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

<a NAME="E1:44_2_1"/>
 <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">X</font> =  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">R</font> =  <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>;<br/>
<a NAME="E1:44_2_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Olive" title="b2">y</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Olive" title="b3">z</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Maroon" title="c6">y</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Maroon" title="c7">z</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> &amp; <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="c7">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> 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="c7">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:44_2_1_1_1"/><span class="lab"><font color="Green" title="E36">B0</font></span>: 
( <font color="Maroon" title="c5">x</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Maroon" title="c6">y</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <font color="Maroon" title="c7">z</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"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> &amp; <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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> &amp; <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="c7">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c7">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">X1</font> =  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>;<br/>
<a NAME="E2:44_2_1_1_1"/><span class="lab"><font color="Green" title="E37">B1</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E35"><span class="lab"><font color="Green" title="E32">LemConRelat01</font></span></a></span>;<br/>
<a NAME="E3:44_2_1_1_1"/>
 <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="poset_1.html#K4" title="POSET_1:func.4">ConFuncs</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="poset_1.html#E37"><span class="lab"><font color="Green" title="E34">LemConRelat03</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:44_2_1_1_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="c7">z</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="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_2_1_1_1"><span class="lab"><font color="Green" title="E36">B0</font></span></a>, <a class="txt" href="#E2:44_2_1_1_1"><span class="lab"><font color="Green" title="E37">B1</font></span></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:44_2_1_1"/>
 <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:44_2_1"/>
 <a href="poset_1.html#K5" title="POSET_1:func.5">ConRelat</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">Q</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
