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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</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">M</font>, <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Olive" title="b1">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b3">X</font></span><br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</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> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c3">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c2">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font> )</span><br/>







<b>assume </b><a NAME="E1:72"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c3">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">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> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font></span><br/>

<a NAME="E2:72"/><b>then </b><i><font color="Green" title="E60">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c3">M</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</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">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Olive" title="b1">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font></span><br/><b>let </b><font color="Maroon" title="c6">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> implies <font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> )</span><br/><b>assume </b><a NAME="E1:72_1"/><i><font color="Green" title="E61">A3</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c6">C</font> is <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> &amp; <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:72_1"/><b>then </b><i><font color="Green" title="E62">A4</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="#E2:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self" title="AFF_4:th.33">Th33</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">q</font>, <font color="Maroon" title="c8">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E4:72_1"/><i><font color="Green" title="E63">A5</font></i>: 
( <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p</font> )
 <b>by </b><i><a class="txt" href="#E2:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/><font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> = 
<font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">M</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E2:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E4:72_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">M</font>
<b>by </b><i><a class="txt" href="#E2:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="aff_4.html#T31" target="_self" title="AFF_4:th.31">Th31</a></i>
;<br/><b>hence </b><a NAME="E6:72_1"/>
<font color="Maroon" title="c5">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c6">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="#E1:72"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E2:72"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="#E1:72_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="#E2:72_1"><i><font color="Green" title="E62">A4</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</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="E4:72"/>
<font color="Maroon" title="c2">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</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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