reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem
  for L being Semilattice, x being Element of L holds x is irreducible
  iff for A being finite non empty Subset of L st x = inf A holds x in A
proof
  let L be Semilattice, I be Element of L;
  thus I is irreducible implies for A being finite non empty Subset of L st I
  = inf A holds I in A
  proof
    defpred P[set] means $1 is non empty & I = "/\"($1,L) implies I in $1;
    assume
A1: for x,y being Element of L st I = x"/\"y holds I = x or I = y;
    let A be finite non empty Subset of L;
A2: now
      let x,B be set such that
A3:   x in A and
A4:   B c= A and
A5:   P[B];
      thus P[B \/ {x}]
      proof
        reconsider a = x as Element of L by A3;
        reconsider C = B as finite Subset of L by A4,XBOOLE_1:1;
        assume that
        B \/ {x} is non empty and
A6:     I = "/\"(B \/ {x},L);
        per cases;
        suppose
A7:       B = {};
          then "/\"(B \/ {a},L) = a by YELLOW_0:39;
          hence thesis by A6,A7,TARSKI:def 1;
        end;
        suppose
A8:       B <> {};
A9:       inf {a} = a by YELLOW_0:39;
A10:      ex_inf_of {a}, L by YELLOW_0:55;
          ex_inf_of C, L by A8,YELLOW_0:55;
          then
A11:      "/\"(B \/ {x},L) = (inf C)"/\"inf {a} by A10,YELLOW_2:4;
          hereby
            per cases by A1,A6,A11,A9;
            suppose
              inf C = I;
              hence thesis by A5,A8,XBOOLE_0:def 3;
            end;
            suppose
A12:          a = I;
              a in {a} by TARSKI:def 1;
              hence thesis by A12,XBOOLE_0:def 3;
            end;
          end;
        end;
      end;
    end;
A13: P[{}];
A14: A is finite;
    P[A] from FINSET_1:sch 2(A14,A13,A2);
    hence thesis;
  end;
  assume
A15: for A being finite non empty Subset of L st I = inf A holds I in A;
  let a,b be Element of L;
  assume I = a"/\"b;
  then I = inf {a,b} by YELLOW_0:40;
  then I in {a,b} by A15;
  hence thesis by TARSKI:def 2;
end;
