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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>thus </b><a NAME="E1:48"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_1.html#K5" target="_self">:]</a></span><b> such that </b><br/><a NAME="E5:48_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E6:48_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:48_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="borsuk_1.html#T45" target="_self">Th45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:48_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E3:48_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="borsuk_1.html#E5:48_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:48_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="borsuk_1.html#E6:48_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="borsuk_1.html#E8:48_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E11:48_1"/>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E5:48_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="borsuk_1.html#E8:48_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="borsuk_1.html#E10:48_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E12:48_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E3:48_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:48_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E8:48_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="borsuk_1.html#E10:48_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T138">ZFMISC_1:138</a></i>;<br/>
<a NAME="E14:48_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E10:48_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<a NAME="E15:48_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E16:48_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E8:48_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="borsuk_1.html#E10:48_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E2:48"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E3:48"/>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E4:48"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E5:48"/>
<span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="borsuk_1.html#K6" target="_self">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1"><a href="borsuk_1.html#K6" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="borsuk_1.html#K6" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E2:48"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>


</div>
