<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM2">Permutation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E2:34"/>
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 <b>and </b><br/><a NAME="E3:34"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> iff ( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) or ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> ) ) ) )
 ;<br/>

<a NAME="E4:34"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E1:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T48">AFF_1:48</a>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E1:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T40">AFF_1:40</a></i>;<br/>
<a NAME="E6:34"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>1</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:34_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/><a NAME="E2:34_1_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="transgeo.html#T110">TRANSGEO:110</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:34_1_1_2_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E2:34_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:34_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_2_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/><a NAME="E4:34_1_2_1_1"/>
not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E5:34_1_2_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_2_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E6:34_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><b>hence </b><a NAME="E7:34_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_2_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:34_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_2_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_2_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="transgeo.html#T2">TRANSGEO:2</a></i>;<br/><a NAME="E4:34_1_2_2_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_2_2_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:34_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_2_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T48">AFF_1:48</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E6:34_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E7:34_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_2_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><b>hence </b><a NAME="E8:34_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_2_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E2:34_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_3"/><i><font color="Green">E42</font></i>: 
( not <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_3"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:34_1_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E6:34_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E5:34_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E7:34_1_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_3_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="homothet.html#E5:34_1_3"><i><font color="Green">E44</font></i></a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/>
<b>hence </b><a NAME="E2:34_1_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E5:34_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="homothet.html#E6:34_1_3"><i><font color="Green">E45</font></i></a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:34_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E10:34_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E9:34_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E11:34_1_3"/>
not <font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="homothet.html#E9:34_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:34_1_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E10:34_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><a NAME="E14:34_1_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="homothet.html#E5:34_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="homothet.html#E7:34_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E15:34_1_3"/><b>then </b><i><font color="Green">E50</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E17:34_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E18:34_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E19:34_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_3_2_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_3_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:34_1_3_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a></i>;<br/><b>hence </b><a NAME="E3:34_1_3_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_3_2_1_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_3_2_1_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E4:34_1_3_2_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E5:34_1_3_2_1_2"/>
not <font color="Maroon">c<sub>15</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:34_1_3_2_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E4:34_1_3_2_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E8:34_1_3_2_1_2"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_3_2_1_2_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:34_1_3_2_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="homothet.html#E7:34_1_3_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<a NAME="E3:34_1_3_2_1_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="homothet.html#E18:34_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3_2_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1_3_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E4:34"><i><font color="Green">E37</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="homothet.html#E15:34_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:34_1_3_2_1_2"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E9:34_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="homothet.html#E15:34_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3_2_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_2_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="homothet.html#E7:34_1_3_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E10:34_1_3_2_1_2"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E11:34_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="homothet.html#E8:34_1_3_2_1_2"><i><font color="Green">E60</font></i></a></i>;<br/><b>hence </b><a NAME="E12:34_1_3_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E7:34_1_3_2_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:34_1_3_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E2:34_1_3_2_1"><i><font color="Green">E55</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_3_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E20:34_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:34_1_3_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_3_3_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="ref" href="transgeo.html#T111">TRANSGEO:111</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_3_3_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:34_1_3_3_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E4:34_1_3_3_1_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E5:34_1_3_3_1_1"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E5:34"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E1:34_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="homothet.html#E1:34_1_3_3_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E6:34_1_3_3_1_1"/>
not <font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_3_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1_3_3_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E2:34_1_3_3_1_1_1"/><b>then </b>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:34_1_3_3_1_1_1"/><b>then </b>
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E4:34_1_3_3_1_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E5:34_1_3_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E4:34"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:34_1_3_3_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E18:34_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="homothet.html#E4:34_1_3_3_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E9:34_1_3_3_1_1"/>
( <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E9:34_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="homothet.html#E15:34_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="homothet.html#E8:34_1_3_3_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E10:34_1_3_3_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>17</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>hence </b><a NAME="E11:34_1_3_3_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E13:34_1_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="homothet.html#E5:34_1_3_3_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="homothet.html#E8:34_1_3_3_1_1"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:34_1_3_3_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:34"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E17:34_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="homothet.html#E3:34_1_3_3_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1_3_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E21:34_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E18:34_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="homothet.html#E19:34_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E2:34_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="homothet.html#E3:34_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:34"/>
<font color="Maroon">c<sub>5</sub></font> <a href="transgeo.html#NR10" target="_self">is_Col</a> 
 <b>by </b><i><a class="ref" href="transgeo.html#T116">TRANSGEO:116</a></i>;<br/>


</div>
