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<span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x1</font> <a href="euclidlp.html#R1" title="EUCLIDLP:pred.1">//</a> <font color="Maroon" title="c5">x2</font> implies <font color="Maroon" title="c5">x2</font> <a href="euclidlp.html#R1" title="EUCLIDLP:pred.1">//</a> <font color="Maroon" title="c4">x1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:35_1_1"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="euclidlp.html#R1" title="EUCLIDLP:pred.1">//</a> <font color="Maroon" title="c5">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x2</font> <a href="euclidlp.html#R1" title="EUCLIDLP:pred.1">//</a> <font color="Maroon" title="c4">x1</font></span><br/>

<a NAME="E2:35_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:35_1_1"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E5:35_1_1"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:35_1_1"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="euclidlp.html#T32" target="_self" title="EUCLIDLP:th.32">Th37</a></span>;<br/>
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c4">x1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35_1_1"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="euclidlp.html#T1" target="_self" title="EUCLIDLP:th.1">Th1</a></span>
<br/>.= 
1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35_1_1"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
<br/>.= 
<font color="Maroon" title="c5">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:35_1_1"/>
<font color="Maroon" title="c5">x2</font> <a href="euclidlp.html#R1" title="EUCLIDLP:pred.1">//</a> <font color="Maroon" title="c4">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1_1"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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