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<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="Red">::  thesis: </font></span></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>  st <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">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/> 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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies  ex <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 <br/>( <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">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">b</font> ) )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>] <span class="kw">means </span>( ( 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">b</font>,$1 &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> $1,$2 &amp; <font color="Maroon" title="c2">a</font>,$1 <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,$2 ) or (  <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">b</font>,$1 &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( 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">b</font>,<font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> $1,$2 &amp; <font color="Olive" title="b1">p</font>,$1 <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">q</font>,$2 ) ) );<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21"/><span class="lab"><font color="Green" title="E20">A1</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="E2:21"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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 <br/>( <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">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">b</font> )</span><br/>

<a NAME="E3:21"/><span class="lab"><font color="Green" title="E22">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>  ex <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="translac.html#E14"><span class="lab"><font color="Green" title="E12">Lm6</font></span></a></span>;<br/>
<a NAME="E4:21"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">x9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">x9</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">x9</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c5">y</font>] implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
( ( 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">b</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> ) or (  <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">b</font>,<font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( 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">b</font>,<font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c5">y</font> ) ) )
 <span class="kw">and </span><br/><a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
( ( 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">b</font>,<font color="Maroon" title="c6">x9</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">x9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> ) or (  <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">b</font>,<font color="Maroon" title="c6">x9</font> &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( 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">b</font>,<font color="Olive" title="b1">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">x9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">q</font>,<font color="Maroon" title="c5">y</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font></span><br/>

<div><span class="lab"><font color="Green" title="E26">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
 <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">b</font>,<font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font></span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
(  <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">b</font>,<font color="Maroon" title="c6">x9</font> or  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c5">y</font> or  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">x9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
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">b</font>,<font color="Maroon" title="c6">x9</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">x9</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">x9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:21_1_1_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E29">A10</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:21_1_1"/>
(  <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">b</font>,<font color="Maroon" title="c4">x</font> or  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> or  not <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1_2"/><span class="lab"><font color="Green" title="E29">A12</font></span>: 
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">b</font>,<font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1_2"/><span class="lab"><font color="Green" title="E30">A13</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:21_1_1_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2"><span class="lab"><font color="Green" title="E30">A13</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E29">A12</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:21_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E28">A9</font></span></a>, <a class="txt" href="translac.html#E20"><span class="lab"><font color="Green" title="E18">Lm11</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="21_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_1_2"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
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">b</font>,<font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font></span><br/><a NAME="E2:21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A15</font></span>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="translac.html#E19"><span class="lab"><font color="Green" title="E17">Lm10</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E3:21_1_2"/><span class="lab"><font color="Green" title="E29">A16</font></span>: 
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="translac.html#E19"><span class="lab"><font color="Green" title="E17">Lm10</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><a NAME="E4:21_1_2"/>
( not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="txt" href="#E1:21_1_2"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="translac.html#E10"><span class="lab"><font color="Green" title="E8">Lm5</font></span></a>, <a class="txt" href="translac.html#E19"><span class="lab"><font color="Green" title="E17">Lm10</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:21_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_2"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E2:21_1_2"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:21_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E26">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:21"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
