
theorem th10:
for X be set, S be Subset-Family of X st
  S is with_empty_element & S is cap-finite-partition-closed
& S is diff-c=-finite-partition-closed
 holds S is semi-diff-closed
proof
   let X be set, S be Subset-Family of X;
   assume that
A0: S is with_empty_element and
A1: S is cap-finite-partition-closed and
A2: S is diff-c=-finite-partition-closed;
   now let S1,S2 be set;
    assume A3: S1 in S & S2 in S;
    per cases;
    suppose X1: S1 \ S2 <> {}; then
     consider x be finite Subset of S such that
L4:   x is a_partition of S1 \ S2 by A1,A2,A3,SRINGS_1:def 2;
L8:  union x = S1 \ S2
   & for A be Subset of S1 \ S2 st A in x holds A <> {}
       & for B be Subset of S1 \ S2 st B in x holds
             A = B or A misses B by L4,EQREL_1:def 4;
L5:  rng (canFS x) c= x;
     rng (canFS x) c= S by XBOOLE_1:1; then
     reconsider F = canFS x as FinSequence of S by FINSEQ_1:def 4;
     now let i,j be object;
      assume L6: i <> j;
      per cases;
      suppose L10: i in dom F & j in dom F; then
       F.i in x & F.j in x by L5,FUNCT_1:3; then
       F.i = F.j or F.i misses F.j by L4,EQREL_1:def 4;
       hence F.i misses F.j by L6,L10,FUNCT_1:def 4;
      end;
      suppose not i in dom F or not j in dom F; then
       F.i = {} or F.j = {} by FUNCT_1:def 2;
       hence F.i misses F.j;
      end;
     end; then
     reconsider F as disjoint_valued FinSequence of S by PROB_2:def 2;
     take F;
     thus Union F = S1 \ S2 by L4,X1,L8,CANFS;
    end;
    suppose T0: S1 \ S2 = {};
     set F = canFS {{}};
     {{}} c= S by A0,SETFAM_1:def 8,ZFMISC_1:31; then
     rng F c= S; then
     reconsider F as FinSequence of S by FINSEQ_1:def 4;
T3:  F = <*{}*> by FINSEQ_1:94;
     now let i,j be object;
      assume i <> j;
      i in dom F implies i=1 by T3,FINSEQ_1:90; then
      F.i = {} by T3,FUNCT_1:def 2;
      hence F.i misses F.j;
     end; then
     reconsider F as disjoint_valued FinSequence of S by PROB_2:def 2;
     take F;
     Union F = union {{}} by CANFS;
     hence S1 \ S2 = Union F by T0;
    end;
   end;
   hence S is semi-diff-closed;
end;
