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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  ex <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 <br/>( <font color="Olive" title="b3">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> &amp; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) ) implies <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> )</span><br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green" title="E21">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  ex <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 <br/>( <font color="Olive" title="b3">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a> &amp; <font color="Olive" title="b3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">b'</font>, <font color="Maroon" title="c7">c'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c7">c'</font> implies <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font> )</span><br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green" title="E22">A2</font></i>: 
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b</font> &amp; not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c6">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c7">c'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/><b>consider </b><font color="Maroon" title="c8">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><b> such that </b><br/><a NAME="E3:21_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>
 <b>and </b><br/><a NAME="E4:21_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a'</font>
 <b>by </b><i><a class="txt" href="#E1:21"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/><a NAME="E5:21_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c6">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:21_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E4:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="translac.html#T5" target="_self" title="TRANSLAC:th.5">Th5</a></i>;<br/><a NAME="E6:21_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c7">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="#E1:21_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:21_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E4:21_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="translac.html#T5" target="_self" title="TRANSLAC:th.5">Th5</a></i>;<br/><a NAME="E7:21_1"/>
<font color="Maroon" title="c8">f</font> is <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a>
 <b>by </b><i><a class="txt" href="#E3:21_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="transgeo.html#D11" title="TRANSGEO:def.11">TRANSGEO:def 11</a></i>;<br/><b>hence </b><a NAME="E8:21_1"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E5:21_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="#E6:21_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="transgeo.html#T85" title="TRANSGEO:th.85">TRANSGEO:85</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
<font color="Maroon" title="c1">AFP</font> is <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>
 <b>by </b><i><a class="ref" href="translac.html#T6" target="_self" title="TRANSLAC:th.6">Th6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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