
theorem Th101:
for F be disjoint_valued FinSequence, n be Nat holds
 Union(F|n) misses F.(n+1)
proof
   let F be disjoint_valued FinSequence, n be Nat;
   assume Union(F|n) meets F.(n+1); then
   consider x be object such that
A1: x in Union(F|n) & x in F.(n+1) by XBOOLE_0:3;
   x in union rng(F|n) by A1,CARD_3:def 4; then
   consider A be set such that
A2: x in A & A in rng(F|n) by TARSKI:def 4;
   consider m be object such that
A3: m in dom(F|n) & A = (F|n).m by A2,FUNCT_1:def 3;
   reconsider m as Element of NAT by A3;
   m <= len(F|n) & len(F|n) <= n by A3,FINSEQ_3:25,FINSEQ_1:86; then
   m <> n+1 by NAT_1:13; then
   F.m misses F.(n+1) by PROB_2:def 2; then
   (F|n).m /\ F.(n+1) = {} by A3,FUNCT_1:47;
   hence contradiction by A1,A2,A3,XBOOLE_0:def 4;
end;
