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;
reserve e1,e2 for ExtReal;
reserve h,h1,h2 for PartFunc of REAL,REAL;

theorem
  Y misses dom h implies h|Y is increasing & h|Y is decreasing &
  h|Y is non-decreasing & h|Y is non-increasing & h|Y is monotone
proof
  assume
A1: Y /\ dom h = {};
  then
  for r1,r2 holds ( r1 in Y /\ dom h & r2 in Y /\ dom h & r1<r2 implies h.
  r1<h.r2);
  hence h|Y is increasing by Th20;
  for r1,r2 holds ( r1 in Y /\ dom h & r2 in Y /\ dom h & r1<r2 implies h.
  r2<h.r1) by A1;
  hence h|Y is decreasing by Th21;
  for r1,r2 holds ( r1 in Y /\ dom h & r2 in Y /\ dom h & r1<r2 implies h.
  r1<= h.r2) by A1;
  hence h|Y is non-decreasing by Th22;
A2: for r1,r2 holds ( r1 in Y /\ dom h & r2 in Y /\ dom h & r1<r2 implies h.
  r2<= h.r1) by A1;
  hence h|Y is non-increasing by Th23;
  thus thesis by A2,Th23;
end;
