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<b>let </b><font color="Maroon" title="c1">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>let </b><font color="Maroon" title="c2">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">x2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K7" title="REAL_NS1:func.7">REAL-US</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">x1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="real_ns1.html#D4" target="_self" title="REAL_NS1:def.4">Def4</a></i>;<br/>
<b>thus </b><span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<b>by </b><i><a class="ref" href="real_ns1.html#T1" target="_self" title="REAL_NS1:th.1">Th1</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_2.html#T12" title="EUCLID_2:th.12">EUCLID_2:12</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_2.html#D1" title="EUCLID_2:def.1">EUCLID_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="real_ns1.html#D5" target="_self" title="REAL_NS1:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c3">x2</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c3">x2</font>
<b>by </b><i><a class="txt" href="real_ns1.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="real_ns1.html#D6" target="_self" title="REAL_NS1:def.6">Def6</a></i>
;<br/>


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