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<div class="add">

<b>let </b><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">q</font>, <font color="Maroon" title="c9">q'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/><b>let </b><font color="Maroon" title="c10">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>;<br/>



















<b>assume </b><b>that </b><br/><a NAME="E1:28"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">K</font> &amp; not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> )
 <b>and </b><br/><a NAME="E2:28"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c1">AFP</font> <a href="aff_2.html#NR7" title="AFF_2:NR.7">satisfies_TDES</a> 
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green" title="E31">A3</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">K</font> )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c9">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c4">x</font> )
 ;<br/>

<a NAME="E5:28"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c10">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 
<b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/>

<b>assume </b><a NAME="E2:28_1"/><i><font color="Green" title="E36">A8</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">y</font>
 ;<br/>

<a NAME="E3:28_1"/>
<font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E1:28_1"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E4:28_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:28_1"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">p'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font>;<br/>
<a NAME="E6:28_1"/><i><font color="Green" title="E38">A10</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="txt" href="homothet.html#E2:28_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="homothet.html#E5:28_1"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E7:28_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">K</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E2:28_1"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:28_1"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E6:28_1"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E35">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_2"/><i><font color="Green" title="E36">A12</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font>
 ;<br/><a NAME="E2:28_2"/>
<font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2_1"/><i><font color="Green" title="E37">A13</font></i>: 
<font color="Maroon" title="c6">p'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">q'</font>
 ;<br/>

<a NAME="E2:28_2_1"/>
<font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="homothet.html#E1:28_2"><i><font color="Green" title="E36">A12</font></i></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></i>;<br/>
<a NAME="E3:28_2_1"/><b>then </b>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">p'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">q'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/>
<a NAME="E4:28_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c9">q'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<a NAME="E5:28_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">p'</font>,<font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></i>;<br/>
<b>hence </b><a NAME="E6:28_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="homothet.html#E1:28_2_1"><i><font color="Green" title="E37">A13</font></i></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:28_2"/>
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E1:28_2"><i><font color="Green" title="E36">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E36">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_3"/><i><font color="Green" title="E37">A15</font></i>: 
( not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 ;<br/><a NAME="E2:28_3"/><b>then </b><i><font color="Green" title="E38">A16</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="homothet.html#E5:28"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="aff_1.html#T47" title="AFF_1:th.47">AFF_1:47</a></i>;<br/><b>set </b><font color="Maroon" title="c11">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font>;<br/><a NAME="E3:28_3"/><i><font color="Green" title="E39">A17</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="txt" href="homothet.html#E2:28_3"><i><font color="Green" title="E38">A16</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E4:28_3"/><b>then </b>
not  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">K</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:28_3"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="aff_1.html#T54" title="AFF_1:th.54">AFF_1:54</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">o</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><b> such that </b><br/><a NAME="E6:28_3"/><i><font color="Green" title="E40">A18</font></i>: 
( <font color="Maroon" title="c12">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c12">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">K</font> )
 <b>by </b><i><a class="txt" href="homothet.html#E5:28"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="homothet.html#E3:28_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/><a NAME="E7:28_3"/><i><font color="Green" title="E41">A19</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="homothet.html#E3:28_3"><i><font color="Green" title="E39">A17</font></i></a>, <a class="txt" href="homothet.html#E6:28_3"><i><font color="Green" title="E40">A18</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/><a NAME="E8:28_3"/><i><font color="Green" title="E42">A20</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></i>;<br/><a NAME="E9:28_3"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">p'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">p'</font> )
 <b>by </b><i><a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E10:28_3"/><b>then </b><i><font color="Green" title="E43">A21</font></i>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">o</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E2:28"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E1:28_3"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="homothet.html#E6:28_3"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="homothet.html#E7:28_3"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="homothet.html#E8:28_3"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="homothet.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a></i>;<br/><a NAME="E11:28_3"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/><a NAME="E12:28_3"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c9">q'</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E2:28"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="homothet.html#E5:28"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="homothet.html#E6:28_3"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="homothet.html#E7:28_3"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="homothet.html#E10:28_3"><i><font color="Green" title="E43">A21</font></i></a>, <a class="txt" href="homothet.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E13:28_3"/>
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:28_4"/><i><font color="Green" title="E37">A22</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c10">K</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font> )
 ;<br/><b>set </b><font color="Maroon" title="c11">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/><a NAME="E2:28_4"/><i><font color="Green" title="E38">A23</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="homothet.html#E1:28_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/><a NAME="E3:28_4"/><i><font color="Green" title="E39">A24</font></i>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">K</font>
 <b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E1:28_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></i>;<br/><a NAME="E4:28_4"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:28_4_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E1:28"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="homothet.html#E1:28_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="aff_1.html#T40" title="AFF_1:th.40">AFF_1:40</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></i>;<br/>
<a NAME="E2:28_4_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:28_4_1"/><i><font color="Green" title="E40">A25</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>

<a NAME="E4:28_4_1"/>
<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c7">y</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E3:28_4"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/>
<b>hence </b><a NAME="E5:28_4_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="homothet.html#E2:28_4"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="homothet.html#E3:28_4_1"><i><font color="Green" title="E40">A25</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:28_4"/>
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E2:28"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="homothet.html#E3:28"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="homothet.html#E1:28_4"><i><font color="Green" title="E37">A22</font></i></a>, <a class="txt" href="homothet.html#E2:28_4"><i><font color="Green" title="E38">A23</font></i></a>, <a class="txt" href="homothet.html#E3:28_4"><i><font color="Green" title="E39">A24</font></i></a>, <a class="ref" href="homothet.html#T7" target="_self" title="HOMOTHET:th.7">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:28"/>
<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">q'</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="homothet.html#E4:28"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="homothet.html#E6:28"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="homothet.html#E7:28"><i><font color="Green" title="E35">A11</font></i></a>, <a class="txt" href="homothet.html#E8:28"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/>


</div>
