reserve i,j,k,n for Nat;
reserve D for non empty set,
  p for Element of D,
  f,g for FinSequence of D;

theorem Th51:
  p in rng f & j+1 in dom(f:-p) implies j+p..f in dom f
proof
  assume that
A1: p in rng f and
A2: j+1 in dom(f:-p);
  j+1 <= len(f:-p) by A2,FINSEQ_3:25;
  then j+1 <= len f - p..f + 1 by A1,Th50;
  then j <= len f - p..f by XREAL_1:6;
  then
A3: j+p..f <= len f by XREAL_1:19;
A4: p..f <= j+p..f by NAT_1:11;
  1 <= p..f by A1,FINSEQ_4:21;
  then 1 <= j+p..f by A4,XXREAL_0:2;
  hence thesis by A3,FINSEQ_3:25;
end;
