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<b>let </b><font color="Maroon">T1</font> be   <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T2</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font>;<br/><b>let </b><font color="Maroon">S2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">S1</font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">S2</font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
( ( not <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> ) or ( <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> ) or ( not <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> ) or ( <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/><i><font color="Green">E26</font></i>: 
( not <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S2</font>
 <b>by </b><i><a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">T1</font> = <font color="Maroon">T1</font>, <font color="Maroon">T2</font> = <font color="Maroon">T2</font> as  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, <a class="txt" href="borsuk_3.html#E3:45_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">S2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> <b>by </b><i><a class="txt" href="borsuk_3.html#E3"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="borsuk_3.html#E4"><i><font color="Green">Lemma29</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">X</font> = <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span> ;<br/><a NAME="E10:45_1_1"/>
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">S2</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">T1</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">S1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T2</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">S2</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> #) <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">X</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">X</font></span>)</span> #)
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:45_1_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/>
( <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; not <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S1</font> = <font color="Maroon">S1</font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T1</font> ;<br/><a NAME="E3:45_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><b>hence </b><a NAME="E4:45_1_2"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_3"><b>suppose </b></a><a NAME="E1:45_1_3"/>
( not <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">S2</font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><a NAME="E3:45_1_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><b>hence </b><a NAME="E4:45_1_3"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_4"><b>suppose </b></a><a NAME="E1:45_1_4"/>
( <font color="Maroon">S1</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">S2</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S2</font> = <font color="Maroon">S2</font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T2</font> ;<br/><a NAME="E3:45_1_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 ;<br/><b>hence </b><a NAME="E4:45_1_4"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">S1</font>,<font color="Maroon">S2</font></span><a href="borsuk_1.html#K6">:]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">T1</font>,<font color="Maroon">T2</font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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