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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> 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">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:12"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:12"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then 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="E4:12"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:12"/><i><font color="Green">E11</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:12"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E8:12"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:12"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:12"/><i><font color="Green">E14</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<i><font color="Green">E15</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal1.html#E4:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topreal1.html#E10:12"><i><font color="Green">E13</font></i></a>, <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">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></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"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></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">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T34">EUCLID:34</a></i>
;<br/>
<a NAME="E13:12"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="topreal1.html#E5:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topreal1.html#E6:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topreal1.html#E11:12"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_2.html#T121">REAL_2:121</a>, <a class="ref" href="real_2.html#T140">REAL_2:140</a></i>;<br/>
<a NAME="E14:12"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal1.html#E12:12"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E15:12"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div>
