theorem Th24:
  h|Y is non-decreasing iff for r1,r2 st r1 in Y /\ dom h & r2 in Y /\ dom h &
  r1 <= r2 holds h.r1 <= h.r2
proof
  thus h|Y is non-decreasing implies for r1,r2 st r1 in Y /\ dom h & r2 in Y
  /\ dom h & r1<=r2 holds h.r1 <= h.r2
  proof
    assume
A1: h|Y is non-decreasing;
    let r1,r2 such that
A2: r1 in Y /\ dom h & r2 in Y /\ dom h and
A3: r1<=r2;
    now
      per cases by A3,XXREAL_0:1;
      suppose
        r1 < r2;
        hence thesis by A1,A2,Th22;
      end;
      suppose
        r1 = r2;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
  assume
A4: for r1,r2 st r1 in Y /\ dom h & r2 in Y /\ dom h & r1<=r2 holds h.
  r1 <= h.r2;
  let r1,r2;
  assume that
A5: r1 in dom(h|Y) & r2 in dom(h|Y) and
A6: r1<r2;
A7: (h|Y).r1 = h.r1 & (h|Y).r2 = h.r2 by A5,FUNCT_1:47;
  r1 in Y /\ dom h & r2 in Y /\ dom h by A5,RELAT_1:61;
  hence thesis by A4,A6,A7;
end;
