
theorem Th72:
for F be disjoint_valued FinSequence, n,m be Nat
  st n < m holds union rng(F|n) misses F.m
proof
   let F be disjoint_valued FinSequence, n,m be Nat;
   assume A1: n < m;
   per cases;
   suppose n >= len F; then
    m > len F by A1,XXREAL_0:2; then
    not m in dom F by FINSEQ_3:25; then
    F.m = {} by FUNCT_1:def 2;
    hence union rng(F|n) misses F.m;
   end;
   suppose A2: n < len F;
    for A be set st A in rng(F|n) holds A misses F.m
    proof
     let A be set;
     assume A in rng(F|n); then
     consider k be object such that
A3:   k in dom(F|n) & A = (F|n).k by FUNCT_1:def 3;
     reconsider k as Element of NAT by A3;
     1 <= k <= len(F|n) by A3,FINSEQ_3:25; then
A4:  k <= n by A2,FINSEQ_1:59; then
     A = F.k by A3,FINSEQ_3:112;
     hence A misses F.m by A1,A4,PROB_2:def 2;
    end;
    hence union rng(F|n) misses F.m by ZFMISC_1:80;
   end;
end;
