
theorem Th61:
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds idsMap (subrelstr B) is sups-preserving
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  set f = idsMap (subrelstr B);
A1: subrelstr B is join-inheriting by Def2;
A2: Bottom L in B by Def8;
  then
A3: Bottom L in the carrier of subrelstr B by YELLOW_0:def 15;
  now
    let X be Subset of InclPoset Ids subrelstr B;
    reconsider supX = sup X as Ideal of subrelstr B by YELLOW_2:41;
    reconsider unionX = union X as Subset of L by WAYBEL13:3;
    reconsider dfuX = downarrow finsups union X as Subset of L by WAYBEL13:3;
    reconsider fuX = finsups union X as Subset of L by WAYBEL13:3;
A4: ex J be Subset of L st supX = J & f.supX = downarrow J by Def11;
    now
      assume ex_sup_of X,InclPoset Ids subrelstr B;
      thus ex_sup_of f.:X,InclPoset Ids L by YELLOW_0:17;
A5:   downarrow finsups union (f.:X) c= downarrow dfuX
      proof
        defpred P[object,object] means
ex I be Element of InclPoset Ids subrelstr B,
z1,z2 be Element of L st z1 = $1 & z2 = $2 & I in X & $2 in I & z1 <= z2;
        let x be object;
        assume
A6:     x in downarrow finsups union (f.:X);
        then reconsider x1 = x as Element of L;
        consider y1 be Element of L such that
A7:     y1 >= x1 and
A8:     y1 in finsups union (f.:X) by A6,WAYBEL_0:def 15;
        y1 in { "\/"(V,L) where V is finite Subset of union (f.:X) :
        ex_sup_of V,L } by A8,WAYBEL_0:def 27;
        then consider Y be finite Subset of union (f.:X) such that
A9:     y1 = "\/"(Y,L) and
        ex_sup_of Y,L;
A10:    for z be object st z in Y ex v be Element of B st P[z,v]
        proof
          let z be object;
          assume z in Y;
          then consider J be set such that
A11:      z in J and
A12:      J in f.:X by TARSKI:def 4;
          consider I be object such that
          I in dom f and
A13:      I in X and
A14:      J = f.I by A12,FUNCT_1:def 6;
          reconsider I as Element of InclPoset Ids subrelstr B by A13;
          f.I is Element of InclPoset Ids L;
          then reconsider J as Element of InclPoset Ids L by A14;
          J is Ideal of L by YELLOW_2:41;
          then reconsider z1 = z as Element of L by A11;
          reconsider I1 = I as Ideal of subrelstr B by YELLOW_2:41;
          consider I2 be Subset of L such that
A15:      I1 = I2 and
A16:      f.I1 = downarrow I2 by Def11;
          consider z2 be Element of L such that
A17:      z2 >= z1 and
A18:      z2 in I2 by A11,A14,A16,WAYBEL_0:def 15;
          reconsider v = z2 as Element of B by A15,A18,YELLOW_0:def 15;
          take v,I,z1,z2;
          thus thesis by A13,A15,A17,A18;
        end;
        consider g be Function of Y,B such that
A19:    for z be object st z in Y holds P[z,g.z] from MONOID_1:sch 1(A10
        );
        reconsider Z = rng g as finite Subset of subrelstr B by YELLOW_0:def 15
;
A20:    dom g = Y by FUNCT_2:def 1;
A21:    "\/"(rng g,L) is_>=_than Y
        proof
          let a be Element of L;
A22:      "\/"(rng g,L) is_>=_than rng g by YELLOW_0:32;
          assume
A23:      a in Y;
          then consider
          I be Element of InclPoset Ids subrelstr B, a1,a2 be Element
          of L such that
A24:      a1 = a and
A25:      a2 = g.a and
          I in X and
          g.a in I and
A26:      a1 <= a2 by A19;
          g.a in rng g by A20,A23,FUNCT_1:def 3;
          then a2 <= "\/"(rng g,L) by A25,A22;
          hence a <= "\/"(rng g,L) by A24,A26,YELLOW_0:def 2;
        end;
A27:    ex_sup_of Z,subrelstr B
        proof
          per cases;
          suppose
            Z is non empty;
            hence thesis by YELLOW_0:54;
