3:def 8;
  then reconsider nSL = SL as non empty Subset of [:L, L:] by A1,A3,
FUNCT_1:def 7;
  set pL = [:L, L:], pg = [:g, g:];
A4: g is infs-preserving directed-sups-preserving by WAYBEL16:def 1;
A5: the carrier of pL=[:the carrier of L,the carrier of L:] by YELLOW_3:def 2;
A6: subrelstr nSL is non empty;
A7: subrelstr SL is directed-sups-inheriting
  proof
    let X be directed Subset of subrelstr SL such that
A8: X <> {} and
A9: ex_sup_of X, pL;
    reconsider X9 = X as directed non empty Subset of pL by A6,A8,YELLOW_2:7;
    pg is directed-sups-preserving by A4,Th12;
    then pg preserves_sup_of X9;
    then
A10: sup (pg.:X9)=pg.sup X9 by A9;
    X c= the carrier of subrelstr SL;
    then X c= SL by YELLOW_0:def 15;
    then
A11: pg.:X c= pg.:SL by RELAT_1:123;
    pg.:SL c= id the carrier of T & ex_sup_of pg.:X9, [:T, T:] by A1,FUNCT_1:75
,YELLOW_0:17;
    then
A12: sup (pg.:X9) in id the carrier of T by A11,Th14,XBOOLE_1:1;
    consider x, y being object such that
A13: x in the carrier of L & y in the carrier of L and
A14: sup X9 = [x, y] by A5,ZFMISC_1:def 2;
    [x, y] in [:dom g, dom g:] by A2,A13,ZFMISC_1:87;
    then [x, y] in dom [:g, g:] by FUNCT_3:def 8;
    then [x, y] in [:g, g:]"(id the carrier of T) by A14,A10,A12,FUNCT_1:def 7;
    hence thesis by A1,A14,YELLOW_0:def 15;
  end;
  subrelstr SL is infs-inheriting
  proof
    let X be Subset of subrelstr SL such that
A15: ex_inf_of X, pL;
    X c= the carrier of subrelstr SL;
    then
A16: X c= SL by YELLOW_0:def 15;
    then reconsider X9 = X as Subset of pL by XBOOLE_1:1;
A17: pg.:SL c= id the carrier of T & ex_inf_of pg.:X9, [:T, T:] by A1,
FUNCT_1:75,YELLOW_0:17;
    pg is infs-preserving by A4,Th9;
    then pg preserves_inf_of X9;
    then
A18: inf (pg.:X9)=pg.inf X9 by A15;
    pg.:X c= pg.:SL by A16,RELAT_1:123;
    then
A19: inf (pg.:X9) in id the carrier of T by A17,Th13,XBOOLE_1:1;
    consider x, y being object such that
A20: x in the carrier of L & y in the carrier of L and
A21: inf X9 = [x, y] by A5,ZFMISC_1:def 2;
    [x, y] in [:dom g, dom g:] by A2,A20,ZFMISC_1:87;
    then [x, y] in dom [:g, g:] by FUNCT_3:def 8;
    then [x, y] in [:g, g:]"(id the carrier of T) by A21,A18,A19,FUNCT_1:def 7;
    hence thesis by A1,A21,YELLOW_0:def 15;
  end;
  hence thesis by A7;
end;
