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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green">E24</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><br/>for <font color="Olive">b<sub>3</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>,<font color="Olive">b<sub>2</sub></font> <a href="aff_1.html#R2">//</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>4</sub></font> being  <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> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="homothet.html#R1" target="_self">is_Sc</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</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> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</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>3</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>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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="E1:23_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; not <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <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><b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="homothet.html#R1" target="_self">is_Sc</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:23"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E4:23_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="transgeo.html#NR10" target="_self">is_Col</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="homothet.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:23_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/><a NAME="E6:23_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="homothet.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E7:23_1"/>
<font color="Maroon">c<sub>10</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>9</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:23_1_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T60">AFF_1:60</a></i>;<br/><a NAME="E3:23_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="homothet.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:23_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>10</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#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T45">AFF_1:45</a></i>;<br/><a NAME="E5:23_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="homothet.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E6:23_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</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#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="homothet.html#E4:23_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="transgeo.html#T108">TRANSGEO:108</a></i>;<br/><b>hence </b><a NAME="E7:23_1_1_1"/>
<font color="Maroon">c<sub>10</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>9</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E5:23_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="homothet.html#E2:23_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="homothet.html#E4:23_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="homothet.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</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="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E4:23_1_1_2"/>
<font color="Maroon">c<sub>10</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>9</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E1:23_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="aff_1.html#T61">AFF_1:61</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23_1_1"/>
<font color="Maroon">c<sub>10</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>9</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E8:23_1"/>
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:23_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="homothet.html#E3:23_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="homothet.html#E4:23_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="homothet.html#E6:23_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="transgeo.html#T107">TRANSGEO:107</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
<font color="Maroon">c<sub>1</sub></font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E2"><i><font color="Green">Lemma2</font></i></a></i>;<br/>


</div>
