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

<b>let </b><font color="Maroon">A</font> be  <a href="integra1.html#V1">closed-interval</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<a NAME="E1:3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> implies <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 ;<br/>

<b>assume </b><a NAME="E2:3_1"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> as   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>, <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E4:3_1"/><i><font color="Green">E41</font></i>: 
 <a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T78">RFUNCT_1:78</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>
;<br/>
<a NAME="E6:3_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E7:3_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:3_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E3:3_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i><a class="ref" href="rfunct_1.html#T79">RFUNCT_1:79</a></i>
<br/>.= 
<font color="Maroon">r</font>

;<br/>
<b>hence </b><a NAME="E5:3_1_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:3_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun2.html#T3">PARTFUN2:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:3"/>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> implies ( <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:3_2"/><b>then </b>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> &amp;  <a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font> is <a href="partfun1.html#V1">total</a> )
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T78">RFUNCT_1:78</a>, <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E3:3_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:3_2"/>
<font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<a NAME="E5:3_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:3_2_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:3_2_1"/>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T79">RFUNCT_1:79</a></i>;<br/>
<b>then </b><font color="Maroon">x</font> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1
<b>by </b><i>, , <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">r</font>

;<br/>
<b>hence </b><a NAME="E6:3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.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>
<a NAME="E6:3_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:3_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:3_2_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:3_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">a</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="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:3_2_2"/>
<span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="ref" href="rfunct_1.html#T79">RFUNCT_1:79</a></i>;<br/>
<a NAME="E6:3_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 1
 
<b>by </b><i>, , <a class="ref" href="seq_1.html#D6">SEQ_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E7:3_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:3_2"/><b>then </b>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E9:3_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
( ( <font color="Maroon">f</font> is <a href="partfun1.html#V1">total</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1">}</a></span> ) iff <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="seq_1.html#K13">(#)</a> <span class="p1">(<span class="default"><a href="rfunct_1.html#K5">chi</a> <font color="Maroon">A</font>,<font color="Maroon">A</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
