<?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">K</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b3">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 <font color="Olive" title="b3">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  &amp; <font color="Olive" title="b1">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; (  for <font color="Olive" title="b4">x</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="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">x</font> ) &amp;  not <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</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">  for <font color="Olive" title="b1">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font><br/>  for <font color="Olive" title="b2">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 <font color="Olive" title="b2">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  &amp; <font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; (  for <font color="Olive" title="b3">x</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="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> ) &amp;  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b2">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">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">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 <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  &amp; <font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; (  for <font color="Olive" title="b2">x</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="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) &amp;  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a>  &amp; <font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; (  for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) &amp;  not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> implies <font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:116"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V8" title="TRANSGEO:attr.8">collineation</a> 
 <span class="kw">and </span><br/><a NAME="E2:116"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c2">K</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E3:116"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
 for <font color="Olive" title="b1">x</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E4:116"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
 not <font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E5:116"/><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</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="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span><br/>

<a NAME="E6:116"/><span class="lab"><font color="Green" title="E84">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:116_1_1"/>
 not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <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="E3:116_1_1"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E4:116_1_1"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E5:116_1_1"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="aff_1.html#T19" title="AFF_1:th.19">AFF_1:19</a></span>;<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">p</font>,<font color="Maroon" title="c6">a</font>);<br/><a NAME="E6:116_1_1"/><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<font color="Maroon" title="c3">p</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">p</font>,<font color="Maroon" title="c6">a</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="E7:116_1_1"/>
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c6">a</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="transgeo.html#T89" target="_self" title="TRANSGEO:th.89">Th113</a></span>;<br/><a NAME="E8:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A11</font></span>: 
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c6">a</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> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a></span>;<br/><a NAME="E9:116_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">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="#E4:116"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E85">A7</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">then </span><span class="kw">consider </span><font color="Maroon" title="c9">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E11:116_1_1"/><span class="lab"><font color="Green" title="E90">A12</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">and </span><br/><a NAME="E12:116_1_1"/><span class="lab"><font color="Green" title="E91">A13</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E13:116_1_1"/><span class="lab"><font color="Green" title="E92">A14</font></span>: 
<font color="Maroon" title="c9">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="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E14:116_1_1"/>
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c6">a</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="transgeo.html#T91" target="_self" title="TRANSGEO:th.91">Th115</a></span>;<br/><a NAME="E15:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A15</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="txt" href="#E8:116_1_1"><span class="lab"><font color="Green" title="E89">A11</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E16:116_1_1"/><span class="lab"><font color="Green" title="E94">A16</font></span>: 
<font color="Maroon" title="c6">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">p</font>,<font color="Maroon" title="c6">a</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="E17:116_1_1"/>
 not <font color="Maroon" title="c9">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1_1_1"/>
<font color="Maroon" title="c9">C</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</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/>

<a NAME="E2:116_1_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E6:116_1_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="txt" href="#E16:116_1_1"><span class="lab"><font color="Green" title="E94">A16</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">then </span><span class="kw">consider </span><font color="Maroon" title="c10">c</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="E19:116_1_1"/><span class="lab"><font color="Green" title="E95">A17</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">and </span><br/><a NAME="E20:116_1_1"/><span class="lab"><font color="Green" title="E96">A18</font></span>: 
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E13:116_1_1"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E21:116_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E20:116_1_1"><span class="lab"><font color="Green" title="E96">A18</font></span></a></span>;<br/><a NAME="E22:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:116_1_1"><span class="lab"><font color="Green" title="E95">A17</font></span></a>, <a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><a NAME="E23:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A19</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:116_1_1"><span class="lab"><font color="Green" title="E95">A17</font></span></a>, <a class="txt" href="#E15:116_1_1"><span class="lab"><font color="Green" title="E93">A15</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c11">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>);<br/><a NAME="E24:116_1_1"/><span class="lab"><font color="Green" title="E98">A20</font></span>: 
<font color="Maroon" title="c7">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">p</font>,<font color="Maroon" title="c7">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="E25:116_1_1"/>
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c7">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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="transgeo.html#T89" target="_self" title="TRANSGEO:th.89">Th113</a></span>;<br/><a NAME="E26:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A21</font></span>: 
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c7">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> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E5:116"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a></span>;<br/><a NAME="E27:116_1_1"/>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">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="#E4:116"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E86">A8</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">then </span><span class="kw">consider </span><font color="Maroon" title="c12">D</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AFP</font><span class="kw"> such that </span><br/><a NAME="E29:116_1_1"/><span class="lab"><font color="Green" title="E100">A22</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">and </span><br/><a NAME="E30:116_1_1"/><span class="lab"><font color="Green" title="E101">A23</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T49" title="AFF_1:th.49">AFF_1:49</a></span>;<br/><a NAME="E31:116_1_1"/><span class="lab"><font color="Green" title="E102">A24</font></span>: 
<font color="Maroon" title="c12">D</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="#E30:116_1_1"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E32:116_1_1"/>
<font color="Maroon" title="c4">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">p</font>,<font color="Maroon" title="c7">b</font>)</span>)</span> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E30:116_1_1"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="transgeo.html#T91" target="_self" title="TRANSGEO:th.91">Th115</a></span>;<br/><a NAME="E33:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A25</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">D</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:116_1_1"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="txt" href="#E26:116_1_1"><span class="lab"><font color="Green" title="E99">A21</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/><a NAME="E34:116_1_1"/><span class="lab"><font color="Green" title="E104">A26</font></span>: 
<font color="Maroon" title="c3">p</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">p</font>,<font color="Maroon" title="c7">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="E35:116_1_1"/>
 not <font color="Maroon" title="c12">D</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1_1_2"/>
<font color="Maroon" title="c12">D</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</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/>

