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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>



<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default">0 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">x'</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default">1 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">x'</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72">RVSUM_1:72</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
;<br/>
<a NAME="E3:1"/>
(  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp;  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseq_2.html#K2">|-&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a>, <a class="ref" href="euclid.html#D9">EUCLID:def 9</a></i>;<br/>
<b>then </b><font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">x'</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T33">RVSUM_1:33</a></i>
<br/>.= 
<font color="Maroon">x</font>

;<br/>
<b>hence </b><a NAME="E5:1"/>
( <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i/>;<br/>


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