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<b>let </b><font color="Maroon">a3</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b3</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:67_1_2"/><i><font color="Green">E50</font></i>: 
(  ex <font color="Olive">a1</font>, <font color="Olive">a2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) &amp;  ex <font color="Olive">a1</font>, <font color="Olive">a2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <span class="p1">(<span class="default"><font color="Olive">a1</font> <a href="real_1.html#K3">+</a> <font color="Olive">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">a01</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a02</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:67_1_2"/><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a01</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a02</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a01</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a02</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 ;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:67_1_2"/><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a2</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">b3</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a001</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a002</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">a003</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:67_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a001</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">a002</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">a003</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a001</font> <a href="real_1.html#K3">+</a> <font color="Maroon">a002</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">a003</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <font color="Olive">b01</font>, <font color="Olive">b02</font>, <font color="Olive">b03</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">b01</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Olive">b02</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">b03</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p3</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b01</font> <a href="real_1.html#K3">+</a> <font color="Olive">b02</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Olive">b03</font> <a href="hidden.html#R1">=</a> 1 holds <br/>( <font color="Olive">b01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a001</font> &amp; <font color="Olive">b02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a002</font> &amp; <font color="Olive">b03</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a003</font> ) ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:67_1_2"/>
( <font color="Maroon">a01</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a001</font> &amp; <font color="Maroon">a02</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a002</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a003</font> )
 
<b>by </b><i>, <a class="ref" href="euclid_3.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E9:67_1_2"/>
<font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font>
 <b>by </b><i>, <a class="ref" href="euclid_3.html#T1" target="_self">Th1</a></i>;<br/>


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