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<b>let </b><font color="Maroon" title="c1">MS</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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font>  st not  <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="b3">c</font> holds <br/> ex <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="c1">MS</font> st <br/>( <font color="Olive" title="b4">d</font>,<font color="Olive" title="b1">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b4">d</font>,<font color="Olive" title="b2">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</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">b</font>,<font color="Maroon" title="c4">c</font> implies  ex <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font> st <br/>( <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> ) )</span><br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E12">A1</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">b</font>,<font color="Maroon" title="c4">c</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">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font> st <br/>( <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<a NAME="E2:12"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="euclmetr.html#T1" target="_self" title="EUCLMETR:th.1">Th1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>;<br/>
<b>set </b><font color="Maroon" title="c6">K</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>;<br/>
<a NAME="E3:12"/><i><font color="Green" title="E14">A3</font></i>: 
(  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )
 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="analmetr.html#D13" title="ANALMETR:def.13">ANALMETR:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font><b> such that </b><br/><a NAME="E5:12"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c7">P</font> )
 <b>by </b><i><a class="ref" href="conmetr.html#T3" title="CONMETR:th.3">CONMETR:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">MS</font><b> such that </b><br/><a NAME="E7:12"/><i><font color="Green" title="E16">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Q</font> &amp;  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c8">Q</font> )
 <b>by </b><i><a class="txt" href="#E3:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="conmetr.html#T3" title="CONMETR:th.3">CONMETR:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">P'</font> = <font color="Maroon" title="c7">P</font>, <font color="Maroon" title="c10">Q'</font> = <font color="Maroon" title="c8">Q</font> as   <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">MS</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></i>;<br/>
<a NAME="E9:12"/>
( <font color="Maroon" title="c7">P</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> &amp; <font color="Maroon" title="c8">Q</font> is <a href="analmetr.html#V4" title="ANALMETR:attr.4">being_line</a> )
 
<b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="#E7:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="analmetr.html#T62" title="ANALMETR:th.62">ANALMETR:62</a></i>;<br/>
<a NAME="E10:12"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
( <font color="Maroon" title="c9">P'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c10">Q'</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T58" title="ANALMETR:th.58">ANALMETR:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">A'</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c12">K'</font> =  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> as   <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">MS</font></span>)</span> <b>by </b><i><a class="ref" href="analmetr.html#T57" title="ANALMETR:th.57">ANALMETR:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">a'</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c14">b'</font> = <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c15">c'</font> = <font color="Maroon" title="c4">c</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">MS</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="E13:12"/><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c11">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="c13">a'</font>,<font color="Maroon" title="c15">c'</font> &amp; <font color="Maroon" title="c12">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="c14">b'</font>,<font color="Maroon" title="c15">c'</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#T56" title="ANALMETR:th.56">ANALMETR:56</a></i>;<br/>
<a NAME="E14:12"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
( <font color="Maroon" title="c13">a'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A'</font> &amp; <font color="Maroon" title="c15">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A'</font> &amp; <font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">K'</font> &amp; <font color="Maroon" title="c15">c'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">K'</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<a NAME="E15:12"/>
not <font color="Maroon" title="c9">P'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">Q'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c9">P'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c10">Q'</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="E2:12_1"/><b>then </b>
<font color="Maroon" title="c7">P</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a> <font color="Maroon" title="c8">Q</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64" title="ANALMETR:th.64">ANALMETR:64</a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b>
 <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R10" title="ANALMETR:pred.10">_|_</a> <font color="Maroon" title="c8">Q</font>
 
<b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="analmetr.html#T73" title="ANALMETR:th.73">ANALMETR:73</a></i>;<br/>
<a NAME="E4:12_1"/><b>then </b>
 <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analmetr.html#R9" title="ANALMETR:pred.9">//</a>  <a href="analmetr.html#K4" title="ANALMETR:func.4">Line</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="#E7:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="analmetr.html#T87" title="ANALMETR:th.87">ANALMETR:87</a></i>;<br/>
<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon" title="c11">A'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c12">K'</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#T64" title="ANALMETR:th.64">ANALMETR:64</a></i>;<br/>
<a NAME="E6:12_1"/><b>then </b>
<font color="Maroon" title="c14">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">A'</font>
 
<b>by </b><i><a class="txt" href="#E14:12"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/>
<a NAME="E7:12_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c15">c'</font>,<font color="Maroon" title="c14">b'</font>
 
<b>by </b><i><a class="txt" href="#E13:12"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="aff_1.html#D2" title="AFF_1:def.2">AFF_1:def 2</a></i>;<br/>
<a NAME="E8:12_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c13">a'</font>,<font color="Maroon" title="c14">b'</font>,<font color="Maroon" title="c15">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="E9:12_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E12">A1</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>
<b>then consider </b><font color="Maroon" title="c16">d'</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">MS</font></span>)</span><b> such that </b><br/><a NAME="E17:12"/><i><font color="Green" title="E20">A9</font></i>: 
( <font color="Maroon" title="c16">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P'</font> &amp; <font color="Maroon" title="c16">d'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q'</font> )
 <b>by </b><i><a class="txt" href="#E10:12"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="aff_1.html#T72" title="AFF_1:th.72">AFF_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">d</font> = <font color="Maroon" title="c16">d'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">MS</font> <b>by </b><i><a class="ref" href="analmetr.html#T47" title="ANALMETR:th.47">ANALMETR:47</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">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="c17">d</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )</span><br/>

<b>thus </b><a NAME="E19:12"/>
( <font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c2">a</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c17">d</font>,<font color="Maroon" title="c3">b</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="#E5:12"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="#E7:12"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="#E14:12"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="#E17:12"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="analmetr.html#T78" title="ANALMETR:th.78">ANALMETR:78</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>
