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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:7"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:7"/><i><font color="Green">E35</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E5:7"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<b>assume </b><a NAME="E6:7"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:7"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 <b>and </b><br/><a NAME="E9:7"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E10:7"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_1.html#T21">SPPOL_1:21</a></i>;<br/>
<a NAME="E11:7"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span>
 
;<br/>
<a NAME="E12:7"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span>
 
;<br/>
<i><font color="Green">E67</font></i>: <font color="Maroon">q</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid.html#T36">EUCLID:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
;<br/>
<i><font color="Green">E68</font></i>: <font color="Maroon">r</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="euclid.html#T36">EUCLID:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
;<br/>
<a NAME="E15:7"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i>, , , , <a class="ref" href="real_2.html#T140">REAL_2:140</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_3"/>
( <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_3_1"><b>suppose </b></a><a NAME="E1:7_3_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:7_3_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> or 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 <b>by </b><i>, , <a class="ref" href="real_2.html#T139">REAL_2:139</a></i>;<br/><a NAME="E3:7_3_1"/><b>then </b>
( 1 <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 1 or 1 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><font color="Maroon">q</font> = 
<span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">r</font>
<b>by </b><i>, , <a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_3_2"><b>suppose </b></a><a NAME="E1:7_3_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><b>hence </b><font color="Maroon">q</font> = 
<font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i>, , </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
