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<div class="add">

<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">H1</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">H</font> be    <a href="seqfunc.html#M1" target="_self">Functional_Sequence</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/>







<b>thus </b><a NAME="E1:14"/>
( <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="seqfunc.html#K9" target="_self">/</a> <font color="Maroon">H</font> implies for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H1</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="rfunct_1.html#K2">/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="seqfunc.html#K9" target="_self">/</a> <font color="Maroon">H</font>
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><font color="Maroon">H1</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K3" target="_self">"</a> </span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rfunct_1.html#K4">^</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rfunct_1.html#K2">/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47">RFUNCT_1:47</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H1</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span> <a href="rfunct_1.html#K2">/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Olive">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>thus </b><font color="Maroon">H1</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rfunct_1.html#K2">/</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="rfunct_1.html#K4">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47">RFUNCT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="seqfunc.html#K3" target="_self">"</a> </span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="seqfunc.html#K9" target="_self">/</a> <font color="Maroon">H</font></span>)</span> <a href="seqfunc.html#K1" target="_self">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">H1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="seqfunc.html#K9" target="_self">/</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/>


</div>
