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<b>let </b><font color="Maroon" title="c1">AFS</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">f</font> being  <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</font><br/> for <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">AFS</font>  st <font color="Olive" title="b1">f</font> <a href="transgeo.html#NR10" title="TRANSGEO:NR.10">is_Col</a>  &amp; <font color="Olive" title="b2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFS</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">K</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFS</font>  st <font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR10" title="TRANSGEO:NR.10">is_Col</a>  &amp; <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </span><br/><b>let </b><font color="Maroon" title="c3">K</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFS</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">f</font> <a href="transgeo.html#NR10" title="TRANSGEO:NR.10">is_Col</a>  &amp; <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  implies <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:120"/><i><font color="Green" title="E78">A1</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="transgeo.html#NR10" title="TRANSGEO:NR.10">is_Col</a> 
 <b>and </b><br/><a NAME="E2:120"/><i><font color="Green" title="E79">A2</font></i>: 
<font color="Maroon" title="c3">K</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">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> </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">AFS</font><b> such that </b><br/><a NAME="E4:120"/><i><font color="Green" title="E80">A3</font></i>: 
( <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> &amp; <font color="Maroon" title="c3">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="c4">a</font>,<font color="Maroon" title="c5">b</font> )
 <b>by </b><i><a class="txt" href="transgeo.html#E2:120"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font>;<br/>
<b>set </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>;<br/>
<a NAME="E5:120"/><i><font color="Green" title="E81">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="transgeo.html#E1:120"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="transgeo.html#E4:120"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="transgeo.html#T113" target="_self" title="TRANSGEO:th.113">Th113</a></i>;<br/>
<a NAME="E6:120"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="transgeo.html#E4:120"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="funct_2.html#T85" title="FUNCT_2:th.85">FUNCT_2:85</a></i>;<br/>
<b>hence </b><a NAME="E7:120"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="transgeo.html#E5:120"><i><font color="Green" title="E81">A4</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</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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