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<div class="add">

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">G</font>, <font color="Maroon" title="c4">H</font> be    <a href="seqfunc.html#M1" title="SEQFUNC:mode.1">Functional_Sequence</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<b>thus </b><a NAME="E1:14"/>
( <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K9" title="SEQFUNC:func.9">/</a> <font color="Maroon" title="c4">H</font> implies for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">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" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K9" title="SEQFUNC:func.9">/</a> <font color="Maroon" title="c4">H</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>thus </b><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K3" title="SEQFUNC:func.3">"</a> </span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E1:14_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D3" target="_self" title="SEQFUNC:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" title="RFUNCT_1:th.47">RFUNCT_1:47</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:14"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b1">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" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>thus </b><font color="Maroon" title="c2">H1</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="txt" href="seqfunc.html#E2:14"><i><font color="Green" title="E11">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rfunct_1.html#K4" title="RFUNCT_1:func.4">^</a> </span>)</span>
<b>by </b><i><a class="ref" href="rfunct_1.html#T47" title="RFUNCT_1:th.47">RFUNCT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="seq_1.html#K8" title="SEQ_1:func.8">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="seqfunc.html#K3" title="SEQFUNC:func.3">"</a> </span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seqfunc.html#D3" target="_self" title="SEQFUNC:def.3">Def3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K9" title="SEQFUNC:func.9">/</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c5">n</font>
<b>by </b><i><a class="ref" href="seqfunc.html#D8" target="_self" title="SEQFUNC:def.8">Def8</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> <a href="seqfunc.html#K9" title="SEQFUNC:func.9">/</a> <font color="Maroon" title="c4">H</font>
 <b>by </b><i><a class="ref" href="seqfunc.html#T2" target="_self" title="SEQFUNC:th.2">Th2</a></i>;<br/>


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