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<div class="add">

<span class="kw">reconsider </span><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#K12" title="BORSUK_1:func.12">space</a> <font color="Maroon" title="c2">D</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></span>;<br/>
<a NAME="E2:86_1_1"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
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#K12" title="BORSUK_1:func.12">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></span>;<br/>
<a NAME="E3:86_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="86_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="borsuk_1.html#D3" target="_self" title="BORSUK_1:def.3">BORSUK_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <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>  ex <font color="Olive" title="b2">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> st <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">H</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">W</font> st <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font></span><br/>



<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T25" target="_self" title="BORSUK_1:th.25">Th25</a></span>;<br/>
<a NAME="E2:86_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<a NAME="E3:86_1_1_1"/><span class="kw">then </span>
<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#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<a NAME="E4:86_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T31" target="_self" title="BORSUK_1:th.31">Th31</a></span>;<br/>
<a NAME="E5:86_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#K12" title="BORSUK_1:func.12">space</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E6:86_1_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T69" target="_self" title="BORSUK_1:th.69">Th69</a></span>;<br/>
<a NAME="E7:86_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E8:86_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:86_1_1_1"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></span>;<br/>
<a NAME="E9:86_1_1_1"/><span class="kw">then </span>
<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#K8" title="RELSET_1:func.8">"</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="ref" href="borsuk_1.html#T31" target="_self" title="BORSUK_1:th.31">Th31</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">N</font> = <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">N</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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></span><br/>

<a NAME="E11:86_1_1_1"/>
( <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> &amp;  <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a>, <a class="ref" href="tops_1.html#T44" title="TOPS_1:th.44">TOPS_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:86_1_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:86_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#K12" title="BORSUK_1:func.12">space</a> <font color="Maroon" title="c2">D</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
