
theorem lemma102:
for X being set, S being with_empty_element semi-diff-closed cap-closed
   Subset-Family of X, A,B,P being set
  st P = DisUnion S & A in P & B in P & A misses B
holds A \/ B in P
proof
   let X be set, S be with_empty_element semi-diff-closed cap-closed
     Subset-Family of X, A,B,P be set;
   assume that
A1: P = DisUnion S and
A2: A in P & B in P & A misses B;
   consider A1 be Subset of X such that
A3: A = A1 &
    ex g be disjoint_valued FinSequence of S st A1 = Union g by A1,A2;
   consider g1 be disjoint_valued FinSequence of S such that
A4: A1 = Union g1 by A3;
   consider B1 be Subset of X such that
A5: B = B1 &
    ex g be disjoint_valued FinSequence of S st B1 = Union g by A1,A2;
   consider g2 be disjoint_valued FinSequence of S such that
A6: B1 = Union g2 by A5;
   set F = g1^g2;
   now let n,m be object;
    assume A7: n <> m;
    per cases;
    suppose A8: n in dom F & m in dom F; then
    reconsider n1=n,m1=m as Nat;
A9:  n1 in dom g1
  or ex k being Nat st k in dom g2 & n1 = len g1 + k by A8,FINSEQ_1:25;
A10: m1 in dom g1
  or ex k being Nat st k in dom g2 & m1 = len g1 + k by A8,FINSEQ_1:25;
     per cases by A9,A10;
     suppose n1 in dom g1 & m1 in dom g1; then
      F.n = g1.n1 & F.m = g1.m1 by FINSEQ_1:def 7;
      hence F.n misses F.m by A7,PROB_2:def 2;
     end;
     suppose A11: n1 in dom g1
           & ex k being Nat st k in dom g2 & m1 = len g1 + k; then
      consider k be Nat such that
A12:   k in dom g2 & m1 = len g1 + k;
      F.n = g1.n1 & F.m = g2.k by A11,A12,FINSEQ_1:def 7; then
A13:  F.n in rng g1 & F.m in rng g2 by A11,A12,FUNCT_1:3;
      now assume F.n meets F.m; then
       consider x be object such that
A14:    x in F.n & x in F.m by XBOOLE_0:3;
       x in union rng g1 & x in union rng g2 by A13,A14,TARSKI:def 4;
       hence contradiction by A2,A3,A5,A4,A6,XBOOLE_0:def 4;
      end;
      hence F.n misses F.m;
     end;
     suppose A15: m1 in dom g1
           & ex k being Nat st k in dom g2 & n1 = len g1 + k; then
      consider k be Nat such that
A16:   k in dom g2 & n1 = len g1 + k;
      F.n = g2.k & F.m = g1.m by A15,A16,FINSEQ_1:def 7; then
A17:  F.n in rng g2 & F.m in rng g1 by A15,A16,FUNCT_1:3;
      now assume F.n meets F.m; then
       consider x be object such that
A18:    x in F.n & x in F.m by XBOOLE_0:3;
       x in union rng g1 & x in union rng g2 by A17,A18,TARSKI:def 4;
       hence contradiction by A2,A3,A5,A4,A6,XBOOLE_0:def 4;
      end;
      hence F.n misses F.m;
     end;
     suppose A19: ex k being Nat st k in dom g2 & n1 = len g1 + k
                & ex k being Nat st k in dom g2 & m1 = len g1 + k; then
      consider k1 be Nat such that
A20:   k1 in dom g2 & n1 = len g1 + k1;
      consider k2 be Nat such that
A21:   k2 in dom g2 & m1 = len g1 + k2 by A19;
      F.n = g2.k1 & F.m = g2.k2 by A20,A21,FINSEQ_1:def 7;
      hence F.n misses F.m by A7,A20,A21,PROB_2:def 2;
     end;
    end;
    suppose not n in dom F or not m in dom F; then
     F.n = {} or F.m = {} by FUNCT_1:def 2;
     hence F.n misses F.m;
    end;
   end; then
   reconsider F as disjoint_valued FinSequence of S by PROB_2:def 2;
   rng F = rng g1 \/ rng g2 by FINSEQ_1:31; then
   Union F = A1 \/ B1 by A4,A6,ZFMISC_1:78;
   hence A \/ B in P by A1,A3,A5;
end;
