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<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="E1:35"/><i><font color="Green">E36</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="E2:35"/><i><font color="Green">E37</font></i>: 
<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="E3:35"/><i><font color="Green">E38</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="E4:35"/><i><font color="Green">E39</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#E1:35"><i><font color="Green">E36</font></i></a>, <a class="ref" href="aff_1.html#T40">AFF_1:40</a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="transgeo.html#NR10" target="_self">is_Col</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E1:35"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E2:35"><i><font color="Green">E37</font></i></a>, <a class="txt" href="homothet.html#E3:35"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E35"><i><font color="Green">Lemma34</font></i></a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green">E41</font></i>: 
for <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 <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</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>1</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:35"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E7:35"/>
for <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> holds <font color="Olive">b<sub>1</sub></font>,<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="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:35"><i><font color="Green">E36</font></i></a>, <a class="txt" href="homothet.html#E3:35"><i><font color="Green">E38</font></i></a>, <a class="txt" href="homothet.html#E34"><i><font color="Green">Lemma33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:35"/>
<font color="Maroon">c<sub>5</sub></font> <a href="homothet.html#R1" target="_self">is_Sc</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:35"><i><font color="Green">E39</font></i></a>, <a class="txt" href="homothet.html#E5:35"><i><font color="Green">E40</font></i></a>, <a class="txt" href="homothet.html#E6:35"><i><font color="Green">E41</font></i></a>, <a class="ref" href="homothet.html#D1" target="_self">Def1</a></i>;<br/>


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