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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c4">Q</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">S</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>







<b>thus </b><a NAME="E1:11"/>
( <font color="Maroon" title="c5">x</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"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Q</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon" title="c5">x</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"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green" title="E7">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> &amp; <font color="Maroon" title="c6">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K2" title="WEIERSTR:func.2">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">Q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="weierstr.html#E3:11_1"><i><font color="Green" title="E7">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">A1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:11_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A1</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E3:11_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="weierstr.html#D2" target="_self" title="WEIERSTR:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E8:11_1"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c9">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> &amp; <font color="Maroon" title="c8">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">A2</font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E6:11_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="weierstr.html#D1" target="_self" title="WEIERSTR:def.1">Def1</a></i>;<br/>
<a NAME="E9:11_1"/>
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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> <font color="Maroon" title="c9">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A2</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145" title="FUNCT_1:th.145">FUNCT_1:145</a></i>;<br/>
<b>hence </b><a NAME="E10:11_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c4">Q</font>
 <b>by </b><i><a class="txt" href="weierstr.html#E3:11_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="weierstr.html#E6:11_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="weierstr.html#E8:11_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


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


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