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<b>let </b><font color="Maroon">seq</font> be   <a href="seq_1.html#NM1">Real_Sequence</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="E1:36"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>thus </b><a NAME="E1:36_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font></span>)</span> implies <font color="Maroon">r</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">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:36_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:36_1_1"/><i><font color="Green">E45</font></i>: 
( <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> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> &amp; <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="partfun2.html#T78">PARTFUN2:78</a></i>;<br/>
<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:36_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:36_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E7:36_1_1"/>
<font color="Maroon">r</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">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:36_1"/>
<font color="Maroon">r</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">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/><a NAME="E5:36_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i/>;<br/><a NAME="E6:36_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E7:36_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="rfunct_2.html#K1" target="_self">*</a> <font color="Maroon">seq</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T8">SUBSET_1:8</a></i>;<br/>


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