<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</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">A</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c2">Y</font>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>





<b>thus </b><a NAME="E1:31"/>
<font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c3">A</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<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>reconsider </b><font color="Maroon" title="c6">uA</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>assume </b><a NAME="E2:31_1"/><i><font color="Green" title="E17">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="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/>

<a NAME="E3:31_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">uA</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31_1"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">YY</font> &amp; <font color="Maroon" title="c7">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 <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="c8">fY</font> = <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">YY</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E7:31_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c8">fY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:31_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="weierstr.html#D1" title="WEIERSTR:def.1">WEIERSTR:def 1</a></i>;<br/>
<a NAME="E8:31_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">YY</font>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:31_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="borsuk_2.html#E5:31_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/>
<b>hence </b><a NAME="E9:31_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E7:31_1"><i><font color="Green" title="E19">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>

<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>assume </b><a NAME="E2:31"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">YY</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:31"/><i><font color="Green" title="E17">A4</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">YY</font> &amp; <font color="Maroon" title="c6">YY</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="weierstr.html#K1" title="WEIERSTR:func.1">"</a> <font color="Maroon" title="c3">A</font> )
 <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">B1</font> = <font color="Maroon" title="c6">YY</font> 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_2.html#E4:31"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font><b> such that </b><br/><a NAME="E7:31"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c8">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c7">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">B</font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E4:31"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="weierstr.html#D1" title="WEIERSTR:def.1">WEIERSTR:def 1</a></i>;<br/>
<a NAME="E8:31"/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E4:31"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E7:31"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E9:31"/><b>then </b><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:31"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E10:31"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E4:31"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E11:31"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:31"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E9:31"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div>
