reserve a,b,c for set;

theorem
  for X,x0 being set st x0 in X for A being Subset of DiscrWithInfin(X,
  x0) st A` is infinite holds Int A = A \ {x0}
proof
  let X,x0 be set such that
A1: x0 in X;
  set T = DiscrWithInfin(X,x0);
  reconsider T as non empty TopSpace by A1;
  reconsider x = x0 as Point of T by A1,Def5;
  let A be Subset of DiscrWithInfin(X,x0);
  reconsider A9 = A as Subset of T;
  assume A` is infinite;
  then Cl A` = A9` \/ {x} by A1,Th26;
  hence Int A = (A9` \/ {x})` by TOPS_1:def 1
    .= (A9 \ {x})`` by SUBSET_1:14
    .= A \ {x0};
end;