for <font color="Olive" title="b1">y</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="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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">y</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="translac.html#E15"><span class="lab"><font color="Green" title="E13">Lm7</font></span></a></span>;<br/>
<a NAME="E6:21"/><span class="lab"><font color="Green" title="E25">A18</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">y9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y9</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="translac.html#E18"><span class="lab"><font color="Green" title="E16">Lm9</font></span></a>, <a class="ref" href="transgeo.html#T97" title="TRANSGEO:th.97">TRANSGEO:97</a></span>;<br/>
<a NAME="E7:21"/>
 ex <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 <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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="b3">y</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>] )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="transgeo.html#S1" title="TRANSGEO:sch.1">TRANSGEO:sch 1</a>(<a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E25">A18</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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><span class="kw"> such that </span><br/><a NAME="E9:21"/><span class="lab"><font color="Green" title="E26">A19</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</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="b2">y</font> iff ( ( 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">b</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b2">y</font> ) or (  <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">b</font>,<font color="Olive" title="b1">x</font> &amp;  ex <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( 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">b</font>,<font color="Olive" title="b3">p</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b4">q</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Olive" title="b1">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">q</font>,<font color="Olive" title="b2">y</font> ) ) ) )
 ;<br/>
<a NAME="E10:21"/><span class="lab"><font color="Green" title="E27">A20</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="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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></span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">y</font> = <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>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_2_1"/><span class="lab"><font color="Green" title="E28">A21</font></span>: 
 <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">b</font>,<font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</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="E3:21_2_1"/><span class="lab"><font color="Green" title="E29">A22</font></span>: 
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">b</font>,<font color="Maroon" title="c7">p</font>
 <span class="kw">and </span><br/><a NAME="E4:21_2_1"/><span class="lab"><font color="Green" title="E30">A23</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E5:21_2_1"/><span class="lab"><font color="Green" title="E31">A24</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">q</font>
 <span class="kw">and </span><br/><a NAME="E6:21_2_1"/><span class="lab"><font color="Green" title="E32">A25</font></span>: 
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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>
 <span class="kw">and </span><br/><a NAME="E7:21_2_1"/>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">q</font>,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>;<br/><a NAME="E8:21_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E1:21_2_1"><span class="lab"><font color="Green" title="E28">A21</font></span></a>, <a class="txt" href="#E3:21_2_1"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="txt" href="#E4:21_2_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="txt" href="#E5:21_2_1"><span class="lab"><font color="Green" title="E31">A24</font></span></a>, <a class="txt" href="#E6:21_2_1"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="txt" href="translac.html#E19"><span class="lab"><font color="Green" title="E17">Lm10</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:21_2_1"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21_2_1"><span class="lab"><font color="Green" title="E30">A23</font></span></a>, <a class="txt" href="#E6:21_2_1"><span class="lab"><font color="Green" title="E32">A25</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21_2"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:21"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/>



<div><span class="lab"><font color="Green" title="E28">A26</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_3_1"/><span class="lab"><font color="Green" title="E29">A27</font></span>: 
 ex <font color="Olive" title="b1">x9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">x9</font> &amp; <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>,<font color="Olive" title="b1">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21_3_1"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
 <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">b</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E3:21_3_1"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
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">b</font>,<font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E4:21_3_1"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E3:21_3_1"><span class="lab"><font color="Green" title="E31">A29</font></span></a></span>;<br/><a NAME="E5:21_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>,<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:21_3_1"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E3:21_3_1"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E1:21_3_1"><span class="lab"><font color="Green" title="E29">A27</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_3_1"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E29">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:21_3_2"/><span class="lab"><font color="Green" title="E30">A31</font></span>: 
 ex <font color="Olive" title="b1">y9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFP</font> st <br/>( <font color="Maroon" title="c5">x</font>,<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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Olive" title="b1">y9</font> &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Olive" title="b1">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:21_3_2"/><span class="lab"><font color="Green" title="E31">A32</font></span>: 
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">b</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E3:21_3_2"/><span class="lab"><font color="Green" title="E32">A33</font></span>: 
