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
  X c= Y & h|Y is non-decreasing implies h|X is non-decreasing
proof
  assume that
A1: X c= Y and
A2: h|Y is non-decreasing;
  now
A3: X /\ dom h c= Y /\ dom h by A1,XBOOLE_1:26;
    let r1,r2;
    assume r1 in X /\ dom h & r2 in X /\ dom h & r1 < r2;
    hence h.r1 <= h.r2 by A2,A3,Th22;
  end;
  hence thesis by Th22;
end;
