reserve x, y, i for object,
  L for up-complete Semilattice;

theorem ::Theorem 2.1 (1) implies (4)
  for L being continuous lower-bounded sup-Semilattice holds SupMap L is
  upper_adjoint
proof
  let L be continuous lower-bounded sup-Semilattice;
  set P = InclPoset(Ids L);
  set r = SupMap L;
  deffunc F(Element of L) = inf(r"(uparrow {$1}));
  ex d being Function of L, InclPoset(Ids L) st for t being Element of L
  holds d.t = F(t) from FUNCT_2:sch 4;
  then consider d being Function of L, InclPoset(Ids L) such that
A1: for t being Element of L holds d.t = inf(r"(uparrow {t}));
  for t being Element of L holds d.t is_minimum_of r"(uparrow t)
  proof
    let t be Element of L;
    set I = inf(r"(uparrow t));
    reconsider I9= I as Ideal of L by YELLOW_2:41;
A2: d.t = inf(r"(uparrow {t})) by A1
      .= I by WAYBEL_0:def 18;
    I in the carrier of P;
    then I in Ids L by YELLOW_1:1;
    then
A3: I in dom r by YELLOW_2:49;
    consider J being Ideal of L such that
A4: t <= sup J and
A5: for K being Ideal of L st t <= sup K holds J c= K by Lm3;
    reconsider J9= J as Element of P by YELLOW_2:41;
A6: for K being Element of P st r"(uparrow t) is_>=_than K holds K <= J9
    proof
      J9 in the carrier of P;
      then J9 in Ids L by YELLOW_1:1;
      then
A7:   J in dom r by YELLOW_2:49;
      let K be Element of P;
      assume
A8:   r"(uparrow t) is_>=_than K;
      t <= r.J9 by A4,YELLOW_2:def 3;
      then r.J in uparrow t by WAYBEL_0:18;
      then J in r"(uparrow t) by A7,FUNCT_1:def 7;
      hence thesis by A8;
    end;
    r"(uparrow t) is_>=_than J9
    proof
      let K be Element of P;
      reconsider K9= K as Ideal of L by YELLOW_2:41;
      assume K in r"(uparrow t);
      then r.K in uparrow t by FUNCT_1:def 7;
      then t <= r.K by WAYBEL_0:18;
      then t <= sup K9 by YELLOW_2:def 3;
      then J9 c= K9 by A5;
      hence J9 <= K by YELLOW_1:3;
    end;
    then t <= sup I9 by A4,A6,YELLOW_0:31;
    then t <= r.I by YELLOW_2:def 3;
    then r.I in uparrow t by WAYBEL_0:18;
    then ex_inf_of r"(uparrow t),InclPoset(Ids L) & I in r"(uparrow t) by A3,
FUNCT_1:def 7,YELLOW_0:17;
    hence thesis by A2,WAYBEL_1:def 6;
  end;
  then [r, d] is Galois by WAYBEL_1:10;
  hence thesis by WAYBEL_1:def 11;
end;
