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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">M</font>, <font color="Olive" title="b2">N</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">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b2">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/>( <font color="Olive" title="b1">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">N</font> iff <font color="Olive" title="b1">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Olive" title="b2">N</font> )</span><br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</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">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  implies ( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> iff <font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</font> ) )</span><br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">N</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> iff <font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</font> )</span><br/>

<div><i><font color="Green" title="E46">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</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">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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">A</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b1">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font></span><br/><b>let </b><font color="Maroon" title="c5">A</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font> implies <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:53_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E2:53_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</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">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font></span><br/><a NAME="E3:53_1_1"/>
<font color="Maroon" title="c3">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:53_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="aff_4.html#E1:53_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E4:53_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="aff_4.html#E2:53_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self" title="AFF_4:th.33">Th33</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="E3:53_1"/>
<font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</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/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E47">A6</font></i>: 
<font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</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">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font></span><br/><b>consider </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</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="E3:53_2"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font>
 <b>and </b><br/><a NAME="E4:53_2"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/><a NAME="E5:53_2"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:53_2"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="aff_4.html#E3:53_2"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="aff_4.html#D4" target="_self" title="AFF_4:def.4">Def4</a></i>;<br/><a NAME="E6:53_2"/>
<font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">M</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="aff_4.html#T27" target="_self" title="AFF_4:th.27">Th27</a></i>;<br/><a NAME="E7:53_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:53_2"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="aff_4.html#T33" target="_self" title="AFF_4:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E8:53_2"/>
<font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:53"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="aff_4.html#D3" target="_self" title="AFF_4:def.3">Def3</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="E4:53"/>
( <font color="Maroon" title="c2">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">N</font> iff <font color="Maroon" title="c2">M</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c3">N</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E2:53"><i><font color="Green" title="E46">A2</font></i></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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