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<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 <font color="Maroon" title="c1">AFP</font> is  <a href="translac.html#V1" title="TRANSLAC:attr.1">Fanoian</a>  &amp; <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  holds <br/> ex <font color="Olive" title="b2">g</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 <br/>( <font color="Olive" title="b2">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b2">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">f</font> )</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"> ( <font color="Maroon" title="c1">AFP</font> is  <a href="translac.html#V1" title="TRANSLAC:attr.1">Fanoian</a>  &amp; <font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  &amp; <font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  implies  ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:25"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c1">AFP</font> is  <a href="translac.html#V1" title="TRANSLAC:attr.1">Fanoian</a> 
 <span class="kw">and </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c1">AFP</font> is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c2">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E27">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">a</font> =  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/><span class="kw">set </span><font color="Maroon" title="c4">b</font> = <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>;<br/><span class="kw">assume </span><a NAME="E1:25_1"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )</span><br/><a NAME="E2:25_1"/><span class="kw">then </span>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="transgeo.html#D11" title="TRANSGEO:def.11">TRANSGEO:def 11</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">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="E4:25_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">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="E6:25_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c6">d</font>
 <span class="kw">and </span><br/><a NAME="E7:25_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c5">c</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></span>;<br/><a NAME="E8:25_1"/>
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></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="E10:25_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E11:25_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="translac.html#T2" target="_self" title="TRANSLAC:th.2">Th4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">f1</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><span class="kw"> such that </span><br/><a NAME="E13:25_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c8">f1</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E14:25_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="translac.html#T7" target="_self" title="TRANSLAC:th.7">Th9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">f2</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><span class="kw"> such that </span><br/><a NAME="E16:25_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
<font color="Maroon" title="c9">f2</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E17:25_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="translac.html#T7" target="_self" title="TRANSLAC:th.7">Th9</a></span>;<br/><a NAME="E18:25_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="transgeo.html#T82" title="TRANSGEO:th.82">TRANSGEO:82</a></span>;<br/><a NAME="E19:25_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>,<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="#E11:25_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E20:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><div><span class="lab"><font color="Green" title="E40">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:25_1_1"/>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>, the   <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"> contradiction</span><br/><a NAME="E2:25_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">c</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E3:25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A18</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E4:25_1_1"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a>, <a class="ref" href="aff_1.html#T7" title="AFF_1:th.7">AFF_1:7</a></span>;<br/><a NAME="E5:25_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E3:25_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="aff_1.html#T8" title="AFF_1:th.8">AFF_1:8</a></span>;<br/><a NAME="E6:25_1_1"/><span class="kw">then </span>
(  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> or  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E20:25_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E7:25_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">c</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> or  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a></span>;<br/><a NAME="E8:25_1_1"/><span class="kw">then </span>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> or  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E9:25_1_1"/><span class="kw">then </span>
 the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E7:25_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="aff_1.html#T4" title="AFF_1:th.4">AFF_1:4</a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E10:25_1_1"/><span class="kw">then </span>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E11:25_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</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">consider </span><font color="Maroon" title="c10">f3</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><span class="kw"> such that </span><br/><a NAME="E23:25_1"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<font color="Maroon" title="c10">f3</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E24:25_1"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
<font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="translac.html#T7" target="_self" title="TRANSLAC:th.7">Th9</a></span>;<br/><a NAME="E25:25_1"/>
<font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:25_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="transgeo.html#T81" title="TRANSGEO:th.81">TRANSGEO:81</a></span>;<br/><a NAME="E26:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="transgeo.html#T82" title="TRANSGEO:th.82">TRANSGEO:82</a></span>;<br/><a NAME="E27:25_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:25_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E28:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E17:25_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E29:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<font color="Maroon" title="c7">p</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:25_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="txt" href="#E14:25_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="translac.html#T11" target="_self" title="TRANSLAC:th.11">Th13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">f4</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><span class="kw"> such that </span><br/><a NAME="E31:25_1"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
<font color="Maroon" title="c11">f4</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E32:25_1"/><span class="lab"><font color="Green" title="E46">A24</font></span>: 
<font color="Maroon" title="c11">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="translac.html#T7" target="_self" title="TRANSLAC:th.7">Th9</a></span>;<br/><a NAME="E33:25_1"/>
<font color="Maroon" title="c11">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:25_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="transgeo.html#T2" title="TRANSGEO:th.2">TRANSGEO:2</a></span>;<br/><a NAME="E34:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:25_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">h</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><span class="kw"> such that </span><br/><a NAME="E36:25_1"/><span class="lab"><font color="Green" title="E48">A26</font></span>: 
