
theorem
  for X be set,
      S be non empty semi-diff-closed cap-closed Subset-Family of X,
      E1,E2 be Element of S
  ex F1,F2,F3 be disjoint_valued FinSequence of S
     st union rng F1 = E1 \ E2
      & union rng F2 = E2 \ E1 & union rng F3 = E1 /\ E2
      & F1^F2^F3 is disjoint_valued FinSequence of S
proof
   let X be set, S be non empty semi-diff-closed cap-closed Subset-Family of X,
       E1,E2 be Element of S;
   consider F1 be disjoint_valued FinSequence of S such that
A2: E1 \ E2 = Union F1 by SRINGS_3:def 1;
   consider F2 be disjoint_valued FinSequence of S such that
A3: E2 \ E1 = Union F2 by SRINGS_3:def 1;
   E1 \ E2 misses E2 \ E1 by XBOOLE_1:82; then
   union rng F1 misses Union F2 by A2,A3,CARD_3:def 4; then
   union rng F1 misses union rng F2 by CARD_3:def 4; then
   reconsider G = F1^F2 as disjoint_valued FinSequence of S by Th07;
   rng G = rng F1 \/ rng F2 by FINSEQ_1:31; then
A4:union rng G = union rng F1 \/ union rng F2 by ZFMISC_1:78;
   Union F1 = union rng F1 & Union F2 = union rng F2 by CARD_3:def 4; then
   Union G = (E1 \ E2) \/ (E2 \ E1) by A2,A3,A4,CARD_3:def 4; then
A5:Union G = E1 \+\ E2 by XBOOLE_0:def 6;
   reconsider E = E1 /\ E2 as Element of S by FINSUB_1:def 2;
   reconsider F3 = <*E*> as FinSequence of S;
   reconsider F3 as disjoint_valued FinSequence of S;
   take F1,F2,F3;
   reconsider F = G^<*E*> as FinSequence of S;
   thus union rng F1 = E1 \ E2 & union rng F2 = E2 \ E1 by A2,A3,CARD_3:def 4;
   rng F3 = {E} by FINSEQ_1:38;
   hence union rng F3 = E1 /\ E2 by ZFMISC_1:25;
A6:Union G misses E by A5,XBOOLE_1:103;
B1:len F = len G + 1 by FINSEQ_2:16;
A7:now let i,j be Nat;
    assume A8: i in dom G & j = len F; then
A9: F.j = E by B1,FINSEQ_1:42;
    F.i = G.i by A8,FINSEQ_1:def 7; then
    F.i c= union rng G by A8,FUNCT_1:3,ZFMISC_1:74; then
    F.i c= Union G by CARD_3:def 4;
    hence F.i misses F.j by A6,A9,XBOOLE_1:63;
   end;
   now let x,y be object;
    assume A10: x <> y;
    per cases;
    suppose A11: x in dom F & y in dom F; then
     reconsider x1=x, y1=y as Nat;
A12: 1 <= x1 & x1 <= len F & 1 <= y1 & y1 <= len F by A11,FINSEQ_3:25;
     per cases;
     suppose A13: x1 = len F; then
      y1 < len F by A10,A12,XXREAL_0:1; then
      y1 <= len G by B1,NAT_1:13;
      hence F.x misses F.y by A7,A13,A12,FINSEQ_3:25;
     end;
     suppose A11: y1 = len F; then
      x1 < len F by A10,A12,XXREAL_0:1; then
      x1 <= len G by B1,NAT_1:13;
      hence F.x misses F.y by A7,A11,A12,FINSEQ_3:25;
     end;
     suppose x1 <> len F & y1 <> len F; then
      x1 < len F & y1 < len F by A12,XXREAL_0:1; then
      x1 <= len G & y1 <= len G by B1,NAT_1:13; then
      x1 in dom G & y1 in dom G by A12,FINSEQ_3:25; then
      F.x = G.x & F.y = G.y by FINSEQ_1:def 7;
      hence F.x misses F.y by A10,PROB_2:def 2;
     end;
    end;
    suppose not x in dom F or not y in dom F; then
     F.x = {} or F.y = {} by FUNCT_1:def 2;
     hence F.x misses F.y;
    end;
   end;
   hence thesis by PROB_2:def 2;
end;
