theorem
  p is x-context_including & i in dom p implies
  (p/.i is non x-omitting iff p/.i is x-context)
  proof
    assume p is x-context_including;
    then consider j such that
A0: j in dom p & p.j is context of x &
    for n,t st n in dom p & n <> j & t = p.n
    holds t is x-omitting;
    assume Z1: i in dom p;
    then
A1: p/.i = p.i by PARTFUN1:def 6;
    thus p/.i is non x-omitting & p/.i is non x-context implies contradiction
    by A0,A1,Z1;
    assume p/.i is x-context;
    hence p/.i is non x-omitting;
  end;