<font color="Maroon" title="c12">h</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">and </span><br/><a NAME="E37:25_1"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="translac.html#T7" target="_self" title="TRANSLAC:th.7">Th9</a></span>;<br/><a NAME="E38:25_1"/>
 not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>, the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="aff_1.html#T6" title="AFF_1:th.6">AFF_1:6</a></span>;<br/><a NAME="E39:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<font color="Maroon" title="c12">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E7:25_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E36:25_1"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E37:25_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="translac.html#T3" target="_self" title="TRANSLAC:th.3">Th5</a></span>;<br/><a NAME="E40:25_1"/>
<font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:25_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="transgeo.html#T2" title="TRANSGEO:th.2">TRANSGEO:2</a></span>;<br/><a NAME="E41:25_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>;<br/><a NAME="E42:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A29</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c12">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:25_1"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E37:25_1"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E26:25_1"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="transgeo.html#T80" title="TRANSGEO:th.80">TRANSGEO:80</a></span>;<br/><a NAME="E43:25_1"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="transgeo.html#T81" title="TRANSGEO:th.81">TRANSGEO:81</a></span>;<br/><a NAME="E44:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:25_1"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="transgeo.html#T82" title="TRANSGEO:th.82">TRANSGEO:82</a></span>;<br/><a NAME="E45:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:25_1"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E39:25_1"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E42:25_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E34:25_1"><span class="lab"><font color="Green" title="E47">A25</font></span></a>, <a class="ref" href="transgeo.html#T80" title="TRANSGEO:th.80">TRANSGEO:80</a></span>;<br/><a NAME="E46:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/><a NAME="E47:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.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#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T61" title="FUNCT_2:th.61">FUNCT_2:61</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c8">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
<br/>.= 
<font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E23:25_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E43:25_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="ref" href="translac.html#T10" target="_self" title="TRANSLAC:th.10">Th12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A31</font></span>: <font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T61" title="FUNCT_2:th.61">FUNCT_2:61</a></span>
<br/>.= 
<font color="Maroon" title="c11">f4</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c10">f3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
;<br/><a NAME="E50:25_1"/>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c10">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c11">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:25_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E32:25_1"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E19:25_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></span>;<br/><a NAME="E51:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">f3</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c11">f4</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:25_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E24:25_1"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="txt" href="#E31:25_1"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="translac.html#T11" target="_self" title="TRANSLAC:th.11">Th13</a></span>;<br/><a NAME="E52:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E23:25_1"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E31:25_1"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E49:25_1"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="translac.html#T10" target="_self" title="TRANSLAC:th.10">Th12</a></span>;<br/><a NAME="E53:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:25_1"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="txt" href="#E21:25_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="aff_1.html#T5" title="AFF_1:th.5">AFF_1:5</a></span>;<br/><a NAME="E54:25_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:25_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="transgeo.html#D11" title="TRANSGEO:def.11">TRANSGEO:def 11</a></span>;<br/><a NAME="E55:25_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c8">f1</font></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>;<br/><a NAME="E56:25_1"/><span class="kw">then </span>
<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="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T61" title="FUNCT_2:th.61">FUNCT_2:61</a></span>;<br/><a NAME="E57:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">f2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="transgeo.html#K1" title="TRANSGEO:func.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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A32</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f2</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</a></span>
<br/>.= 
<font color="Maroon" title="c9">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f1</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:25_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E17:25_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="transgeo.html#T2" title="TRANSGEO:th.2">TRANSGEO:2</a></span>
;<br/><a NAME="E59:25_1"/>
<font color="Maroon" title="c9">f2</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Maroon" title="c9">f2</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="transgeo.html#T82" title="TRANSGEO:th.82">TRANSGEO:82</a></span>;<br/><span class="kw">hence </span><a NAME="E60:25_1"/>
 ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E16:25_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E58:25_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="transgeo.html#T80" title="TRANSGEO:th.80">TRANSGEO:80</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">g</font> =  <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>;<br/><a NAME="E1:25_2"/><span class="lab"><font color="Green" title="E28">A33</font></span>: 
(  <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> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <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="transgeo.html#K1" title="TRANSGEO:func.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#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#T17" title="FUNCT_2:th.17">FUNCT_2:17</a>, <a class="ref" href="transgeo.html#T77" title="TRANSGEO:th.77">TRANSGEO:77</a></span>;<br/><span class="kw">assume </span><a NAME="E2:25_2"/>
<font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )</span><br/><span class="kw">hence </span><a NAME="E3:25_2"/>
 ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_2"><span class="lab"><font color="Green" title="E28">A33</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>
<span class="kw">hence </span><a NAME="E6:25"/>
 ex <font color="Olive" title="b1">g</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 <br/>( <font color="Olive" title="b1">g</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">g</font> <a href="transgeo.html#K1" title="TRANSGEO:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c2">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E27">A4</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>
