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<b>let </b><font color="Maroon">S</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">AS</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="analoaf.html#L1">AffinStruct</a> ;<br/>



<b>assume </b><a NAME="E1:48"/>
<font color="Maroon">AS</font> <a href="hidden.html#R1">=</a>  <a href="diraf.html#K2" target="_self">Lambda</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">AS</font> <a href="hidden.html#R1">=</a>  <a href="analoaf.html#G1">AffinStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="diraf.html#K1" target="_self">lambda</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font></span>)</span> #)
 
;<br/>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">b'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">c'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>, <font color="Maroon">d'</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/>



<b>assume </b><a NAME="E3:48"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c'</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d'</font> )
 ;<br/>

<b>thus </b><a NAME="E4:48"/>
( <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c'</font>,<font color="Maroon">d'</font> implies <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AS</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a><br/>

<b>assume </b><a NAME="E2:48_1"/>
not <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a>,<a class="ref" href="diraf.html#D4" target="_self">DIRAF:def 4</a><br/>

<b>hence </b><a NAME="E3:48_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font>
 <b>by </b><i>, , , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E5:48"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="diraf.html#R2" target="_self">'||'</a> <font color="Maroon">c</font>,<font color="Maroon">d</font>
 ;<br/>

<a NAME="E6:48"/><b>then </b>
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">d</font> or <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">d</font>,<font color="Maroon">c</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E7:48"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">d</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">d</font>,<font color="Maroon">c</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:48"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">d'</font></span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="analoaf.html#U1">CONGR</a> of <font color="Maroon">AS</font>
 <b>by </b><i>, , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="analoaf.html#D2">ANALOAF:def 2</a><br/>


</div>
