<?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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font>;<br/>
<a NAME="E1:16"/><i><font color="Green">E30</font></i>: 
 <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<a NAME="E2:16"/><i><font color="Green">E31</font></i>: 
 <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<b>assume </b><a NAME="E3:16"/><i><font color="Green">E46</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="E4:16"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <a href="tarski.html#R1">c=</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="E5:16"/><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> as   <a href="binop_1.html#NM2">BinOp</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">B2</font> = the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E7:16"/>
 <a href="relat_1.html#K1">dom</a> the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:16"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:16_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">y1</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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">y2</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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:16_1"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1'</font> = <font color="Maroon">y1</font>, <font color="Maroon">y2'</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">y11</font> = <font color="Maroon">y1</font>, <font color="Maroon">y22</font> = <font color="Maroon">y2</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <b>by </b><i>, <a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/>
<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">y1'</font> <a href="substlat.html#K2">\/</a> <font color="Maroon">y2'</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">y11</font> <a href="substlat.html#K2">\/</a> <font color="Maroon">y22</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E10:16_1"/>
<font color="Maroon">x</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="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B2</font> = the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> as   <a href="binop_1.html#NM2">BinOp</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<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="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> <b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>;<br/><a NAME="E2:16_2"/>
( <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> )
 ;<br/><b>then 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="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font> <b>by </b><i/>;<br/><b>thus </b><font color="Maroon">B1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">a'</font> <a href="substlat.html#K2">\/</a> <font color="Maroon">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>
<br/>.= 
 <a href="substlat.html#K3">mi</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="substlat.html#K2">\/</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="substlat.html#D4">SUBSTLAT:def 4</a></i>
<br/>.= 
<font color="Maroon">B2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">B2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:16"/>
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V'</font>,<font color="Maroon">C'</font></span>)</span> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5">SubstLatt</a> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
