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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> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>thus </b><a NAME="E2:29"/>
( <span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1"/>
<span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="euclid.html#K5" target="_self">0*</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0

;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T59">RVSUM_1:59</a></i>;<br/>


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

<b>assume </b><a NAME="E3:29"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#T58">RVSUM_1:58</a></i>
<br/>.= 
 <a href="euclid.html#K5" target="_self">0*</a> <font color="Maroon">c<sub>1</sub></font>

;<br/>
<b>hence </b><a NAME="E5:29"/>
<span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="euclid.html#T10" target="_self">Th10</a></i>;<br/>


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