reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;

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;