<a NAME="E2:116_1_1_2"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c7">b</font>) <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:116_1_1"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:116_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="txt" href="#E34:116_1_1"><span class="lab"><font color="Green" title="E104">A26</font></span></a>, <a class="txt" href="#E24:116_1_1"><span class="lab"><font color="Green" title="E98">A20</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">then </span><span class="kw">consider </span><font color="Maroon" title="c13">d</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="E37:116_1_1"/><span class="lab"><font color="Green" title="E105">A27</font></span>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">and </span><br/><a NAME="E38:116_1_1"/><span class="lab"><font color="Green" title="E106">A28</font></span>: 
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E31:116_1_1"><span class="lab"><font color="Green" title="E102">A24</font></span></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></span>;<br/><a NAME="E39:116_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E38:116_1_1"><span class="lab"><font color="Green" title="E106">A28</font></span></a></span>;<br/><a NAME="E40:116_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:116_1_1"><span class="lab"><font color="Green" title="E105">A27</font></span></a>, <a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/><a NAME="E41:116_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A29</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c12">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:116_1_1"><span class="lab"><font color="Green" title="E105">A27</font></span></a>, <a class="txt" href="#E33:116_1_1"><span class="lab"><font color="Green" title="E103">A25</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/><a NAME="E42:116_1_1"/><span class="lab"><font color="Green" title="E108">A30</font></span>: 
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">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="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="ref" href="aff_1.html#T36" title="AFF_1:th.36">AFF_1:36</a></span>;<br/><a NAME="E43:116_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:116_1_1_3"/><span class="lab"><font color="Green" title="E109">A31</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</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/>

<a NAME="E2:116_1_1_3"/>
( <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">C</font> &amp; <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">D</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:116_1_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E29:116_1_1"><span class="lab"><font color="Green" title="E100">A22</font></span></a>, <a class="txt" href="#E23:116_1_1"><span class="lab"><font color="Green" title="E97">A19</font></span></a>, <a class="txt" href="#E41:116_1_1"><span class="lab"><font color="Green" title="E107">A29</font></span></a>, <a class="ref" href="transgeo.html#T86" target="_self" title="TRANSGEO:th.86">Th110</a></span>;<br/>
<a NAME="E3:116_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:116_1_1"><span class="lab"><font color="Green" title="E90">A12</font></span></a>, <a class="txt" href="#E29:116_1_1"><span class="lab"><font color="Green" title="E100">A22</font></span></a>, <a class="txt" href="#E13:116_1_1"><span class="lab"><font color="Green" title="E92">A14</font></span></a>, <a class="txt" href="#E31:116_1_1"><span class="lab"><font color="Green" title="E102">A24</font></span></a>, <a class="txt" href="#E1:116_1_1_3"><span class="lab"><font color="Green" title="E109">A31</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/>
<a NAME="E4:116_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">a</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="c3">p</font>,<font color="Maroon" title="c7">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:116_1_1"><span class="lab"><font color="Green" title="E91">A13</font></span></a>, <a class="txt" href="#E30:116_1_1"><span class="lab"><font color="Green" title="E101">A23</font></span></a>, <a class="ref" href="aff_1.html#T44" title="AFF_1:th.44">AFF_1:44</a></span>;<br/>
<a NAME="E5:116_1_1_3"/><span class="kw">then </span>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">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="c3">p</font>,<font color="Maroon" title="c7">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:116_1_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="txt" href="#E34:116_1_1"><span class="lab"><font color="Green" title="E104">A26</font></span></a>, <a class="ref" href="aff_1.html#T45" title="AFF_1:th.45">AFF_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:116_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E3:116_1_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E4:116_1_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="txt" href="#E5:116_1_1"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="txt" href="#E42:116_1_1"><span class="lab"><font color="Green" title="E108">A30</font></span></a>, <a class="txt" href="#E6:116_1_1"><span class="lab"><font color="Green" title="E88">A10</font></span></a>, <a class="txt" href="#E16:116_1_1"><span class="lab"><font color="Green" title="E94">A16</font></span></a>, <a class="txt" href="#E24:116_1_1"><span class="lab"><font color="Green" title="E98">A20</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</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="E44:116_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <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="E2:116_1"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</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="txt" href="#E3:116"><span class="lab"><font color="Green" title="E81">A3</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="E7:116"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font></span><br/>

<a NAME="E1:116_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</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="txt" href="#E6:116"><span class="lab"><font color="Green" title="E84">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:116_2"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</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="E8:116"/>
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</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>
