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<b>let </b><font color="Maroon">p'</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/><b>let </b><font color="Maroon">q'</font> be   <a href="afinsq_1.html#NM1">XFinSequence</a>;<br/>



<a NAME="E1:1"/><i><font color="Green">E51</font></i>: 
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> <a href="tarski.html#R1">c=</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> &amp;  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> )
 
<b>by </b><i><a class="ref" href="afinsq_1.html#T24">AFINSQ_1:24</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E2:1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font></span>)</span> as   <a href="afinsq_1.html#NM1">XFinSequence</a> <b>by </b><i><a class="ref" href="afinsq_1.html#T12">AFINSQ_1:12</a></i>;<br/>
<a NAME="E4:1"/>
 <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/>
<a NAME="E5:1"/><b>then </b><i><font color="Green">E53</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font>
 ;<br/><a NAME="E2:1_1"/>
<font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E3:1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> )
 <b>by </b><i>, <a class="ref" href="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E4:1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a>, <a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><a NAME="E5:1_1"/><b>then </b>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/><b>hence </b><a NAME="E7:1_1"/>
<font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:1"/>
<span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/>


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