
theorem Th63: :: PROPOSITION 4.3.(i)
  for L be lower-bounded continuous sup-Semilattice for B be
  with_bottom CLbasis of L holds [supMap subrelstr B,baseMap B] is Galois
proof
  let L be lower-bounded continuous sup-Semilattice;
  let B be with_bottom CLbasis of L;
  now
    let x be Element of L, y be Element of InclPoset Ids subrelstr B;
    reconsider I = y as Ideal of subrelstr B by YELLOW_2:41;
    reconsider J = I as non empty directed Subset of L by YELLOW_2:7;
A1: ex_sup_of waybelow x /\ B,L by YELLOW_0:17;
    thus x <= (supMap subrelstr B).y implies (baseMap B).x <= y
    proof
A2:   downarrow J /\ B c= J
      proof
        let z be object;
        assume
A3:     z in downarrow J /\ B;
        then reconsider z1 = z as Element of L;
        z in B by A3,XBOOLE_0:def 4;
        then reconsider z2 = z as Element of subrelstr B by YELLOW_0:def 15;
        z in downarrow J by A3,XBOOLE_0:def 4;
        then consider v1 be Element of L such that
A4:     v1 >= z1 and
A5:     v1 in J by WAYBEL_0:def 15;
        reconsider v2 = v1 as Element of subrelstr B by A5;
        z2 <= v2 by A4,YELLOW_0:60;
        hence thesis by A5,WAYBEL_0:def 19;
      end;
      assume x <= (supMap subrelstr B).y;
      then x <= "\/"(I,L) by Def10;
      then
A6:   x <= sup downarrow J by WAYBEL_0:33,YELLOW_0:17;
      waybelow x c= downarrow J
      proof
        let z be object;
        assume
A7:     z in waybelow x;
        then reconsider z1 = z as Element of L;
        z1 << x by A7,WAYBEL_3:7;
        hence thesis by A6,WAYBEL_3:20;
      end;
      then waybelow x /\ B c= downarrow J /\ B by XBOOLE_1:26;
      then waybelow x /\ B c= y by A2;
      then (baseMap B).x c= y by Def12;
      hence thesis by YELLOW_1:3;
    end;
A8: ex_sup_of J,L by YELLOW_0:17;
    thus (baseMap B).x <= y implies x <= (supMap subrelstr B).y
    proof
      assume (baseMap B).x <= y;
      then (baseMap B).x c= y by YELLOW_1:3;
      then waybelow x /\ B c= y by Def12;
      then sup (waybelow x /\ B) <= sup J by A8,A1,YELLOW_0:34;
      then x <= "\/"(I,L) by Def7;
      hence thesis by Def10;
    end;
  end;
  hence thesis by WAYBEL_1:8;
end;
