<?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">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>,<font color="Maroon">D</font>;<br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green">E29</font></i>: 
<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">f</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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/>

<b>thus </b><a NAME="E2:43"/>
( <font color="Maroon">f</font> <a href="partfun2.html#R1" target="_self">is_constant_on</a> <font color="Maroon">X</font> implies  ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">d</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">f</font> <a href="partfun2.html#R1" target="_self">is_constant_on</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:43_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</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">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E4:43_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">y</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:43_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>then </b><font color="Maroon">d</font> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E7:43_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E5:43_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:43_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <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">f</font></span>)</span>;<br/>
<a NAME="E3:43_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</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">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> ;<br/>
<a NAME="E5:43_1_2"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</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">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">y</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:43_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


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


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

<b>given </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><a NAME="E3:43"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>take </b>
<font color="Maroon">d</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="partfun2.html#D3" target="_self">PARTFUN2:def 3</a><br/>

<b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E4:43"/>
<font color="Maroon">c</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">f</font></span>)</span>
 ;<br/>

<a NAME="E5:43"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c</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">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E6:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/>


</div>
