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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:48_1_1_1_1"/><i><font color="Green">E29</font></i>: 
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="relat_2.html#V8">transitive</a> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E2:48_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="relat_2.html#V1">reflexive</a>
 <b>by </b><i><a class="ref" href="relat_2.html#T22">RELAT_2:22</a></i>;<br/><b>hence </b><a NAME="E3:48_1_1_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>3</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:48_1_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="eqrel_1.html#T11">EQREL_1:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:48_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:48_1_1"/><b>then </b>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E3:48_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="orders_1.html#T98">ORDERS_1:98</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1_1"/>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> is <a href="partfun1.html#V1">total</a>
 <b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>assume </b><a NAME="E3:48_1_1_2"/><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:48_1_1_2_1"/>
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> is <a href="relat_2.html#V8">transitive</a> &amp; <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>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:48_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a>, <a class="ref" href="pua2mss1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E2:48_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</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> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:48_1_1_2"/>
<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>2</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:48_1_1"/><b>then </b>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R3">is_symmetric_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D3">RELAT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:48_1_1"/>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V3">symmetric</a>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:48_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="relat_2.html#D11">RELAT_2:def 11</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:48_1_1_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>assume </b><a NAME="E3:48_1_1_3"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> &amp; <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>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:48_1_1_3_1"/>
( <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> is <a href="partfun1.html#V1">total</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> is <a href="relat_2.html#V3">symmetric</a> &amp; <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> is <a href="relat_2.html#V8">transitive</a> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:48_1_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a>, <a class="ref" href="pua2mss1.html#T22" target="_self">Th22</a></i>;<br/><b>hence </b><a NAME="E2:48_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K6" target="_self">DomRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pua2mss1.html#K8" target="_self">|^</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T13">EQREL_1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:48_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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>  <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="pua2mss1.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:48_1_1"/><b>then </b>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_1"/>
 <a href="pua2mss1.html#K9" target="_self">LimDomRel</a> <font color="Maroon">c<sub>1</sub></font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="txt" href="pua2mss1.html#E3:48_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>


</div>
