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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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>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/>







<b>assume </b><a NAME="E1:3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</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> or <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</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>
 ;<br/><div class="add"><a NAME="E2:3_1_1_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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;  <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#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:3_1_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:3_1_1_1"/><b>then </b>
(  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp;  <a href="afinsq_1.html#K1">len</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="tarski.html#R1">c=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T56">CARD_1:56</a></i>;<br/><a NAME="E5:3_1_1_1"/><b>then </b>
(  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">k</font> &amp;  <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> <a href="tarski.html#R1">c=</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:3_1_1_1"/><b>then </b>
( <font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font> &amp; <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="card_fin.html#K3">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_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="card_fin.html#K3">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</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>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2"/><b>then </b>
( <font color="Maroon">n</font> <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> &amp;  <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#T20">AFINSQ_1:20</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E3:3_1_1_2"/><b>then </b>
<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">q'</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E4:3_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a></i>;<br/><a NAME="E5:3_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</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> <font color="Maroon">k</font> &amp;  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <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>, <a class="ref" href="afinsq_1.html#T14">AFINSQ_1:14</a>, <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:3_1_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:3_1_1_2_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</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> )
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:3_1_1_2_1"/><b>then </b>
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</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> <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="hidden.html#R1">=</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> )
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E4:3_1_1_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">m</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#K8">/\</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:3_1_1_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</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">m</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:3_1_1_2_1_1_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> or <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:3_1_1_2_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</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="euler_1.html#T1">EULER_1:1</a></i>;<br/><a NAME="E3:3_1_1_2_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> &amp; <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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> )
 <b>by </b><i><a class="ref" href="afinsq_1.html#D4">AFINSQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:3_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="3_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:3_1_1_2_1_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">p'</font>
 ;<br/><div class="add"><a NAME="E2:3_1_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</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> )
 <b>by </b><i>, , , , <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E3:3_1_1_2_1_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</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">q'</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span></span>)</span> <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">m</font> &amp;  <a href="afinsq_1.html#K1">len</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="hidden.html#R1">=</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">m</font> <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> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</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#T22">AFINSQ_1:22</a>, , <a class="ref" href="afinsq_1.html#T20">AFINSQ_1:20</a></i>;<br/><a NAME="E4:3_1_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> is   <a href="nat_1.html#NM1">Nat</a> )
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T61">NAT_1:61</a></i>;<br/><a NAME="E5:3_1_1_2_1_1_1_2"/><b>then </b>
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font> &amp;  <a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a>  <a href="afinsq_1.html#K1">len</a> <font color="Maroon">q'</font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T1">EULER_1:1</a>, <a class="ref" href="afinsq_1.html#D1">AFINSQ_1:def 1</a></i>;<br/><a NAME="E6:3_1_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1">len</a> <font color="Maroon">p'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">k</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2">dom</a> <font color="Maroon">q'</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E7:3_1_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T71">FUNCT_1:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:3_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <font color="Maroon">q'</font></span>)</span> <a href="card_fin.html#K3">|</a> <font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p2">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:3_1_1_2"/>
<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="card_fin.html#K3">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="afinsq_1.html#T11">AFINSQ_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
<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="card_fin.html#K3">|</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="ordinal4.html#K1">^</a> <span class="p1">(<span class="default"><font color="Maroon">q'</font> <a href="card_fin.html#K3">|</a> <font color="Maroon">k</font></span>)</span>
 ;<br/>


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