 <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">b</font>,<font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E4:21_3_2"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:21_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:21_3_2"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E2:21_3_2"><span class="lab"><font color="Green" title="E31">A32</font></span></a>, <a class="txt" href="#E3:21_3_2"><span class="lab"><font color="Green" title="E32">A33</font></span></a>, <a class="txt" href="#E1:21_3_2"><span class="lab"><font color="Green" title="E30">A31</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E30">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_3_3"/><span class="lab"><font color="Green" title="E31">A35</font></span>: 
(  <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">b</font>,<font color="Maroon" title="c5">x</font> &amp;  <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">b</font>,<font color="Maroon" title="c6">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E2:21_3_3"/><span class="kw">then </span>
(  <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">b</font>,<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> &amp;  <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">b</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E27">A20</font></span></a>, <a class="ref" href="aff_1.html#T18" title="AFF_1:th.18">AFF_1:18</a></span>;<br/><a NAME="E3:21_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A36</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T19" title="AFF_1:th.19">AFF_1:19</a></span>;<br/><a NAME="E4:21_3_3"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_3_3"><span class="lab"><font color="Green" title="E31">A35</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">hence </span><a NAME="E5:21_3_3"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="txt" href="#E3:21_3_3"><span class="lab"><font color="Green" title="E32">A36</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="21_3_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:21_3_4"/><span class="lab"><font color="Green" title="E31">A37</font></span>: 
( 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">b</font>,<font color="Maroon" title="c5">x</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">b</font>,<font color="Maroon" title="c6">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E2:21_3_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A38</font></span>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a></span>;<br/><a NAME="E3:21_3_4"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E1:21_3_4"><span class="lab"><font color="Green" title="E31">A37</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_3_4"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E1:21_3_4"><span class="lab"><font color="Green" title="E31">A37</font></span></a>, <a class="txt" href="#E2:21_3_4"><span class="lab"><font color="Green" title="E32">A38</font></span></a>, <a class="ref" href="translac.html#T6" target="_self" title="TRANSLAC:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:21_3"/>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</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>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_3"><span class="lab"><font color="Green" title="E30">A34</font></span></a>, <a class="txt" href="#E1:21_3"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="txt" href="#E2:21_3"><span class="lab"><font color="Green" title="E29">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A39</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="transgeo.html#V6" title="TRANSGEO:attr.6">dilatation</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="transgeo.html#T85" title="TRANSGEO:th.85">TRANSGEO:85</a></span>;<br/>
<a NAME="E13:21"/><span class="lab"><font color="Green" title="E29">A40</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="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_4"/><span class="lab"><font color="Green" title="E30">A41</font></span>: 
 <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">b</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T16" title="AFF_1:th.16">AFF_1:16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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="E3:21_4"/><span class="lab"><font color="Green" title="E31">A42</font></span>: 
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">b</font>,<font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="aff_1.html#T22" title="AFF_1:th.22">AFF_1:22</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">q</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="E5:21_4"/><span class="lab"><font color="Green" title="E32">A43</font></span>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T47" title="DIRAF:th.47">DIRAF:47</a></span>;<br/>
<a NAME="E6:21_4"/>
( <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">q</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c6">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_4"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="ref" href="aff_1.html#T13" title="AFF_1:th.13">AFF_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:21_4"/>
<font color="Maroon" title="c4">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">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E26">A19</font></span></a>, <a class="txt" href="#E3:21_4"><span class="lab"><font color="Green" title="E31">A42</font></span></a>, <a class="txt" href="#E5:21_4"><span class="lab"><font color="Green" title="E32">A43</font></span></a>, <a class="txt" href="#E1:21_4"><span class="lab"><font color="Green" title="E30">A41</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:21"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="Olive" title="b1">x</font>,<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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font>,<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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span><br/>



<a NAME="E1:21_5"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E27">A20</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:21_5"/>
<font color="Maroon" title="c5">x</font>,<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="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="aff_1.html#T14" title="AFF_1:th.14">AFF_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">f</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="#E12:21"><span class="lab"><font color="Green" title="E28">A39</font></span></a>, <a class="ref" href="transgeo.html#T100" title="TRANSGEO:th.100">TRANSGEO:100</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:21"/>
 ex <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 <br/>( <font color="Olive" title="b1">f</font> is  <a href="transgeo.html#V7" title="TRANSGEO:attr.7">translation</a>  &amp; <font color="Olive" title="b1">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">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E29">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
