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<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">h</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>assume </b><a NAME="E1:50"/><i><font color="Green">E44</font></i>: 
<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">h</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E2:50"/><b>then </b>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r2</font>
 
;<br/>
<b>hence </b><a NAME="E3:50"/>
<font color="Maroon">h</font> <a href="rfunct_2.html#R1" target="_self">is_increasing_on</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>

<a NAME="E4:50"/>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">h</font> <a href="rfunct_2.html#R2" target="_self">is_decreasing_on</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>

<a NAME="E6:50"/>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:50"/>
<font color="Maroon">h</font> <a href="rfunct_2.html#R3" target="_self">is_non_decreasing_on</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>

<a NAME="E8:50"/>
for <font color="Olive">r1</font>, <font color="Olive">r2</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">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">h</font></span>)</span> &amp; <font color="Olive">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">r1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:50"/>
<font color="Maroon">h</font> <a href="rfunct_2.html#R4" target="_self">is_non_increasing_on</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>

<b>hence </b><a NAME="E10:50"/>
<font color="Maroon">h</font> <a href="rfunct_2.html#R5" target="_self">is_monotone_on</a> <font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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