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<b>let </b><font color="Maroon" title="c1">SAS</font> be   <a href="semi_af1.html#NM1" title="SEMI_AF1:NM.1">Semi_Affine_Space</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">o</font>, <font color="Olive" title="b2">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>  st  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p</font></span><br/><b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font> implies <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> )</span><br/>





<b>assume </b><a NAME="E1:108"/><i><font color="Green" title="E85">A1</font></i>: 
 <a href="semi_af1.html#R5" title="SEMI_AF1:pred.5">qtrap</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</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">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font></span><br/>

<b>assume </b><a NAME="E2:108"/><i><font color="Green" title="E86">A2</font></i>: 
not <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</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="E3:108"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E2:108_1"/><i><font color="Green" title="E87">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T130" target="_self" title="SEMI_AF1:th.130">Th130</a></i>;<br/>
<a NAME="E3:108_1"/>
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> or <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">z</font> )
 
<b>by </b><i><a class="txt" href="#E2:108_1"><i><font color="Green" title="E87">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:108_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font> st <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">b</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>then consider </b><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SAS</font><b> such that </b><br/><a NAME="E5:108"/><i><font color="Green" title="E87">A4</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ;<br/>
<b>consider </b><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">SAS</font><b> such that </b><br/><a NAME="E7:108"/><i><font color="Green" title="E88">A5</font></i>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="#E5:108"><i><font color="Green" title="E87">A4</font></i></a>, <a class="ref" href="semi_af1.html#T26" target="_self" title="SEMI_AF1:th.26">Th26</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">SAS</font><b> such that </b><br/><a NAME="E9:108"/><i><font color="Green" title="E89">A6</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">b</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  implies ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="semi_af1.html#R1" title="SEMI_AF1:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> ) )
 <b>by </b><i><a class="txt" href="#E1:108"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="semi_af1.html#D9" target="_self" title="SEMI_AF1:def.9">Def9</a></i>;<br/>
<a NAME="E10:108"/>
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> implies ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> ) )
 
<b>by </b><i><a class="txt" href="#E2:108"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="#E9:108"><i><font color="Green" title="E89">A6</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self" title="SEMI_AF1:def.2">Def2</a></i>;<br/>
<a NAME="E11:108"/><b>then </b><i><font color="Green" title="E90">A7</font></i>: 
( ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">d</font> ) or ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> ) )
 
<b>by </b><i><a class="txt" href="#E7:108"><i><font color="Green" title="E88">A5</font></i></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T14" target="_self" title="SEMI_AF1:th.14">Th14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:108_2"/><i><font color="Green" title="E91">A8</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="c6">d</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</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="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> )</span><br/><a NAME="E2:108_2"/><b>then </b>
<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a></i>;<br/><b>hence </b><a NAME="E3:108_2"/>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> &amp; not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="#E1:108_2"><i><font color="Green" title="E91">A8</font></i></a>, <a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</a>, <a class="ref" href="semi_af1.html#T18" target="_self" title="SEMI_AF1:th.18">Th18</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="E13:108"/>
contradiction
 <b>by </b><i><a class="txt" href="#E11:108"><i><font color="Green" title="E90">A7</font></i></a>, <a class="ref" href="semi_af1.html#D1" target="_self" title="SEMI_AF1:def.1">Def1</a>, <a class="ref" href="semi_af1.html#T17" target="_self" title="SEMI_AF1:th.17">Th17</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>
