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<span class="kw">let </span><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a>  implies <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/>
<font color="Maroon" title="c1">P</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a> </span><br/>

<a NAME="E2:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c1">P</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D10" target="_self" title="RELAT_2:def.10">Def10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span> implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font> )</span><br/><div><span class="lab"><font color="Green" title="E22">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:18_1_1"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/><a NAME="E4:18_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_1"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E6:18_1_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="relat_2.html#D2" target="_self" title="RELAT_2:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E23">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:18_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:18_1_2"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></span>;<br/><a NAME="E4:18_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_2"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E5:18_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><span class="kw">hence </span><a NAME="E6:18_1_2"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="relat_2.html#D2" target="_self" title="RELAT_2:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:18_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span><br/><a NAME="E4:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D6" title="RELAT_1:def.6">RELAT_1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E5:18_1"/>
 not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E2:18_1"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:18"/><span class="kw">then </span>
<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font> <a href="relat_2.html#R2" title="RELAT_2:pred.2">is_irreflexive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D2" target="_self" title="RELAT_2:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:18"/>
<font color="Maroon" title="c1">P</font> <a href="xboole_0.html#K4" title="XBOOLE_0:func.4">\</a> <font color="Maroon" title="c2">R</font> is  <a href="relat_2.html#V2" title="RELAT_2:attr.2">irreflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D10" target="_self" title="RELAT_2:def.10">Def10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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