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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">S</font> being  <a href="kurato_2.html#NM1" title="KURATO_2:NM.1">SetSequence</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">S</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> &amp; <font color="Olive" title="b1">S</font> is <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> &amp; ( not <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> implies <font color="Olive" title="b1">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> ) )</span><br/>

<b>take </b><font color="Maroon" title="c2">S</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> &amp; ( not <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> implies <font color="Maroon" title="c2">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> ) )</span><br/>

<div><i><font color="Green" title="E13">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/><a NAME="E1:16_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:16_1"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/><a NAME="E1:16_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E2:16_2"/>
<font color="Maroon" title="c2">S</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c3">i</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
( <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V4" title="COMPL_SP:attr.4">open</a> &amp; <font color="Maroon" title="c2">S</font> is <a href="compl_sp.html#V5" title="COMPL_SP:attr.5">closed</a> )
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="compl_sp.html#D5" target="_self" title="COMPL_SP:def.5">Def5</a>, <a class="ref" href="compl_sp.html#D6" target="_self" title="COMPL_SP:def.6">Def6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> implies <font color="Maroon" title="c2">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> )</span><br/>

<b>assume </b><a NAME="E4:16"/><i><font color="Green" title="E14">A3</font></i>: 
not <font color="Maroon" title="c1">T</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a></span><br/>

<a NAME="E5:16"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">S</font> holds <br/> not <font color="Maroon" title="c2">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="txt" href="#E4:16"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon" title="c2">S</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D15" title="FUNCT_1:def.15">FUNCT_1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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