          end;
          suppose
            Z is empty;
            hence thesis by YELLOW_0:42;
          end;
        end;
        rng g c= union X
        proof
          let a be object;
          assume a in rng g;
          then consider b be object such that
A28:      b in dom g and
A29:      a = g.b by FUNCT_1:def 3;
          ex I be Element of InclPoset Ids subrelstr B, b1,b2 be Element
          of L st b1 = b & b2 = g.b & I in X & g.b in I & b1 <= b2 by A19,A28;
          hence thesis by A29,TARSKI:def 4;
        end;
        then "\/"(Z,subrelstr B) in { "\/"(V,subrelstr B) where V is finite
        Subset of union X : ex_sup_of V,subrelstr B } by A27;
        then
A30:    "\/"(rng g,subrelstr B) in finsups union X by WAYBEL_0:def 27;
        "\/"(Z,L) in the carrier of subrelstr B
        proof
          per cases;
          suppose
            Z is non empty;
            hence thesis by A1,WAYBEL21:15;
          end;
          suppose
            Z is empty;
            hence thesis by A2,YELLOW_0:def 15;
          end;
        end;
        then reconsider xl = "\/"(Z,L) as Element of subrelstr B;
        reconsider srg = "\/"(rng g,subrelstr B) as Element of L by YELLOW_0:58
;
A31:    ex_sup_of Z,L by YELLOW_0:17;
A32:    now
          let b be Element of subrelstr B;
          reconsider b1 = b as Element of L by YELLOW_0:58;
          assume
A33:      b is_>=_than Z;
          b1 is_>=_than Z
          by A33,YELLOW_0:59;
          then "\/"(Z, L) <= b1 by A31,YELLOW_0:30;
          hence xl <= b by YELLOW_0:60;
        end;
A34:    "\/"(Z, L) is_>=_than Z by A31,YELLOW_0:30;
        xl is_>=_than Z
        proof
          let b be Element of subrelstr B;
          reconsider b1 = b as Element of L by YELLOW_0:58;
          assume b in Z;
          then b1 <= "\/"(Z, L) by A34;
          hence b <= xl by YELLOW_0:60;
        end;
        then "\/"(Z,subrelstr B) = "\/"(Z,L) by A32,YELLOW_0:30;
        then "\/"(Y,L) <= srg by A21,YELLOW_0:32;
        then
A35:    x1 <= srg by A7,A9,YELLOW_0:def 2;
        finsups union X c= downarrow finsups union X by WAYBEL_0:16;
        hence thesis by A35,A30,WAYBEL_0:def 15;
      end;
      now
        let x be set;
        assume
A36:    x in X;
        then reconsider x1 = x as Ideal of subrelstr B by YELLOW_2:41;
        consider x2 be Subset of L such that
A37:    x1 = x2 and
A38:    f.x1 = downarrow x2 by Def11;
        x in the carrier of InclPoset Ids subrelstr B by A36;
        then x1 in dom f by FUNCT_2:def 1;
        then
A39:    f.x1 in f.:X by A36,FUNCT_1:def 6;
        thus x c= union (f.:X)
        proof
          let y be object;
          assume
A40:      y in x;
          x c= downarrow x2 by A37,WAYBEL_0:16;
          hence thesis by A38,A39,A40,TARSKI:def 4;
        end;
      end;
      then union X c= union (f.:X) by ZFMISC_1:76;
      then
A41:  finsups unionX c= finsups union (f.:X) by Th2;
      finsups union X c= finsups unionX by A3,A1,Th5;
      then finsups union X c= finsups union (f.:X) by A41;
      then
A42:  downarrow fuX c= downarrow finsups union (f.:X) by YELLOW_3:6;
      downarrow finsups union X c= downarrow fuX by Th11;
      then dfuX c= downarrow finsups union (f.:X) by A42;
      then downarrow dfuX c= downarrow downarrow finsups union (f.:X) by
YELLOW_3:6;
      then
A43:  downarrow dfuX c= downarrow finsups union (f.:X) by Th7;
      thus sup (f.:X) = downarrow finsups union (f.:X) by Th6
        .= downarrow dfuX by A5,A43
        .= f.sup X by A4,Th6;
    end;
    hence f preserves_sup_of X by WAYBEL_0:def 31;
  end;
  hence thesis by WAYBEL_0:def 33;
end;
