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<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"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">A</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> holds <br/> ex <font color="Olive" title="b3">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">K</font> &amp; <font color="Olive" title="b1">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b3">K</font> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">A</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> holds <br/> ex <font color="Olive" title="b2">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b2">K</font> )</span><br/><b>let </b><font color="Maroon" title="c3">a</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"> ( <font color="Maroon" title="c2">A</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> implies  ex <font color="Olive" title="b1">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b1">K</font> ) )</span><br/>





<b>assume </b><a NAME="E1:4"/>
<font color="Maroon" title="c2">A</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a>
 ; <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">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Olive" title="b1">K</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:4"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#D13" title="ANALMETR:def.13">ANALMETR:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E5:4"/><i><font color="Green" title="E11">A2</font></i>: 
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="ref" href="analmetr.html#T51" title="ANALMETR:th.51">ANALMETR:51</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">K</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">d</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font> &amp; <font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c7">K</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c9">d'</font> = <font color="Maroon" title="c6">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="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="E7:4"/>
<font color="Maroon" title="c7">K</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">d'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T56" title="ANALMETR:th.56">ANALMETR:56</a></i>;<br/>
<b>hence </b><a NAME="E8:4"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">K</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c7">K</font></span><br/>

<b>thus </b><a NAME="E9:4"/>
<font color="Maroon" title="c2">A</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c7">K</font>
 <b>by </b><i><a class="txt" href="#E3:4"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="#E5:4"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="analmetr.html#T63" title="ANALMETR:th.63">ANALMETR:63</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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