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

<b>let </b><font color="Maroon">Z</font> be  <a href="unialg_2.html#V2">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon">A</font> =  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #);<br/>
<b>reconsider </b><font color="Maroon">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E2:38"/><i><font color="Green">E32</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span>,<font color="Maroon">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A1</font> being <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) st <br/>( <font color="Olive">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Olive">A1</font> );<br/>
<a NAME="E3:38"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)  ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #);<br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">x</font> as  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E2:38_1"/>
 <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_1"/><b>then </b>
 <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">B</font> is  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">B</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="msualg_2.html#K14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E5:38_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #), <a href="msualg_2.html#K14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span><b> such that </b><br/><a NAME="E5:38"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span> ;<br/>
<a NAME="E7:38"/>
<font color="Maroon">f</font> is    <a href="lattice4.html#M1">Homomorphism</a> of  <a href="unialg_2.html#K12">UnSubAlLattice</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #), <a href="msualg_2.html#K17">MSSubAlLattice</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>, <font color="Maroon">b1</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12">UnSubAlLattice</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D1">LATTICE4:def 1</a><br/>

<b>reconsider </b><font color="Maroon">a2</font> = <font color="Maroon">a1</font>, <font color="Maroon">b2</font> = <font color="Maroon">b1</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) ;<br/>
<b>reconsider </b><font color="Maroon">a3</font> = <font color="Maroon">a2</font>, <font color="Maroon">b3</font> = <font color="Maroon">b2</font> as  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">a3</font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">b3</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E5:38_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A3</font> being  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E7:38_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a3</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A3</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A4</font> being  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E9:38_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">A4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b3</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A4</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:38_2"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g4</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A3</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i/>;<br/>
<a NAME="E12:38_2"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A4</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g3</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A4</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i/>;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">b1</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10">UniAlg_join</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
<br/>.= 
 <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="unialg_2.html#K8">"\/"</a> <font color="Maroon">b3</font></span>)</span>
<b>by </b><i><a class="ref" href="mssublat.html#T1" target="_self">Th1</a>, <a class="ref" href="unialg_2.html#D16">UNIALG_2:def 16</a></i>
<br/>.= 
<font color="Maroon">g4</font> <a href="msualg_2.html#K13">"\/"</a> <font color="Maroon">g3</font>
<b>by </b><i><a class="ref" href="mssublat.html#T5" target="_self">Th5</a>, , </i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="mssublat.html#T5" target="_self">Th5</a>, , <a class="ref" href="msualg_2.html#D21">MSUALG_2:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1">LATTICES:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">a3</font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">b3</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E16:38_2"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">a3</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i/>;<br/>
<a NAME="E18:38_2"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">b3</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">g2</font> =  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">b3</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#V8">non-empty</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E21:38_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">b1</font></span>)</span> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11">UniAlg_meet</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a2</font>,<font color="Maroon">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
<br/>.= 
 <a href="msualg_1.html#K10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="unialg_2.html#K5">/\</a> <font color="Maroon">b3</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="unialg_2.html#D17">UNIALG_2:def 17</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="msualg_2.html#K11">/\</a> <font color="Maroon">g2</font>
<b>by </b><i/>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="mssublat.html#T5" target="_self">Th5</a>, , <a class="ref" href="msualg_2.html#D22">MSUALG_2:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2">LATTICES:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E23:38_2"/>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a1</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mssublat.html#E2:38"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="lattice4.html#M1">Homomorphism</a> of  <a href="unialg_2.html#K12">UnSubAlLattice</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #), <a href="msualg_2.html#K17">MSSubAlLattice</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D5">LATTICE4:def 5</a><br/>

<a NAME="E9:38"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E10:38"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D2">LATTICE4:def 2</a>,<a class="ref" href="lattice4.html#D4">LATTICE4:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:38_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:38_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:38_3_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><b>reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">x1</font>, <font color="Maroon">y2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="mssublat.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">A1</font> being  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E6:38_3_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">A2</font> being  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E8:38_3_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A2</font> )
 <b>by </b><i/>;<br/><i><font color="Green">E64</font></i>:  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A1</font> = 
 <a href="msualg_1.html#K7">MSSign</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
<b>by </b><i/>
<br/>.= 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A2</font>
<b>by </b><i/>
;<br/><b>thus </b><font color="Maroon">x1</font> = 
 <a href="msualg_1.html#K13">1-Alg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_1.html#T15">MSUALG_1:15</a></i>
<br/>.= 
<font color="Maroon">x2</font>
<b>by </b><i><a class="txt" href="mssublat.html#E2:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="mssublat.html#E3:38"><i><font color="Green">E33</font></i></a>, <a class="txt" href="mssublat.html#E5:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="mssublat.html#T6" target="_self">Th6</a>, <a class="ref" href="msualg_1.html#T15">MSUALG_1:15</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38_3"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:38"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D3">LATTICE4:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <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="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E2:38_4"/>
<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="msualg_2.html#K17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="msualg_2.html#D20">MSUALG_2:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C</font> =  <a href="msualg_2.html#K2">Constants</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">y</font> <b>by </b><i><a class="ref" href="msualg_2.html#T11">MSUALG_2:11</a></i>;<br/>
<a NAME="E5:38_4"/>
( <font color="Maroon">C</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">y</font> &amp; <font color="Maroon">C</font> is <a href="relat_1.html#V2">non-empty</a> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:38_4"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">y</font> is <a href="relat_1.html#V2">non-empty</a>
 
<b>by </b><i><a class="ref" href="pboole.html#T143">PBOOLE:143</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="msualg_1.html#D8">MSUALG_1:def 8</a></i>;<br/>
<a NAME="E8:38_4"/>
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">y</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="msualg_1.html#K13">1-Alg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10">MSAlg</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)</span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:38_4"/><b>then </b>
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">y</font> is    <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T15">MSUALG_1:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> =  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="unialg_2.html#K9">Sub</a> <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E11:38_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="mssublat.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being  <a href="unialg_1.html#V3">strict</a>  <a href="unialg_2.html#M3">SubAlgebra</a> of  <a href="unialg_1.html#G1">UAStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">Z</font> #)<b> such that </b><br/><a NAME="E13:38_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K10">MSAlg</a> <font color="Maroon">A1</font> )
 <b>by </b><i/>;<br/>
<a NAME="E14:38_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E15:38_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="mssublat.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E12:38"/>
verum
 ;<br/>


</div>
