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<b>let </b><font color="Maroon" title="c1">AS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="analoaf.html#L1" title="ANALOAF:struct.1">AffinStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="analoaf.html#NM2" title="ANALOAF:NM.2">OAffinPlane</a>  st <font color="Maroon" title="c1">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Olive" title="b1">S</font> holds <br/>for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font>, <font color="Olive" title="b5">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">z</font>,<font color="Olive" title="b5">t</font> holds <br/> ex <font color="Olive" title="b6">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b6">u</font> &amp; <font color="Olive" title="b4">z</font>,<font color="Olive" title="b5">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">z</font>,<font color="Olive" title="b6">u</font> )</span><br/><b>let </b><font color="Maroon" title="c2">S</font> be   <a href="analoaf.html#NM2" title="ANALOAF:NM.2">OAffinPlane</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">S</font> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font>, <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not <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="Olive" title="b3">z</font>,<font color="Olive" title="b4">t</font> holds <br/> ex <font color="Olive" title="b5">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <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="Olive" title="b1">x</font>,<font color="Olive" title="b5">u</font> &amp; <font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">z</font>,<font color="Olive" title="b5">u</font> ) )</span><br/>



<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="diraf.html#K2" title="DIRAF:func.2">Lambda</a> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font>, <font color="Olive" title="b4">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not <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="Olive" title="b3">z</font>,<font color="Olive" title="b4">t</font> holds <br/> ex <font color="Olive" title="b5">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <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="Olive" title="b1">x</font>,<font color="Olive" title="b5">u</font> &amp; <font color="Olive" title="b3">z</font>,<font color="Olive" title="b4">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">z</font>,<font color="Olive" title="b5">u</font> )</span><br/>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Olive" title="b1">u</font> ) )</span><br/>

<b>assume </b><a NAME="E2:55"/><i><font color="Green" title="E37">A2</font></i>: 
not <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Olive" title="b1">u</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c9">z'</font> = <font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c10">t'</font> = <font color="Maroon" title="c6">t</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="diraf.html#E1:55"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E4:55"/>
not <font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c8">y'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c9">z'</font>,<font color="Maroon" title="c10">t'</font>
 
<b>by </b><i><a class="txt" href="diraf.html#E1:55"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="diraf.html#E2:55"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="diraf.html#T45" target="_self" title="DIRAF:th.45">Th45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">u'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E6:55"/><i><font color="Green" title="E38">A3</font></i>: 
( <font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c8">y'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c11">u'</font> &amp; <font color="Maroon" title="c9">z'</font>,<font color="Maroon" title="c10">t'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c9">z'</font>,<font color="Maroon" title="c11">u'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T50" target="_self" title="DIRAF:th.50">Th50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> <b>by </b><i><a class="txt" href="diraf.html#E1:55"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E8:55"/>
( <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c12">u</font> &amp; <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c12">u</font> )
 
<b>by </b><i><a class="txt" href="diraf.html#E1:55"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="diraf.html#E6:55"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="diraf.html#T45" target="_self" title="DIRAF:th.45">Th45</a></i>;<br/>
<b>hence </b><a NAME="E9:55"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c6">t</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">z</font>,<font color="Olive" title="b1">u</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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