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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">X</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="E1:55"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">h</font> <a href="rfunct_2.html#R3" target="_self">is_non_decreasing_on</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">h</font> <a href="rfunct_2.html#R4" target="_self">is_non_increasing_on</a> <font color="Maroon">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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>  or <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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> 
 ;<br/><div class="add"><a NAME="E2:55_1_1_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E3:55_1_1_1"/>
<font color="Maroon">h</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T58">PARTFUN2:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_2"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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>;<br/><a NAME="E3:55_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">r1</font> = <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>assume </b><a NAME="E1:55_1_1_2_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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>
 ;<br/><a NAME="E2:55_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:55_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:55_1_1_2_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">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">h</font></span>)</span> &amp; <font color="Maroon">p</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">h</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:55_1_1_2_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:55_1_1_2_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:55_1_1_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">x</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="Maroon">x</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">h</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> or <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:55_1_1_2_1_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:55_1_1_2_1_1_1_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E4:55_1_1_2_1_1_1_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:55_1_1_2_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E3:55_1_1_2_1_1_1_1_2"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E4:55_1_1_2_1_1_1_1_2"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:55_1_1_2_1_1"/>
<font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:55_1_1_2_1"/>
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> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font></span>)</span> <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> holds <br/><font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:55_1_1_2"/>
<font color="Maroon">h</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="partfun2.html#T76">PARTFUN2:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:55"/>
<font color="Maroon">h</font> <a href="partfun2.html#R1">is_constant_on</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X</font>
 ;<br/>


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