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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>









<span class="kw">reconsider </span><font color="Maroon" title="c12">a9</font> = <font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c13">b9</font> = <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c14">c9</font> = <font color="Maroon" title="c10">c</font>, <font color="Maroon" title="c15">d9</font> = <font color="Maroon" title="c11">d</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="c7">POS</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E2:121"/><span class="lab"><font color="Green" title="E73">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<div><span class="lab"><font color="Green" title="E74">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><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/><span class="kw">set </span><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/><span class="kw">assume </span><a NAME="E1:121_1"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:121_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:121"><span class="lab"><font color="Green" title="E73">A1</font></span></a>, <a class="ref" href="analmetr.html#T63" target="_self" title="ANALMETR:th.63">Th63</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:121_1"/><span class="lab"><font color="Green" title="E75">A3</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T88" target="_self" title="ANALMETR:th.88">Th88</a></span>;<br/><a NAME="E5:121_1"/>
(  <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:121_1"><span class="lab"><font color="Green" title="E75">A3</font></span></a>, <a class="ref" href="analmetr.html#D12" target="_self" title="ANALMETR:def.12">Def12</a></span>;<br/><span class="kw">hence </span><a NAME="E6:121_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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:121"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c12">a9</font>,<font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c12">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E5:121"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" target="_self" title="ANALMETR:th.55">Th55</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E76">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:121_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T80" target="_self" title="ANALMETR:th.80">Th80</a></span>;<br/><a NAME="E3:121_2"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D11" target="_self" title="ANALMETR:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:121_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:121"><span class="lab"><font color="Green" title="E75">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:121"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c14">c9</font>,<font color="Maroon" title="c15">d9</font>,<font color="Maroon" title="c14">c9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<a NAME="E8:121"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A6</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T55" target="_self" title="ANALMETR:th.55">Th55</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:121_3"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:121_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T80" target="_self" title="ANALMETR:th.80">Th80</a></span>;<br/><a NAME="E3:121_3"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D11" target="_self" title="ANALMETR:def.11">Def11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:121_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:121"><span class="lab"><font color="Green" title="E77">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:121"/>
 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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:121"><span class="lab"><font color="Green" title="E76">A5</font></span></a>, <a class="txt" href="#E3:121"><span class="lab"><font color="Green" title="E74">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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