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<a NAME="E1:75_1_1"/><i><font color="Green" title="E57">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">F</font> =  <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">D</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Maroon" title="c2">D</font></span>)</span> ;<br/>
<a NAME="E3:75_1_1"/>
<font color="Maroon" title="c3">F</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">W</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_1.html#D3" target="_self" title="BORSUK_1:def.3">BORSUK_1:def 3</a><br/><b>let </b><font color="Maroon" title="c5">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c6">H</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="borsuk_1.html#E1:75_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T25" target="_self" title="BORSUK_1:th.25">Th25</a></i>;<br/>
<a NAME="E2:75_1_1_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:75_1_1_1"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:75_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T69" target="_self" title="BORSUK_1:th.69">Th69</a></i>;<br/>
<a NAME="E4:75_1_1_1"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c6">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E5:75_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E6:75_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<a NAME="E7:75_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:75_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T31" target="_self" title="BORSUK_1:th.31">Th31</a></i>;<br/>
<a NAME="E8:75_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E4:75_1_1_1"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/>
<a NAME="E9:75_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:75_1_1"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T31" target="_self" title="BORSUK_1:th.31">Th31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font></span>)</span> as    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">N</font>
;<br/>

<a NAME="E11:75_1_1_1"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></i>;<br/>
<a NAME="E12:75_1_1_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E13:75_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="borsuk_1.html#E11:75_1_1_1"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:75_1_1"/>
 <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">D</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Maroon" title="c2">D</font></span>)</span>
 ;<br/>


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