reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem Th46:
  for L being lower-bounded sup-Semilattice for A being non empty
Subset of InclPoset(Ids L) holds ex_inf_of A,InclPoset(Ids L) & inf A = meet A
proof
  let L be lower-bounded sup-Semilattice;
  let A be non empty Subset of InclPoset(Ids L);
  set P = InclPoset(Ids L);
  reconsider A9= A as non empty Subset of Ids L by YELLOW_1:1;
  meet A9 is Ideal of L by Th45;
  then reconsider I = meet A as Element of P by Th41;
A1: for b being Element of P st b is_<=_than A holds I >= b
  proof
    let b be Element of P;
    assume
A2: A is_>=_than b;
A3: for J being set st J in A holds b c= J
    by A2,YELLOW_1:3;
    b c= I
    proof
      let c be object;
      assume
A4:   c in b;
      for J being set st J in A holds c in J
      proof
        let J be set;
        assume J in A;
        then b c= J by A3;
        hence thesis by A4;
      end;
      hence thesis by SETFAM_1:def 1;
    end;
    hence thesis by YELLOW_1:3;
  end;
  I is_<=_than A
  proof
    let y be Element of P;
    assume
A5: y in A;
    I c= y
    by A5,SETFAM_1:def 1;
    hence I <= y by YELLOW_1:3;
  end;
  hence thesis by A1,YELLOW_0:31;
end;
