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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <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"> for <font color="Olive" title="b1">D</font> being   <a href="borsuk_1.html#M1" title="BORSUK_1:mode.1">u.s.c._decomposition</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K13" title="BORSUK_1:func.13">space</a> <font color="Olive" title="b1">D</font></span>)</span><br/> for <font color="Olive" title="b3">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b2">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b3">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b2">t</font></span><br/><b>let </b><font color="Maroon" title="c2">D</font> be    <a href="borsuk_1.html#M1" title="BORSUK_1:mode.1">u.s.c._decomposition</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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><br/> for <font color="Olive" title="b2">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Olive" title="b1">t</font></span><br/><b>let </b><font color="Maroon" title="c3">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds  <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font></span><br/><b>let </b><font color="Maroon" title="c4">G</font> be    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font></span><br/>







<a NAME="E1:86"/><i><font color="Green" title="E66">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/>
<a NAME="E2:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T30" target="_self" title="BORSUK_1:th.30">Th30</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:86"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c5">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:86"/><i><font color="Green" title="E68">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">G</font> )
 <b>and </b><br/><a NAME="E6:86"/><i><font color="Green" title="E69">A4</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Olive" title="b1">B</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c5">W</font> holds <br/><font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font>
 <b>by </b><i><a class="txt" href="#E1:86"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="borsuk_1.html#D13" target="_self" title="BORSUK_1:def.13">Def13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Q</font> = <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font>;<br/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K1" title="BORSUK_1:func.1">proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:86"><i><font color="Green" title="E66">A1</font></i></a>, <a class="ref" href="borsuk_1.html#T31" target="_self" title="BORSUK_1:th.31">Th31</a></i>
<br/>.= 
<font color="Maroon" title="c5">W</font>
<b>by </b><i><a class="txt" href="#E6:86"><i><font color="Green" title="E69">A4</font></i></a>, <a class="ref" href="borsuk_1.html#T33" target="_self" title="BORSUK_1:th.33">Th33</a></i>
;<br/>
<a NAME="E8:86"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</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="#E4:86"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E9:86"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</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="txt" href="#E1:86"><i><font color="Green" title="E66">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="E10:86"/><b>then </b><i><font color="Green" title="E70">A5</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</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="E11:86"/><i><font color="Green" title="E71">A6</font></i>: 
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">G</font>
 
<b>by </b><i><a class="txt" href="#E5:86"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E12:86"/><i><font color="Green" title="E72">A7</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T72" target="_self" title="BORSUK_1:th.72">Th72</a></i>;<br/>
<a NAME="E13:86"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="txt" href="#E5:86"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E14:86"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="txt" href="#E12:86"><i><font color="Green" title="E72">A7</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<a NAME="E15:86"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c5">W</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E16:86"/><b>then </b>
<font color="Maroon" title="c3">t</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"><span class="p2">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:86"><i><font color="Green" title="E70">A5</font></i></a>, <a class="txt" href="#E11:86"><i><font color="Green" title="E71">A6</font></i></a>, <a class="ref" href="tops_1.html#T54" title="TOPS_1:th.54">TOPS_1:54</a></i>;<br/>
<b>hence </b><a NAME="E17:86"/>
<span class="p1">(<span class="default"><a href="borsuk_1.html#K14" title="BORSUK_1:func.14">Proj</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">t</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</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>
