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<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E31">A2</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T59" target="_self" title="FUNCT_3:th.59">Th59</a></i>;<br/>
<a NAME="E4:53"/>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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="E1:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E2:53_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="txt" href="funct_3.html#E1:53"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="funct_3.html#D5" target="_self" title="FUNCT_3:def.5">Def5</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#D5" target="_self" title="FUNCT_3:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:53"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K7" title="FUNCT_3:func.7">pr1</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="funct_3.html#E3:53"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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