theorem Th43:
  x in rng p & n in dom(p |-- x) implies n + x..p in dom p
proof
  assume that
A1: x in rng p and
A2: n in dom(p |-- x);
  reconsider m = len p - x..p as Element of NAT by A1,Th22;
  n in Seg m by A1,A2,Th42;
  then n <= len p - x..p by FINSEQ_1:1;
  then
A3: n + x..p <= len p by XREAL_1:19;
  1 <= n by A2,FINSEQ_3:25;
  then 1 <= n + x..p by NAT_1:12;
  hence thesis by A3,FINSEQ_3:25;
end;
