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

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<b>thus </b><a NAME="E1:65"/>
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R1" title="RFUNCT_2:pred.1">is_increasing_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D2" target="_self" title="RFUNCT_2:def.2">RFUNCT_2:def 2</a><br/><b>let </b><font color="Maroon" title="c6">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E2:65_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 ;<br/>

<a NAME="E3:65_1"/><b>then </b><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E4:65_1"/><i><font color="Green" title="E33">A4</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E5:65_1"/><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:65_1"/><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="rfunct_2.html#D2" target="_self" title="RFUNCT_2:def.2">Def2</a></i>;<br/>
<a NAME="E7:65_1"/>
<font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="rfunct_2.html#D2" target="_self" title="RFUNCT_2:def.2">Def2</a></i>;<br/>
<a NAME="E8:65_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E6:65_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:65_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:65_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:65"/>
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green" title="E30">A7</font></i>: 
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R2" title="RFUNCT_2:pred.2">is_decreasing_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D3" target="_self" title="RFUNCT_2:def.3">RFUNCT_2:def 3</a><br/><b>let </b><font color="Maroon" title="c6">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E2:65_2"/><i><font color="Green" title="E31">A8</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 ;<br/>

<a NAME="E3:65_2"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E4:65_2"/><i><font color="Green" title="E33">A10</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E5:65_2"/><i><font color="Green" title="E34">A11</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:65_2"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="rfunct_2.html#D3" target="_self" title="RFUNCT_2:def.3">Def3</a></i>;<br/>
<a NAME="E7:65_2"/>
<font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="rfunct_2.html#D3" target="_self" title="RFUNCT_2:def.3">Def3</a></i>;<br/>
<a NAME="E8:65_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E6:65_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/>
<a NAME="E9:65_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:65_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_2"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E3:65"/>
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_3"/><i><font color="Green" title="E30">A13</font></i>: 
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R3" title="RFUNCT_2:pred.3">is_non_decreasing_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">RFUNCT_2:def 4</a><br/><b>let </b><font color="Maroon" title="c6">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E2:65_3"/><i><font color="Green" title="E31">A14</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 ;<br/>

<a NAME="E3:65_3"/><b>then </b><i><font color="Green" title="E32">A15</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E4:65_3"/><i><font color="Green" title="E33">A16</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_3"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E5:65_3"/><i><font color="Green" title="E34">A17</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E2:65_3"><i><font color="Green" title="E31">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:65_3"/><i><font color="Green" title="E35">A18</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_3"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_3"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_3"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_3"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">Def4</a></i>;<br/>
<a NAME="E7:65_3"/>
<font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E1:65_3"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="rfunct_2.html#E2:65_3"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="rfunct_2.html#E3:65_3"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="rfunct_2.html#E4:65_3"><i><font color="Green" title="E33">A16</font></i></a>, <a class="ref" href="rfunct_2.html#D4" target="_self" title="RFUNCT_2:def.4">Def4</a></i>;<br/>
<a NAME="E8:65_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E6:65_3"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:65_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_3"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:65_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E5:65_3"><i><font color="Green" title="E34">A17</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E4:65"/><i><font color="Green" title="E30">A19</font></i>: 
( <font color="Maroon" title="c3">h1</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c4">h2</font> <a href="rfunct_2.html#R4" title="RFUNCT_2:pred.4">is_non_increasing_on</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c5">r1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_2.html#D5" target="_self" title="RFUNCT_2:def.5">RFUNCT_2:def 5</a><br/><b>let </b><font color="Maroon" title="c6">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E5:65"/><i><font color="Green" title="E31">A20</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 ;<br/>

<a NAME="E6:65"/><b>then </b><i><font color="Green" title="E32">A21</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E7:65"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">h1</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">h2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:65"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="rfunct_2.html#T68" target="_self" title="RFUNCT_2:th.68">Th68</a></i>;<br/>
<a NAME="E8:65"/><i><font color="Green" title="E34">A23</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">r2</font> )
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E5:65"><i><font color="Green" title="E31">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E9:65"/><i><font color="Green" title="E35">A24</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:65"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="rfunct_2.html#E5:65"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="rfunct_2.html#E6:65"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="rfunct_2.html#E7:65"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="rfunct_2.html#D5" target="_self" title="RFUNCT_2:def.5">Def5</a></i>;<br/>
<a NAME="E10:65"/>
<font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E4:65"><i><font color="Green" title="E30">A19</font></i></a>, <a class="txt" href="rfunct_2.html#E5:65"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="rfunct_2.html#E6:65"><i><font color="Green" title="E32">A21</font></i></a>, <a class="txt" href="rfunct_2.html#E7:65"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="rfunct_2.html#D5" target="_self" title="RFUNCT_2:def.5">Def5</a></i>;<br/>
<a NAME="E11:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E9:65"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E12:65"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_2.html#E8:65"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E13:65"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <font color="Maroon" title="c4">h2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">r1</font>
 <b>by </b><i><a class="txt" href="rfunct_2.html#E8:65"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>


</div>
