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<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>let </b><font color="Maroon">x1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" target="_self">REAL-NS</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">x2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" target="_self">REAL-US</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x1</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>thus </b><span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><font color="Maroon">x1</font></span><a href="normsp_1.html#K1">.||</a></span> <a href="square_1.html#K5">^2</a>  = 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">x</font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T12">EUCLID_2:12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" target="_self">Euclid_scalar</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">x</font>,<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x2</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Maroon">x2</font>
<b>by </b><i>, </i>
;<br/>


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