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<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>, <font color="Maroon">c<sub>5</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>thus </b><a NAME="E1:8"/>
( <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>3</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_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>3</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>hence </b><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T2" target="_self">Th2</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> implies <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>3</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>


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