<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">AFP</font> be   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">AFP</font>  st (  for <font color="Olive" title="b2">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  holds <br/><font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  ) holds <br/><font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/><span class="kw">let </span><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">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">AFP</font>  st <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  ) implies <font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:115"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
 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">AFP</font>  st <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> </span><br/>

<a NAME="E2:115"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <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="Olive" title="b1">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="Olive" title="b2">b</font></span>)</span>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> implies <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <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="c3">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="c4">b</font></span>)</span>) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">y</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">b</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">K</font> =  <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="c3">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="c4">b</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">M</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:115_1"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <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="c3">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="c4">b</font></span>)</span>)</span><br/>

<a NAME="E2:115_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T58" title="FUNCT_2:th.58">FUNCT_2:58</a></span>;<br/>
<a NAME="E3:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A4</font></span>: 
 <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="c3">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="c4">b</font></span>)</span>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E4:115_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E5:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E79">A1</font></span></a></span>;<br/>
<a NAME="E6:115_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E7:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A6</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/>
<a NAME="E8:115_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<a NAME="E9:115_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/>
<a NAME="E10:115_1"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="c4">b</font></span>)</span>) &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c3">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="c4">b</font></span>)</span>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:115_1"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <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="c3">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="c4">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_1"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E3:115_1"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E5:115_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E7:115_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E9:115_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a>, <a class="ref" href="funct_2.html#T58" title="FUNCT_2:th.58">FUNCT_2:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:115"/><span class="lab"><font color="Green" title="E81">A8</font></span>: 
 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">AFP</font>  st <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  holds <br/><font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  implies <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">K</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">assume </span><a NAME="E1:115_2"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E3:115_2"/><span class="lab"><font color="Green" title="E82">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">and </span><br/><a NAME="E4:115_2"/><span class="lab"><font color="Green" title="E83">A10</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">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="c5">x</font>,<font color="Maroon" title="c6">y</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<a NAME="E5:115_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_2"><span class="lab"><font color="Green" title="E83">A10</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E7:115_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E8:115_2"/><span class="lab"><font color="Green" title="E84">A11</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T87" target="_self" title="TRANSGEO:th.87">Th111</a></span>;<br/>
<a NAME="E9:115_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_2"><span class="lab"><font color="Green" title="E83">A10</font></span></a>, <a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E11:115_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E12:115_2"/><span class="lab"><font color="Green" title="E85">A12</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T87" target="_self" title="TRANSGEO:th.87">Th111</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c7">b</font>);<br/>
<a NAME="E13:115_2"/>
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c7">b</font>) is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c7">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E3:115_2"><span class="lab"><font color="Green" title="E82">A9</font></span></a>, <a class="txt" href="#E4:115_2"><span class="lab"><font color="Green" title="E83">A10</font></span></a>, <a class="txt" href="#E12:115_2"><span class="lab"><font color="Green" title="E85">A12</font></span></a>, <a class="txt" href="#E8:115_2"><span class="lab"><font color="Green" title="E84">A11</font></span></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:115_2"/>
<font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T88" target="_self" title="TRANSGEO:th.88">Th112</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:115"/><span class="lab"><font color="Green" title="E82">A13</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">K</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">M</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font>;<br/>
<span class="kw">assume </span><a NAME="E1:115_3"/><span class="lab"><font color="Green" title="E83">A14</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/>

<a NAME="E2:115_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E3:115_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A15</font></span>: 
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115"><span class="lab"><font color="Green" title="E81">A8</font></span></a></span>;<br/>
<a NAME="E4:115_3"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_3"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E5:115_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A17</font></span>: 
<font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:115"><span class="lab"><font color="Green" title="E81">A8</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:115_3_1"/><span class="lab"><font color="Green" title="E86">A18</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font></span><br/><span class="kw">assume </span><a NAME="E2:115_3_1"/>
 not <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E4:115_3_1"/><span class="lab"><font color="Green" title="E87">A19</font></span>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_3"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="txt" href="#E3:115_3"><span class="lab"><font color="Green" title="E84">A15</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>;<br/><a NAME="E5:115_3_1"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_3_1"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><span class="kw">hence </span><a NAME="E6:115_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_3"><span class="lab"><font color="Green" title="E83">A14</font></span></a>, <a class="txt" href="#E1:115_3_1"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="transgeo.html#T88" target="_self" title="TRANSGEO:th.88">Th112</a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:115_3"/>
<font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_3"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:115"/><span class="lab"><font color="Green" title="E83">A20</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <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">AFP</font>  st <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">d</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">y</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">b</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">z</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">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>;<br/>
<span class="kw">assume </span><a NAME="E1:115_4"/><span class="lab"><font color="Green" title="E84">A21</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c11">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>);<br/><span class="kw">set </span><font color="Maroon" title="c12">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>);<br/><span class="kw">set </span><font color="Maroon" title="c13">K</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c14">M</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span>;<br/><a NAME="E1:115_4_1"/><span class="lab"><font color="Green" title="E85">A22</font></span>: 
( <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><span class="kw">assume </span><a NAME="E2:115_4_1"/><span class="lab"><font color="Green" title="E86">A23</font></span>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font></span><br/><a NAME="E3:115_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A24</font></span>: 
( <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">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="c4">b</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</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="c6">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T58" title="FUNCT_2:th.58">FUNCT_2:58</a></span>;<br/><a NAME="E4:115_4_1"/>
( <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <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="c3">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="c4">b</font></span>)</span>) &amp; <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span> <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="c5">c</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="c6">d</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:115"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E2:115_4_1"><span class="lab"><font color="Green" title="E86">A23</font></span></a></span>;<br/><a NAME="E5:115_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A25</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_4"><span class="lab"><font color="Green" title="E84">A21</font></span></a>, <a class="txt" href="#E3:115_4_1"><span class="lab"><font color="Green" title="E87">A24</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><a NAME="E6:115_4_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E7:115_4_1"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115"><span class="lab"><font color="Green" title="E82">A13</font></span></a>, <a class="txt" href="#E1:115_4_1"><span class="lab"><font color="Green" title="E85">A22</font></span></a>, <a class="txt" href="#E5:115_4_1"><span class="lab"><font color="Green" title="E88">A25</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:115_4"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:115"/><span class="lab"><font color="Green" title="E84">A26</font></span>: 
 for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:115_5"/><span class="lab"><font color="Green" title="E85">A27</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font></span><br/>

