
theorem Lm1:
for X be set, S be Subset-Family of X, S1,S2 be set st
  S1 in S & S2 in S & S is semi-diff-closed holds
  ex x be finite Subset of S st x is a_partition of S1 \ S2
proof
   let X be set, S be Subset-Family of X, S1,S2 be set;
   assume S1 in S & S2 in S & S is semi-diff-closed; then
   consider F be disjoint_valued FinSequence of S such that
Y2: S1 \ S2 = Union F;
   reconsider x = rng F \ {{}} as finite Subset of S;
   take x;
   now let p be object;
    assume U1: p in x; then
    p in S; then
    reconsider p1 = p as Subset of X;
    p in rng F & not p in {{}} by U1,XBOOLE_0:def 5; then
    p1 c= S1 \ S2 by Y2,TARSKI:def 4;
    hence p in bool (S1 \ S2);
   end; then
Y5:x c= bool (S1 \ S2);
Y3:union x = S1 \ S2 by Y2,PARTIT1:2;
   now let A be Subset of S1 \ S2;
    assume A in x; then
Y6: A in rng F & not A in {{}} by XBOOLE_0:def 5;
    hence A <> {} by TARSKI:def 1;
    now let B be Subset of S1 \ S2;
     assume B in x; then
     B in rng F & not B in {{}} by XBOOLE_0:def 5; then
     (ex i be Nat st i in dom F & F.i = A)
   & (ex j be Nat st j in dom F & F.j = B) by Y6,FINSEQ_2:10;
     hence A=B or A misses B by PROB_2:def 2;
    end;
    hence for B be Subset of S1 \ S2 st B in x holds A = B or A misses B;
   end;
   hence x is a_partition of S1 \ S2 by Y3,Y5,EQREL_1:def 4;
end;
