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<div class="add">

<a NAME="E1:32_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">XSeq</font> <a href="prob_1.html#K1">.</a> 0
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:32_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:32_1_1"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>proof </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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_1_2_1_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_1_1"><b>case </b></a><a NAME="E1:32_1_1_2_1_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:32_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_1_2_1_1_2"><b>case </b></a><a NAME="E1:32_1_1_2_1_1_2"/>
 ex <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:32_1_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:32_1_1_2_1_1_2"/>
 <a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">XSeq</font> <a href="funct_2.html#R2">=</a>  <a href="prob_3.html#K5" target="_self">@Partial_Union</a> <font color="Maroon">XSeq</font>
 ;<br/><a NAME="E6:32_1_1_2_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font> &amp; <font color="Maroon">XSeq</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font> )
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/><a NAME="E7:32_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">XSeq</font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K2" target="_self">Partial_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T44">PROB_1:44</a></i>;<br/><b>hence </b><a NAME="E8:32_1_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_1_1_2"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Si</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_1_1"/>
 <a href="prob_3.html#K3" target="_self">Partial_Diff_Union</a> <font color="Maroon">XSeq</font> is    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Si</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D9">PROB_1:def 9</a></i>;<br/>


</div>
