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<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">q'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/>





















<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <b>and </b><br/><a NAME="E3:27"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">AFP</font> <a href="aff_2.html#NR7" target="_self">satisfies_TDES</a> 
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>and </b><br/><a NAME="E5:27"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>and </b><br/><a NAME="E6:27"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> )
 ;<br/>

<a NAME="E7:27"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">M</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:27_1"/>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><b>hence </b><a NAME="E3:27_1"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:27_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E3:27_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">B'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>;<br/><b>set </b><font color="Maroon">D</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">A'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font>;<br/><b>set </b><font color="Maroon">B''</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font>;<br/><b>set </b><font color="Maroon">C'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>;<br/><b>set </b><font color="Maroon">D'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font>;<br/><a NAME="E4:27_2"/><i><font color="Green">E56</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E5:27_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E6:27_2"/><i><font color="Green">E58</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T51">AFF_1:51</a></i>;<br/><a NAME="E7:27_2"/><i><font color="Green">E59</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; not <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E8:27_2"/><i><font color="Green">E60</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">q'</font>,<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> )
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E9:27_2"/><i><font color="Green">E62</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">p'</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">q'</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>(  <a href="aff_1.html#R1">LIN</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> &amp; <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/><b>then consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:27_2_1"/><i><font color="Green">E64</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="translac.html#T2">TRANSLAC:2</a></i>;<br/><a NAME="E4:27_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon">N</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E6:27_2_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E7:27_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">N</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i><a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E8:27_2_1"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_2_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E2:27_2_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:27_2_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E4:27_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="homothet.html#E1:27"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E10:27_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> )
 <b>by </b><i>, <a class="txt" href="homothet.html#E7:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E11:27_2_1"/>
( <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> )
 <b>by </b><i>, , , , , , , , <a class="txt" href="homothet.html#E1:27"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="homothet.html#E5:27"><i><font color="Green">E47</font></i></a>, <a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E12:27_2_1"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">d</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">N</font> &amp; <font color="Maroon">M</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">N</font> &amp; <font color="Maroon">M</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , , , , , <a class="txt" href="homothet.html#E1:27"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="homothet.html#E5:27"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E13:27_2_1"/><b>then </b>
( <font color="Maroon">d</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">d</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">z</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">M</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , , , , , <a class="txt" href="homothet.html#E1:27"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="homothet.html#E5:27"><i><font color="Green">E47</font></i></a>, <a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="aff_1.html#T30">AFF_1:30</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><b>hence </b><a NAME="E14:27_2_1"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>  holds <br/>(  not  <a href="aff_1.html#R1">LIN</a> <font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font> or  not <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> or  not <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> or  not <font color="Olive">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">c</font> )
 ;<br/><div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<br/><a NAME="E2:27_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:27_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:27_2_2_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E5:27_2_2_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:27_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">q'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E7:27_2_2_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E8:27_2_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E82</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_2"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<br/><a NAME="E2:27_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:27_2_2_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">y</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:27_2_2_2"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E5:27_2_2_2"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E7:27"><i><font color="Green">E49</font></i></a>, <a class="ref" href="aff_1.html#T11">AFF_1:11</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E84</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<br/><a NAME="E2:27_2_2_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:27_2_2_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">x</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E4:27_2_2_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><a NAME="E5:27_2_2_3"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E6:27_2_2_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>,<font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E7:27_2_2_3"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E8:27_2_2_3"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> )
 ;<br/><div><i><font color="Green">E87</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_4_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:27_2_2_4_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">q</font>
 <b>by </b><i>, <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E3:27_2_2_4_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:27_2_2_4_1"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><b>hence </b><a NAME="E5:27_2_2_4_1"/>
contradiction
 <b>by </b><i>, , , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_2_4_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:27_2_2_4_2"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, <a class="ref" href="aff_1.html#T14">AFF_1:14</a></i>;<br/><a NAME="E3:27_2_2_4_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E4:27_2_2_4_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">y</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/><a NAME="E5:27_2_2_4_2"/><b>then </b>
<font color="Maroon">q'</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="homothet.html#E8:27"><i><font color="Green">E50</font></i></a>, <a class="txt" href="homothet.html#E1:27_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="aff_1.html#T39">AFF_1:39</a></i>;<br/><b>hence </b><a NAME="E6:27_2_2_4_2"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:27_2_2_4"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="homothet.html#E2:27"><i><font color="Green">E44</font></i></a>, <a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:27_2_2"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="txt" href="homothet.html#E2:27"><i><font color="Green">E44</font></i></a>, <a class="txt" href="homothet.html#E4:27"><i><font color="Green">E46</font></i></a>, <a class="txt" href="homothet.html#E5:27"><i><font color="Green">E47</font></i></a>, <a class="txt" href="homothet.html#E6:27"><i><font color="Green">E48</font></i></a>, <a class="txt" href="homothet.html#E7:27"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:27_2"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:27"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:27"/>
<font color="Maroon">q</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i/>;<br/>


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