<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E15">A1</font></i>: 
 <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:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>











<b>assume </b><a NAME="E2:9"/><i><font color="Green" title="E16">A2</font></i>: 
( 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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/>

<b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c9">a1'</font> = <font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c11">b1'</font> = <font color="Maroon" title="c5">b1</font>, <font color="Maroon" title="c12">c'</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c13">c1'</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> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<a NAME="E4:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c9">a1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">A'</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><b> such that </b><br/><a NAME="E6:9"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c14">A'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A'</font> &amp; <font color="Maroon" title="c9">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A'</font> &amp; <font color="Maroon" title="c9">a1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E7:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">b1'</font>,<font color="Maroon" title="c11">b1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">M'</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><b> such that </b><br/><a NAME="E9:9"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c15">M'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M'</font> &amp; <font color="Maroon" title="c11">b1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M'</font> &amp; <font color="Maroon" title="c11">b1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E10:9"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">c1'</font>,<font color="Maroon" title="c13">c1'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">N'</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><b> such that </b><br/><a NAME="E12:9"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c16">N'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c12">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N'</font> &amp; <font color="Maroon" title="c13">c1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N'</font> &amp; <font color="Maroon" title="c13">c1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E13:9"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c14">A'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">M'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<a NAME="E1:9_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_1"/>
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:9_1"/><b>then </b>
for <font color="Olive" title="b1">x'</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">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A'</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Olive" title="b1">x'</font> )
 
<b>by </b><i><a class="txt" href="#E6:9"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c14">A'</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">a'</font>,<font color="Maroon" title="c9">a1'</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="E4:9_1"/>
<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>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E5:9_1"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">b1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E6:9_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">b1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">A'</font>
 
<b>by </b><i><a class="txt" href="#E1:9_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="#E3:9_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></i>;<br/>
<a NAME="E7:9_1"/><i><font color="Green" title="E24">A10</font></i>: 
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">b1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1_2"/>
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_1_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:9_1_2"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E4:9_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:9_1_2"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">b'</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="E6:9_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:9_1"/><b>then </b>
for <font color="Olive" title="b1">x'</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">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">M'</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">b1'</font>,<font color="Olive" title="b1">x'</font> )
 
<b>by </b><i><a class="txt" href="#E9:9"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<a NAME="E9:9_1"/><b>then </b>
<font color="Maroon" title="c15">M'</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">b'</font>,<font color="Maroon" title="c11">b1'</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/>
<b>hence </b><a NAME="E10:9_1"/>
<font color="Maroon" title="c14">A'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c15">M'</font>
 <b>by </b><i><a class="txt" href="#E6:9_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="#E7:9_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E14:9"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c14">A'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">N'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<a NAME="E1:9_2"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2_1"/>
<font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_2_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">b'</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:9_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:9_2"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c14">A'</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">a'</font>,<font color="Maroon" title="c9">a1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:9_2_2"/>
for <font color="Olive" title="b1">x'</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">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">A'</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Olive" title="b1">x'</font> )
 
<b>by </b><i><a class="txt" href="#E6:9"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E1:9_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E2:9_2_2"/>
<font color="Maroon" title="c14">A'</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">a'</font>,<font color="Maroon" title="c9">a1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:9_2"/>
<font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">c1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:9_2_3"/>
<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>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<b>hence </b><a NAME="E2:9_2_3"/>
<font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">c1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:9_2"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
<font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">c1'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c14">A'</font>
 
<b>by </b><i><a class="txt" href="#E1:9_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="#E2:9_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="aff_1.html#D4" title="AFF_1:def.4">AFF_1:def 4</a></i>;<br/>
<a NAME="E5:9_2"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c12">c'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">c1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2_4"/>
<font color="Maroon" title="c12">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">c1'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_2_4"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T81" title="ANALMETR:th.81">ANALMETR:81</a></i>;<br/>
<a NAME="E3:9_2_4"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="analmetr.html#D11" title="ANALMETR:def.11">ANALMETR:def 11</a></i>;<br/>
<a NAME="E4:9_2_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>;<br/>
<a NAME="E5:9_2_4"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c12">c'</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="E6:9_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:9_2"/>
<font color="Maroon" title="c16">N'</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">c'</font>,<font color="Maroon" title="c13">c1'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:9_2_5"/>
for <font color="Olive" title="b1">x'</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">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">N'</font> iff  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">c'</font>,<font color="Maroon" title="c13">c1'</font>,<font color="Olive" title="b1">x'</font> )
 
<b>by </b><i><a class="txt" href="#E12:9"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E5:9_2"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></i>;<br/>
<b>hence </b><a NAME="E2:9_2_5"/>
<font color="Maroon" title="c16">N'</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">c'</font>,<font color="Maroon" title="c13">c1'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:9_2"/>
<font color="Maroon" title="c14">A'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c16">N'</font>
 <b>by </b><i><a class="txt" href="#E4:9_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="#E5:9_2"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="aff_1.html#D5" title="AFF_1:def.5">AFF_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:9"/><i><font color="Green" title="E22">A16</font></i>: 
( <font color="Maroon" title="c14">A'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">M'</font> &amp; <font color="Maroon" title="c14">A'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">N'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_3"/>
( <font color="Maroon" title="c14">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">M'</font> or <font color="Maroon" title="c14">A'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">N'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:9_3"/><b>then </b>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c10">b'</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c12">c'</font> )
 
<b>by </b><i><a class="txt" href="#E6:9"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E9:9"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E12:9"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:9_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T55" title="ANALMETR:th.55">ANALMETR:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E16:9"/><i><font color="Green" title="E23">A17</font></i>: 
<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c11">b1'</font>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E17:9"/>
<font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c12">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1'</font>,<font color="Maroon" title="c13">c1'</font>
 
<b>by </b><i><a class="txt" href="#E2:9"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>;<br/>
<a NAME="E18:9"/><b>then </b>
<font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c12">c'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">b1'</font>,<font color="Maroon" title="c13">c1'</font>
 
<b>by </b><i><a class="txt" href="#E1:9"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E6:9"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E9:9"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E12:9"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="#E13:9"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="#E14:9"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="#E15:9"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="#E16:9"><i><font color="Green" title="E23">A17</font></i></a>, <a class="ref" href="aff_2.html#D11" title="AFF_2:def.11">AFF_2:def 11</a></i>;<br/>
<b>hence </b><a NAME="E19:9"/>
<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>
 <b>by </b><i><a class="ref" href="analmetr.html#T48" title="ANALMETR:th.48">ANALMETR:48</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
