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

<b>let </b><font color="Maroon">X1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>, <font color="Maroon">X2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:77_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> iff  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 <b>and </b><br/><a NAME="E2:77_1_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> iff  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 ) )
 ;<br/>

<a NAME="E3:77_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:77_1_2_1"/>
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:77_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:77_1_2_1_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:77_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:77_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E3:77_1_2_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="borsuk_1.html#K7">]</a></span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Olive">a</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:77_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:77_1_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X2</font>
 ;<br/>


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