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

<b>let </b><font color="Maroon" title="c7">POS</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"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/> ex <font color="Olive" title="b5">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">p</font> )</span><br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c11">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font> implies  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> ) )</span><br/>









<b>assume </b><a NAME="E1:99"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c13">b'</font> = <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c14">c'</font> = <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c15">d'</font> = <font color="Maroon" title="c11">d</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K3" title="ANALMETR:func.3">Af</a> <font color="Maroon" title="c7">POS</font></span>)</span> ;<br/>
<a NAME="E3:99"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">c'</font>,<font color="Maroon" title="c15">d'</font>,<font color="Maroon" title="c14">c'</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c12">a'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></i>;<br/>
<a NAME="E4:99"/><b>then </b><i><font color="Green" title="E74">A2</font></i>: 
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c10">c</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c8">a</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T55" target="_self" title="ANALMETR:th.55">Th55</a></i>;<br/>
<div><i><font color="Green" title="E75">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )</span><br/><a NAME="E2:99_1"/><b>then </b>
<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T80" target="_self" title="ANALMETR:th.80">Th80</a></i>;<br/><a NAME="E3:99_1"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11" target="_self" title="ANALMETR:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E4:99_1"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )
 <b>by </b><i><a class="txt" href="#E4:99"><i><font color="Green" title="E74">A2</font></i></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>
<div><i><font color="Green" title="E76">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_2"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )</span><br/><a NAME="E2:99_2"/><b>then </b>
<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T80" target="_self" title="ANALMETR:th.80">Th80</a></i>;<br/><a NAME="E3:99_2"/><b>then </b>
 <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c8">a</font>
 <b>by </b><i><a class="ref" href="analmetr.html#D11" target="_self" title="ANALMETR:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E4:99_2"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )
 <b>by </b><i><a class="txt" href="#E4:99"><i><font color="Green" title="E74">A2</font></i></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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:99_3"/><i><font color="Green" title="E77">A5</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">b</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )</span><br/><b>set </b><font color="Maroon" title="c16">M</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>;<br/><b>set </b><font color="Maroon" title="c17">N</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>;<br/><a NAME="E2:99_3"/>
 <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>
 <b>by </b><i><a class="txt" href="#E1:99"><i><font color="Green" title="E73">A1</font></i></a>, <a class="txt" href="#E1:99_3"><i><font color="Green" title="E77">A5</font></i></a>, <a class="ref" href="analmetr.html#T63" target="_self" title="ANALMETR:th.63">Th63</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font><b> such that </b><br/><a NAME="E4:99_3"/><i><font color="Green" title="E78">A6</font></i>: 
( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T88" target="_self" title="ANALMETR:th.88">Th88</a></i>;<br/><a NAME="E5:99_3"/>
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c18">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c18">p</font> )
 <b>by </b><i><a class="txt" href="#E4:99_3"><i><font color="Green" title="E78">A6</font></i></a>, <a class="ref" href="analmetr.html#D12" target="_self" title="ANALMETR:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E6:99_3"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )
 ; <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="E8:99"/>
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">POS</font> st <br/>(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>,<font color="Olive" title="b1">p</font> &amp;  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">p</font> )
 <b>by </b><i><a class="txt" href="#E5:99"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="#E6:99"><i><font color="Green" title="E76">A4</font></i></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>
