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<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>





<b>set </b><font color="Maroon" title="c4">k1</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c5">k2</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>;<br/>
<b>set </b><font color="Maroon" title="c6">k3</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">k4</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">k5</font> =  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">k6</font> =  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>;<br/>
<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<a NAME="E2:16"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1"/>
( <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> or <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="kurato_1.html#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1"/>
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_1"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_2"/>
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_2"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_3"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_3"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T26" title="TOPS_1:th.26">TOPS_1:26</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_3"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="kurato_1.html#E1:16_1_1_3"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_4"/>
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_4"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_5"><b>suppose </b></a><a NAME="E1:16_1_1_5"/><i><font color="Green" title="E9">A3</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_5"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T26" title="TOPS_1:th.26">TOPS_1:26</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_5"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="kurato_1.html#E1:16_1_1_5"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_6"><b>suppose </b></a><a NAME="E1:16_1_1_6"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_6"/><b>then </b><i><font color="Green" title="E10">A5</font></i>: 
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T26" title="TOPS_1:th.26">TOPS_1:26</a></i>;<br/><a NAME="E3:16_1_1_6"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p3">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tdlat_1.html#T5" title="TDLAT_1:th.5">TDLAT_1:5</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_6"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="kurato_1.html#E2:16_1_1_6"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="kurato_1.html#E1:16_1_1_6"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_7"><b>suppose </b></a><a NAME="E1:16_1_1_7"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:16_1_1_7"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T58" title="TOPS_1:th.58">TOPS_1:58</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_7"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="kurato_1.html#E1:16_1_1_7"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="enumset1.html#D5" title="ENUMSET1:def.5">ENUMSET1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c3">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_1.html#K5" title="KURATO_1:func.5">Kurat7Set</a> <font color="Maroon" title="c2">A</font> )
 ;<br/>


</div>
