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<b>let </b><font color="Maroon" title="c3">R1</font>, <font color="Maroon" title="c4">S1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">R</font> be  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c3">R1</font>;<br/><b>let </b><font color="Maroon" title="c6">S</font> be  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c4">S1</font>;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">R1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">S1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font>, <font color="Maroon" title="c9">p3</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_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a>,<a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a><br/>

<b>set </b><font color="Maroon" title="c10">RS</font> = <font color="Maroon" title="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>;<br/>
<b>set </b><font color="Maroon" title="c11">D</font> =  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E2:7"/><i><font color="Green" title="E4">A2</font></i>: 
( <font color="Maroon" title="c7">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> &amp; <font color="Maroon" title="c8">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> &amp; <font color="Maroon" title="c9">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcs_0.html#K1" title="PCS_0:func.1">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:7"/><i><font color="Green" title="E5">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</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="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 <b>and </b><br/><a NAME="E4:7"/><i><font color="Green" title="E6">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">p3</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="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( <font color="Maroon" title="c9">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font> or <font color="Maroon" title="c9">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font> )
 </span><b>by </b><i><a class="txt" href="pcs_0.html#E2:7"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c9">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font>
 ;<br/><div class="add"><a NAME="E2:7_1_1"/><b>then </b>
<font color="Maroon" title="c8">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">R1</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:7"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pcs_0.html#E4:7"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pcs_0.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E3:7_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</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="c5">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">p3</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="c5">R</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:7"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pcs_0.html#E3:7"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="pcs_0.html#E4:7"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pcs_0.html#E1:7_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="pcs_0.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:7_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">p3</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="c5">R</font>
 <b>by </b><i><a class="ref" href="relat_2.html#T48" title="RELAT_2:th.48">RELAT_2:48</a></i>;<br/><b>hence </b><a NAME="E5:7_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">p3</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="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c9">p3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font>
 ;<br/><div class="add"><a NAME="E2:7_1_2"/><b>then </b>
<font color="Maroon" title="c8">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font>
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:7"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pcs_0.html#E4:7"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pcs_0.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E3:7_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</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="c6">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">p2</font>,<font color="Maroon" title="c9">p3</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="c6">S</font> )
 <b>by </b><i><a class="txt" href="pcs_0.html#E1:7"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="pcs_0.html#E3:7"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="pcs_0.html#E4:7"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="pcs_0.html#E1:7_1_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="pcs_0.html#E2"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/><a NAME="E4:7_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">p3</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="c6">S</font>
 <b>by </b><i><a class="ref" href="relat_2.html#T48" title="RELAT_2:th.48">RELAT_2:48</a></i>;<br/><b>hence </b><a NAME="E5:7_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c9">p3</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="c5">R</font> <a href="pcs_0.html#K2" title="PCS_0:func.2">\/</a> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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