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 Th18:
  for h being PartFunc of W,REAL holds
  rng h is real-bounded & upper_bound (rng h) = lower_bound (rng h)
  implies h is constant
proof
  let h be PartFunc of W,REAL;
  assume
A1: rng h is real-bounded & upper_bound (rng h) = lower_bound (rng h);
  assume not h is constant;
  then consider x1,x2 being object such that
A2: x1 in dom h & x2 in dom h and
A3: h.x1 <> h.x2;
  h.x1 in rng h & h.x2 in rng h by A2,FUNCT_1:def 3;
  hence contradiction by A1,A3,SEQ_4:12;
end;
