<?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> <a href="aff_2.html#NR11" title="AFF_2:NR.11">satisfies_des</a>  implies  <a href="conmetr.html#NR8" title="CONMETR:NR.8">des_holds_in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:10"/><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> <a href="aff_2.html#NR11" title="AFF_2:NR.11">satisfies_des</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="conmetr.html#NR8" title="CONMETR:NR.8">des_holds_in</a> <font color="Maroon" title="c1">X</font></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="E2:10"/><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="E4:10"/>
 <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="E6:10"/><i><font color="Green" title="E17">A3</font></i>: 
( <font color="Maroon" title="c14">A'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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="E7:10"/>
 <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="E9:10"/><i><font color="Green" title="E18">A4</font></i>: 
( <font color="Maroon" title="c15">M'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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="E10:10"/>
 <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="E12:10"/><i><font color="Green" title="E19">A5</font></i>: 
( <font color="Maroon" title="c16">N'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_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="E13:10"/><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="10_1"><b>proof </b></a><div class="add">

<a NAME="E1:10_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="10_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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="E2:10_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="E3:10_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:10"><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="E2:10_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="conmetr.html#E6:10"><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="E3:10_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="E4:10_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="conmetr.html#E2:10"><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="E5:10_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="E6:10_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="conmetr.html#E1:10_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="conmetr.html#E3:10_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="E7:10_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="10_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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="E2:10_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="conmetr.html#E2:10"><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="E3:10_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="E4:10_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="E5:10_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="E6:10_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:10"><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="E8:10_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="conmetr.html#E9:10"><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="E9:10_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="E10:10_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="conmetr.html#E6:10_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="conmetr.html#E7:10_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="E14:10"/><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="10_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_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="10_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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="E2:10_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="E3:10_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:10"><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="E2:10_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="10_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:10_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="conmetr.html#E6:10"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="conmetr.html#E1:10_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="E2:10_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="E3:10_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="10_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:10_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="conmetr.html#E2:10"><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="E2:10_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="E4:10_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="conmetr.html#E1:10_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="conmetr.html#E2:10_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="E5:10_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="10_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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="E2:10_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="conmetr.html#E2:10"><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="E3:10_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="E4:10_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="E5:10_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="E6:10_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:10"><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="E6:10_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="10_2_5"><b>proof </b></a><div class="add">

<a NAME="E1:10_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="conmetr.html#E12:10"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="conmetr.html#E5:10_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="E2:10_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="E7:10_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="conmetr.html#E4:10_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="conmetr.html#E5:10_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="E15:10"/><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="10_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_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="E2:10_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="conmetr.html#E6:10"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="conmetr.html#E9:10"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E12:10"><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="E3:10_3"/>
contradiction
 <b>by </b><i><a class="txt" href="conmetr.html#E2:10"><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="E16:10"/><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="conmetr.html#E2:10"><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="E17:10"/>
<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="conmetr.html#E2:10"><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="E18:10"/><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="conmetr.html#E1:10"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="conmetr.html#E6:10"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="conmetr.html#E9:10"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="conmetr.html#E12:10"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="conmetr.html#E13:10"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="conmetr.html#E14:10"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="conmetr.html#E15:10"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="conmetr.html#E16:10"><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="E19:10"/>
<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>
