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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">V'</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">C'</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">K</font> =  <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font>;<br/>
<b>assume </b><a NAME="E1:21"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V'</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C'</font> )
 ;<br/>

<a NAME="E2:21"/><i><font color="Green">E31</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:21"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:21"/><b>then </b><i><font color="Green">E46</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 ;<br/><a NAME="E2:21_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E5:21_1"/>
<font color="Maroon">a'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b'</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E6:21_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a'</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b'</font> &amp; <font color="Olive">y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">x</font> )
 <b>by </b><i/>;<br/><a NAME="E7:21_1"/><b>then </b>
<font color="Maroon">a1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E8:21_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:21"/><b>then </b>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">K</font> =  <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> as    <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font> <b>by </b><i>, <a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E8:21"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">K</font>
 ;<br/><a NAME="E2:21_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:21_2_1"/>
<font color="Maroon">p</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E5:21_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Olive">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E7:21_2_1"/>
<font color="Maroon">p'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:21_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E9:21_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>
 ;<br/><a NAME="E2:21_2_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font>, <font color="Maroon">q</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:21_2_2"/>
<font color="Maroon">p</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E5:21_2_2"/><b>then </b><i><font color="Green">E96</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">p</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">q</font> &amp; <font color="Olive">b</font> <a href="tarski.html#R1">c=</a> <font color="Olive">a</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">x</font>, <font color="Maroon">q'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:21_2_2"/>
<font color="Maroon">p'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">q'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:21_2_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="wellord1.html#K2">|_2</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">K</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D3">RELAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:21"/>
 <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of  <a href="heyting3.html#K1" target="_self">SubstPoset</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#D14">YELLOW_0:def 14</a></i>;<br/>


</div>
