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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:37"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:37"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:37"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:37"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E2:37_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:37_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:37"><i><font color="Green">E39</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E5:37_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="txt" href="euclidlp.html#E4:37_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="euclidlp.html#E2:37_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:37_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:37"><i><font color="Green">E40</font></i></a>, <a class="txt" href="euclidlp.html#E3:37"><i><font color="Green">E41</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>


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