reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem
  for h being PartFunc of W,REAL holds
  h.:Y is real-bounded & upper_bound(h.:Y) = lower_bound(h.:Y) implies
  h|Y is constant
proof
  let h be PartFunc of W,REAL;
  rng (h|Y) = h.:Y by RELAT_1:115;
  hence thesis by Th18;
end;
