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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>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">B</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font>;<br/>





<a NAME="E1:76"/><i><font color="Green" title="E57">A1</font></i>: 
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#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font> :  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</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 <font color="Maroon" title="c1">X</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D10" target="_self" title="BORSUK_1:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E2:76"/>
(  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">B</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 <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c3">B</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#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c2">D</font></span>)</span> )
 ;<br/>

<b>assume </b><a NAME="E3:76"/>
<font color="Maroon" title="c3">B</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#K14" title="BORSUK_1:func.14">space</a> <font color="Maroon" title="c2">D</font></span>)</span>
 ;<br/>

<a NAME="E4:76"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">D</font> st <br/>( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Olive" title="b1">A</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 <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="txt" href="borsuk_1.html#E1:76"><i><font color="Green" title="E57">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:76"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">B</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 <font color="Maroon" title="c1">X</font>
 ;<br/>


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