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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="relat_1.html#NM1">Relation</a>;<br/>



<b>assume </b><a NAME="E1:18"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 ;<br/>

<a NAME="E2:18"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:18_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D6">RELAT_1:def 6</a></i>;<br/><div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><a NAME="E4:18_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="relat_2.html#E3:18_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:18_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E6:18_1_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="relat_2.html#E2:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:18_1_2"/><i><font color="Green">E24</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><a NAME="E4:18_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="relat_2.html#E3:18_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:18_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30">RELAT_1:30</a></i>;<br/><b>hence </b><a NAME="E6:18_1_2"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="relat_2.html#E2:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:18_1"/>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="relat_2.html#E2:18_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="relat_2.html#E3:18_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:18"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font> <a href="relat_2.html#R2" target="_self">is_irreflexive_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>hence </b><a NAME="E5:18"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="relat_2.html#V2" target="_self">irreflexive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#D10" target="_self">Def10</a></i>;<br/>


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