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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</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="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>





<b>thus </b><a NAME="E1:73"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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">(<span class="default"><a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c7">z</font> <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"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:73_1"/><i><font color="Green" title="E36">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:73_1"/><i><font color="Green" title="E37">A3</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="funcop_1.html#E3:73_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="funcop_1.html#D1" target="_self" title="FUNCOP_1:def.1">Def1</a></i>;<br/>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>

<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="funcop_1.html#E3:73_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="funcop_1.html#E4:73_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="funcop_1.html#D1" target="_self" title="FUNCOP_1:def.1">Def1</a></i>
;<br/>
<a NAME="E7:73_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="funcop_1.html#E5:73_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:73_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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">(<span class="default"><a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>


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

<b>assume </b><a NAME="E2:73"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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">(<span class="default"><a href="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 ;<br/>

<a NAME="E3:73"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</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="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:73"/><i><font color="Green" title="E35">A4</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E6:73"/><i><font color="Green" title="E36">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c7">z</font> <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"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="funcop_1.html#E5:73"><i><font color="Green" title="E35">A4</font></i></a>, <a class="ref" href="funcop_1.html#D1" target="_self" title="FUNCOP_1:def.1">Def1</a></i>;<br/>
<a NAME="E8:73"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="funcop_1.html#E5:73"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="funcop_1.html#E6:73"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="funcop_1.html#D1" target="_self" title="FUNCOP_1:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E9:73"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">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="funcop_1.html#K9" title="FUNCOP_1:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funcop_1.html#K10" title="FUNCOP_1:func.10">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="funcop_1.html#E7:73"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div>