<a NAME="E2:115_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E3:115_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A28</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E79">A1</font></span></a></span>;<br/>
<a NAME="E4:115_5"/>
<font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_5"><span class="lab"><font color="Green" title="E85">A27</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/>
<a NAME="E5:115_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A30</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115"><span class="lab"><font color="Green" title="E79">A1</font></span></a></span>;<br/>
<a NAME="E6:115_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A31</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T41" title="AFF_1:th.41">AFF_1:41</a></span>;<br/>
<a NAME="E7:115_5"/>
( <font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> implies <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:115_5_1"/><span class="lab"><font color="Green" title="E89">A32</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font></span><br/>

<span class="kw">assume </span><a NAME="E2:115_5_1"/>
 not <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E4:115_5_1"/><span class="lab"><font color="Green" title="E90">A33</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E5:115_5_1"/><span class="lab"><font color="Green" title="E91">A34</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_5"><span class="lab"><font color="Green" title="E87">A30</font></span></a>, <a class="txt" href="#E3:115_5"><span class="lab"><font color="Green" title="E86">A28</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E7:115_5_1"/><span class="lab"><font color="Green" title="E92">A35</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">and </span><br/><a NAME="E8:115_5_1"/><span class="lab"><font color="Green" title="E93">A36</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115_5_1"><span class="lab"><font color="Green" title="E91">A34</font></span></a>, <a class="ref" href="transgeo.html#T87" target="_self" title="TRANSGEO:th.87">Th111</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E10:115_5_1"/><span class="lab"><font color="Green" title="E94">A37</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">and </span><br/><a NAME="E11:115_5_1"/><span class="lab"><font color="Green" title="E95">A38</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:115_5_1"><span class="lab"><font color="Green" title="E90">A33</font></span></a>, <a class="ref" href="transgeo.html#T87" target="_self" title="TRANSGEO:th.87">Th111</a></span>;<br/>
<a NAME="E12:115_5_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:115_5_1"><span class="lab"><font color="Green" title="E95">A38</font></span></a>, <a class="txt" href="#E8:115_5_1"><span class="lab"><font color="Green" title="E93">A36</font></span></a>, <a class="ref" href="funct_2.html#T58" title="FUNCT_2:th.58">FUNCT_2:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:115_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_5"><span class="lab"><font color="Green" title="E85">A27</font></span></a>, <a class="txt" href="#E1:115_5_1"><span class="lab"><font color="Green" title="E89">A32</font></span></a>, <a class="txt" href="#E10:115_5_1"><span class="lab"><font color="Green" title="E94">A37</font></span></a>, <a class="txt" href="#E7:115_5_1"><span class="lab"><font color="Green" title="E92">A35</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:115_5"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115_5"><span class="lab"><font color="Green" title="E88">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:115"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <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">AFP</font>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> implies <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:115_6"/><span class="lab"><font color="Green" title="E85">A39</font></span>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">C</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>);<br/><span class="kw">set </span><font color="Maroon" title="c8">A</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>);<br/><span class="kw">set </span><font color="Maroon" title="c9">K</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c10">M</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span>;<br/><a NAME="E1:115_6_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E2:115_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A40</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><a NAME="E3:115_6_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E4:115_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A41</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><a NAME="E5:115_6_1"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E6:115_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A42</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><a NAME="E7:115_6_1"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T15" title="AFF_1:th.15">AFF_1:15</a></span>;<br/><a NAME="E8:115_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A43</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><span class="kw">assume </span><a NAME="E9:115_6_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font></span><br/><a NAME="E10:115_6_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:115_6"><span class="lab"><font color="Green" title="E85">A39</font></span></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></span>;<br/><span class="kw">hence </span><a NAME="E11:115_6_1"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:115"><span class="lab"><font color="Green" title="E84">A26</font></span></a>, <a class="txt" href="#E2:115_6_1"><span class="lab"><font color="Green" title="E86">A40</font></span></a>, <a class="txt" href="#E4:115_6_1"><span class="lab"><font color="Green" title="E87">A41</font></span></a>, <a class="txt" href="#E8:115_6_1"><span class="lab"><font color="Green" title="E89">A43</font></span></a>, <a class="txt" href="#E6:115_6_1"><span class="lab"><font color="Green" title="E88">A42</font></span></a>, <a class="ref" href="aff_1.html#T39" title="AFF_1:th.39">AFF_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:115_6"/>
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">a</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">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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">c</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T3" title="AFF_1:th.3">AFF_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:115"/>
<font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:115"><span class="lab"><font color="Green" title="E83">A20</font></span></a>, <a class="ref" href="transgeo.html#T83" target="_self" title="TRANSGEO:th.83">Th107</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
