reserve n,i,k,m for Nat;
reserve r,r1,r2,s,s1,s2 for Real;
reserve p,p1,p2,q1,q2 for Point of TOP-REAL n;

theorem
  for P,Q being Subset of TOP-REAL n st p is_extremal_in P & Q c= P & p in Q
  holds p is_extremal_in Q
proof
  let P,Q be Subset of TOP-REAL n;
  assume that
A1: p is_extremal_in P and
A2: Q c= P;
  assume p in Q;
  hence p in Q;
  let p1,p2;
  assume
A3: p in LSeg(p1,p2);
  assume LSeg(p1,p2) c= Q;
  then LSeg(p1,p2) c= P by A2;
  hence thesis by A1,A3;
end;
