<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  implies <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="conmetr.html#D8" target="_self" title="CONMETR:def.8">CONMETR:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">b1</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">b</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">a1</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">b1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">c1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b3">b1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a1</font>,<font color="Olive" title="b5">c1</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b1</font>,<font color="Olive" title="b5">c1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">c1</font> implies <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c7">c1</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
 not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E3:9"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b1</font>
 <span class="kw">and </span><br/><a NAME="E5:9"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font>
 <span class="kw">and </span><br/><a NAME="E6:9"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">b1</font>
 <span class="kw">and </span><br/><a NAME="E7:9"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">c1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c7">c1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">a9</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c9">a19</font> = <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c10">b9</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c11">b19</font> = <font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c12">c9</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c13">c19</font> = <font color="Maroon" title="c7">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T35" title="ANALMETR:th.35">ANALMETR:35</a></span>;<br/>
<a NAME="E9:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">A9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:9"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<font color="Maroon" title="c14">A9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E12:9"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font>
 <span class="kw">and </span><br/><a NAME="E13:9"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c9">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font>
 <span class="kw">and </span><br/><a NAME="E14:9"/>
<font color="Maroon" title="c9">a19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E15:9"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c10">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c11">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/>
<a NAME="E16:9"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E17:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">b19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/>
<a NAME="E18:9"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a19</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_1"/>
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:9_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:9"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:9"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E20:9"/><span class="kw">then </span>
<font color="Maroon" title="c14">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E21:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">b19</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:9"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E17:9"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></span>;<br/>
<a NAME="E22:9"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c12">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c13">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/>
<a NAME="E23:9"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E24:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">c19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>;<br/>
<a NAME="E25:9"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a19</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_2"/>
<font color="Maroon" title="c8">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_2"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:9_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:9"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A9</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:9"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E27:9"/><span class="kw">then </span>
<font color="Maroon" title="c14">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E28:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">c19</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">A9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:9"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="txt" href="#E24:9"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></span>;<br/>
<a NAME="E29:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">c19</font>,<font color="Maroon" title="c13">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">N9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E31:9"/><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c15">N9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E32:9"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N9</font>
 <span class="kw">and </span><br/><a NAME="E33:9"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c13">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N9</font>
 <span class="kw">and </span><br/><a NAME="E34:9"/>
<font color="Maroon" title="c13">c19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E35:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">b19</font>,<font color="Maroon" title="c11">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">M9</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E37:9"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c16">M9</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E38:9"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M9</font>
 <span class="kw">and </span><br/><a NAME="E39:9"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c11">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M9</font>
 <span class="kw">and </span><br/><a NAME="E40:9"/>
<font color="Maroon" title="c11">b19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<a NAME="E41:9"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
( <font color="Maroon" title="c14">A9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">M9</font> &amp; <font color="Maroon" title="c14">A9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">N9</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_3"/>
( <font color="Maroon" title="c14">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">M9</font> or <font color="Maroon" title="c14">A9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">N9</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_3"/><span class="kw">then </span>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c10">b9</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c12">c9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:9"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E38:9"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E32:9"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:9_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E42:9"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">c19</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_4"/>
<font color="Maroon" title="c12">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E3:9_4"/><span class="kw">then </span>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></span>;<br/>
<a NAME="E4:9_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/>
<a NAME="E5:9_4"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c12">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E43:9"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">N9</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">c19</font>,<font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:9"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E32:9"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E33:9"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E44:9"/><span class="kw">then </span>
<font color="Maroon" title="c15">N9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c12">c9</font>,<font color="Maroon" title="c13">c19</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E45:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<font color="Maroon" title="c14">A9</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c15">N9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:9"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="txt" href="#E42:9"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></span>;<br/>
<a NAME="E46:9"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b19</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_5"/>
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="analmetr.html#T59" title="ANALMETR:th.59">ANALMETR:59</a></span>;<br/>
<a NAME="E3:9_5"/><span class="kw">then </span>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D10" title="ANALMETR:def.10">ANALMETR:def 10</a></span>;<br/>
<a NAME="E4:9_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>;<br/>
<a NAME="E5:9_5"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a9</font>,<font color="Maroon" title="c9">a19</font>,<font color="Maroon" title="c10">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="analmetr.html#T40" title="ANALMETR:th.40">ANALMETR:40</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E47:9"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">M9</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">b19</font>,<font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:9"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E38:9"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E39:9"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="aff_1.html#T21" title="AFF_1:th.21">AFF_1:21</a>, <a class="ref" href="aff_1.html#T25" title="AFF_1:th.25">AFF_1:25</a></span>;<br/>
<a NAME="E48:9"/><span class="kw">then </span>
<font color="Maroon" title="c16">M9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">b19</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></span>;<br/>
<a NAME="E49:9"/><span class="kw">then </span>
<font color="Maroon" title="c14">A9</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c16">M9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:9"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E46:9"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></span>;<br/>
<a NAME="E50:9"/><span class="kw">then </span>
<font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c12">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b19</font>,<font color="Maroon" title="c13">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E11:9"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E12:9"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E13:9"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E37:9"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E38:9"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E39:9"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="txt" href="#E31:9"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E32:9"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E33:9"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="txt" href="#E45:9"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E41:9"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E15:9"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="txt" href="#E22:9"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E51:9"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c7">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" title="ANALMETR:th.36">ANALMETR:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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