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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">x1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x3</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>







<a NAME="E1:94"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K3">+</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E2:94"/><i><font color="Green">E66</font></i>: 
( 1 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> = 
<font color="Maroon">x1</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>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a>, <a class="ref" href="euclidlp.html#T29" target="_self">Th29</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E4:94"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:94"/><i><font color="Green">E89</font></i>: 
<span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E6:94"/>
( 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; 1 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> = 
<font color="Maroon">x2</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>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E8:94"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E9:94"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default">0 <a href="real_1.html#K3">+</a> 0</span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="hidden.html#R1">=</a> 1
 
;<br/>
<a NAME="E10:94"/>
( 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; 0 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; 1 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> )
 
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K9">*</a> <font color="Maroon">x3</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
<br/>.= 
<font color="Maroon">x3</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<b>hence </b><a NAME="E12:94"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="euclidlp.html#K4" target="_self">plane</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#T30" target="_self">Th30</a>, <a class="ref" href="euclidlp.html#T32" target="_self">Th32</a>, <a class="ref" href="euclidlp.html#T33" target="_self">Th33</a></i>;<br/>


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