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
  for x being set st x in rng f & p in rng f & x..f<=p..f holds x in rng(f-:p)
proof
  let x be set;
  assume that
A1: x in rng f and
A2: p in rng f and
A3: x..f<=p..f;
  set g = f-:p, i = x..f;
  1 <= i by A1,FINSEQ_4:21;
  then
A4: i in Seg (p..f) by A3;
  Seg len g = dom g by FINSEQ_1:def 3;
  then
A5: i in dom g by A2,A4,Th42;
  then g.i in rng g by FUNCT_1:def 3;
  then g/.i in rng g by A5,PARTFUN1:def 6;
  then
A6: f/.i in rng g by A2,A4,Th43;
  i in dom f by A1,FINSEQ_4:20;
  then f.i in rng g by A6,PARTFUN1:def 6;
  hence thesis by A1,FINSEQ_4:19;
end;
