<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">h1</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">h2</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon">h1</font> <a href="rfunct_2.html#R1" target="_self">is_increasing_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">h2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> implies <font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font> <a href="rfunct_2.html#R1" target="_self">is_increasing_on</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">h1</font> <a href="rfunct_2.html#R1" target="_self">is_increasing_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">h2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:66_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T76">PARTFUN2:76</a></i>;<br/>
<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D2" target="_self">RFUNCT_2:def 2</a><br/><b>let </b><font color="Maroon">r2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E4:66_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/>

<a NAME="E5:66_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:66_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:66_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:66_1"/>
<font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E9:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E10:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:66_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:66"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">h1</font> <a href="rfunct_2.html#R2" target="_self">is_decreasing_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">h2</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:66"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T76">PARTFUN2:76</a></i>;<br/>
<b>let </b><font color="Maroon">r1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D3" target="_self">RFUNCT_2:def 3</a><br/><b>let </b><font color="Maroon">r2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E5:66"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/>

<a NAME="E6:66"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:66"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">h2</font></span>)</span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:66"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:66"/>
<font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E10:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E11:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:66"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E14:66"/>
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="seq_1.html#K6">+</a> <font color="Maroon">h2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font>
 <b>by </b><i>, <a class="ref" href="seq_1.html#D3">SEQ_1:def 3</a></i>;<br/>


</div>
