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
  (h1|X is increasing & h2|Y is constant implies (h1+h2)|(X /\ Y) is
  increasing) & (h1|X is decreasing & h2|Y is constant implies (h1+h2)|(X /\ Y)
  is decreasing)
proof
  thus h1|X is increasing & h2|Y is constant implies (h1+h2)|(X /\ Y) is
  increasing
  proof
    assume that
A1: h1|X is increasing and
A2: h2|Y is constant;
    consider r being Element of REAL such that
A3: for p being Element of REAL st p in Y /\ dom h2 holds h2.p = r
         by A2,PARTFUN2:57;
    now
      let r1,r2;
      assume that
A4:   r1 in X /\ Y /\ dom (h1+h2) and
A5:   r2 in X /\ Y /\ dom (h1+h2) and
A6:   r1<r2;
      r1 in X /\ dom h1 & r2 in X /\ dom h1 by A4,A5,Th36;
      then h1.r1 < h1.r2 by A1,A6,Th20;
      then
A7:   h1.r1 + r < h1.r2 + r by XREAL_1:6;
      r1 in Y /\ dom h2 by A4,Th36;
      then
A8:   h1.r1 + h2.r1 < h1.r2 + r by A3,A7;
A9:   r1 in dom (h1+h2) by A4,XBOOLE_0:def 4;
      r2 in Y /\ dom h2 by A5,Th36;
      then h1.r1 + h2.r1 < h1.r2 + h2.r2 by A3,A8;
      then
A10:  (h1+h2).r1 < h1.r2 + h2.r2 by A9,VALUED_1:def 1;
      r2 in dom (h1+h2) by A5,XBOOLE_0:def 4;
      hence (h1+h2).r1 < (h1+h2).r2 by A10,VALUED_1:def 1;
    end;
    hence thesis by Th20;
  end;
  assume that
A11: h1|X is decreasing and
A12: h2|Y is constant;
  consider r being Element of REAL such that
A13: for p being Element of REAL st p in Y /\ dom h2 holds h2.p = r
by A12,PARTFUN2:57;
  now
    let r1,r2;
    assume that
A14: r1 in X /\ Y /\ dom (h1+h2) and
A15: r2 in X /\ Y /\ dom (h1+h2) and
A16: r1<r2;
    r1 in X /\ dom h1 & r2 in X /\ dom h1 by A14,A15,Th36;
    then h1.r2 < h1.r1 by A11,A16,Th21;
    then
A17: h1.r2 + r < h1.r1 + r by XREAL_1:6;
    r2 in Y /\ dom h2 by A15,Th36;
    then
A18: h1.r2 + h2.r2 < h1.r1 + r by A13,A17;
A19: r2 in dom (h1+h2) by A15,XBOOLE_0:def 4;
    r1 in Y /\ dom h2 by A14,Th36;
    then h1.r2 + h2.r2 < h1.r1 + h2.r1 by A13,A18;
    then
A20: (h1+h2).r2 < h1.r1 + h2.r1 by A19,VALUED_1:def 1;
    r1 in dom (h1+h2) by A14,XBOOLE_0:def 4;
    hence (h1+h2).r2 < (h1+h2).r1 by A20,VALUED_1:def 1;
  end;
  hence thesis by Th21;
end;
