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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="analmetr.html#NM2" title="ANALMETR:NM.2">OrtAfPl</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  iff <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a>  )</span><br/>

<a NAME="E1:21"/>
( <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a>  implies  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">M</font>, <font color="Olive" title="b3">N</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)<br/>  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font>, <font color="Olive" title="b6">c</font>, <font color="Olive" title="b7">a1</font>, <font color="Olive" title="b8">b1</font>, <font color="Olive" title="b9">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b1">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b7">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">M</font> &amp; <font color="Olive" title="b6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b9">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b1">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b2">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b3">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</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">M</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">N</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b5">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a1</font>,<font color="Olive" title="b8">b1</font> &amp; <font color="Olive" title="b4">a</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">a1</font>,<font color="Olive" title="b9">c1</font> holds <br/><font color="Olive" title="b5">b</font>,<font color="Olive" title="b6">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b8">b1</font>,<font color="Olive" title="b9">c1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">a1</font>, <font color="Olive" title="b5">b1</font>, <font color="Olive" title="b6">c1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #)  st <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b4">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b5">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b6">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</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">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a1</font>,<font color="Olive" title="b5">b1</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a1</font>,<font color="Olive" title="b6">c1</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b1</font>,<font color="Olive" title="b6">c1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c10">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</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">M</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">N</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c10">c1</font> implies <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">c1</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:21_1_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E2:21_1_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="aff_1.html#R5" title="AFF_1:pred.5">//</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E3:21_1_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E4:21_1_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c8">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">and </span><br/><a NAME="E5:21_1_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E6:21_1_1"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 <span class="kw">and </span><br/><a NAME="E7:21_1_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E8:21_1_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c10">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E9:21_1_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E10:21_1_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E11:21_1_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E12:21_1_1"/><span class="lab"><font color="Green" title="E21">A13</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">M</font>
 <span class="kw">and </span><br/><a NAME="E13:21_1_1"/><span class="lab"><font color="Green" title="E22">A14</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="c4">N</font>
 <span class="kw">and </span><br/><a NAME="E14:21_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c9">b1</font>
 <span class="kw">and </span><br/><a NAME="E15:21_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c10">c1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">c1</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">a9</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c12">a19</font> = <font color="Maroon" title="c8">a1</font>, <font color="Maroon" title="c13">b9</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c14">b19</font> = <font color="Maroon" title="c9">b1</font>, <font color="Maroon" title="c15">c9</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c16">c19</font> = <font color="Maroon" title="c10">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/><a NAME="E17:21_1_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">c1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
 not <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">c1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_1"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">a1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_1_1_1_1_1"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E1:21_1_1_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E2:21_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A21</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c1</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:21_1_1_1_1_1"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">c1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:21_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E11:21_1_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E2:21_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T45" onmouseover="rs('aff_1/T45')" onmouseout="rh()">AFF_1:45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:21_1_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:21_1_1_1_1_2"/>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E1:21_1_1_1_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="aff_1.html#D1" onmouseover="rs('aff_1/D1')" onmouseout="rh()">AFF_1:def 1</a></span>;<br/>
<a NAME="E2:21_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A22</font></span>: 
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c5">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="aff_1.html#T6" onmouseover="rs('aff_1/T6')" onmouseout="rh()">AFF_1:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:21_1_1_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">b1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:21_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:21_1_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E2:21_1_1_1_1_2"><span class="lab"><font color="Green" title="E29">A22</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="aff_1.html#T45" onmouseover="rs('aff_1/T45')" onmouseout="rh()">AFF_1:45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E4:21_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E2:21_1_1_1_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="aff_1.html#T2" onmouseover="rs('aff_1/T2')" onmouseout="rh()">AFF_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:21_1_1_1"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
(  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c13">b9</font> &amp;  not  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c15">c9</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:21_1_1_1_2"/>
(  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c13">b9</font> or  <a href="analmetr.html#R5" title="ANALMETR:pred.5">LIN</a> <font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c15">c9</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:21_1_1_1_2"/><span class="kw">then </span>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c6">b</font> or  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a1</font>,<font color="Maroon" title="c7">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T40" onmouseover="rs('analmetr/T40')" onmouseout="rh()">ANALMETR:40</a></span>;<br/>
<a NAME="E3:21_1_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:21_1_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E2:21_1_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="aff_1.html#T25" onmouseover="rs('aff_1/T25')" onmouseout="rh()">AFF_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E12:21_1_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E13:21_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="aff_1.html#T45" onmouseover="rs('aff_1/T45')" onmouseout="rh()">AFF_1:45</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:21_1_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c10">c1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E7:21_1_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E8:21_1_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a></span>;<br/>
<a NAME="E5:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A24</font></span>: 
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">c9</font>,<font color="Maroon" title="c16">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/>
<a NAME="E6:21_1_1_1"/>
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">a1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c9">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:21_1_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E4:21_1_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:21_1_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E6:21_1_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="aff_1.html#T39" onmouseover="rs('aff_1/T39')" onmouseout="rh()">AFF_1:39</a></span>;<br/>
<a NAME="E7:21_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A25</font></span>: 
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c12">a19</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c14">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/>
<a NAME="E8:21_1_1_1"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c15">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c16">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/>
<a NAME="E9:21_1_1_1"/>
<font color="Maroon" title="c11">a9</font>,<font color="Maroon" title="c13">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c12">a19</font>,<font color="Maroon" title="c14">b19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>;<br/>
<a NAME="E10:21_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">b9</font>,<font color="Maroon" title="c15">c9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c14">b19</font>,<font color="Maroon" title="c16">c19</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E3:21_1_1_1"><span class="lab"><font color="Green" title="E27">A23</font></span></a>, <a class="txt" href="#E7:21_1_1_1"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E5:21_1_1_1"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E8:21_1_1_1"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="ref" href="conmetr.html#D8" onmouseover="rs('conmetr/D8')" onmouseout="rh()">CONMETR:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:21_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="analmetr.html#T36" onmouseover="rs('analmetr/T36')" onmouseout="rh()">ANALMETR:36</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E18:21_1_1"/>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c10">c1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:21_1"/>
 <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aff_2.html#D11" onmouseover="rs('aff_2/D11')" onmouseout="rh()">AFF_2:def 11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:21"/>
(  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">U1</a> of <font color="Maroon" title="c1">X</font>, the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c1">X</font> #) is  <a href="aff_2.html#V11" title="AFF_2:attr.11">translational</a>  iff <font color="Maroon" title="c1">X</font> is  <a href="conmetr.html#V8" title="CONMETR:attr.8">satisfying_des</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="conmetr.html#T8" onmouseover="rs('conmetr/T8')" onmouseout="rh()">CONMETR:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
