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

<b>let </b><font color="Maroon" title="c1">Z</font> be  <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/>

<b>set </b><font color="Maroon" title="c2">A</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #);<br/>
<b>reconsider </b><font color="Maroon" title="c3">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7" title="MSUALG_1:th.7">MSUALG_1:7</a></i>;<br/>
<a NAME="E2:37"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G1" title="MSUALG_1:aggr.1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span>,<font color="Maroon" title="c3">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6" title="MSUALG_1:func.6">signature</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16" title="MSUALG_1:th.16">MSUALG_1:16</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A1</font> being <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) st <br/>( <font color="Olive" title="b1">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Olive" title="b1">A1</font> );<br/>
<a NAME="E3:37"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)  ex <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #);<br/>

<b>reconsider </b><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c4">x</font> as  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E2:37_1"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c5">B</font>
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<a NAME="E3:37_1"/><b>then </b>
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c5">B</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">y</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c5">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">y</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">B</font>
;<br/>

<b>thus </b><a NAME="E5:37_1"/>
( <font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c5">B</font> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #), <a href="msualg_2.html#K14" title="MSUALG_2:func.14">MSSub</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span><b> such that </b><br/><a NAME="E5:37"/><i><font color="Green" title="E28">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="mssublat.html#E3:37"><i><font color="Green" title="E27">A2</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span> ;<br/>
<a NAME="E7:37"/>
<font color="Maroon" title="c5">f</font> is    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #), <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">b1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D1" title="LATTICE4:def.1">LATTICE4:def 1</a><br/>

<b>reconsider </b><font color="Maroon" title="c8">a2</font> = <font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c9">b2</font> = <font color="Maroon" title="c7">b1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">a3</font> = <font color="Maroon" title="c8">a2</font>, <font color="Maroon" title="c11">b3</font> = <font color="Maroon" title="c9">b2</font> as  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c10">a3</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c11">b3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">A1</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E5:37_2"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c13">A1</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">A3</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E7:37_2"/><i><font color="Green" title="E30">A5</font></i>: 
( <font color="Maroon" title="c14">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a3</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c14">A3</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">A4</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E9:37_2"/><i><font color="Green" title="E31">A6</font></i>: 
( <font color="Maroon" title="c15">A4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b3</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c15">A4</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<a NAME="E10:37_2"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c14">A3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">g4</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c14">A3</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<a NAME="E12:37_2"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c15">A4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">g3</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c15">A4</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<i><font color="Green" title="E32">A7</font></i>: <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c7">b1</font></span>)</span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K10" title="UNIALG_2:func.10">UniAlg_join</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
<br/>.= 
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a3</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c11">b3</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E5:37_2"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="unialg_2.html#D16" title="UNIALG_2:def.16">UNIALG_2:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c16">g4</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c17">g3</font>
<b>by </b><i><a class="txt" href="mssublat.html#E7:37_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="mssublat.html#E9:37_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="mssublat.html#T30" target="_self" title="MSSUBLAT:th.30">Th30</a></i>
<br/>.= 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E7:37_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="mssublat.html#E9:37_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="msualg_2.html#D21" title="MSUALG_2:def.21">MSUALG_2:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c10">a3</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c11">b3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E16:37_2"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c10">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">g1</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c10">a3</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<a NAME="E18:37_2"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c11">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">g2</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c11">b3</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">A1</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E21:37_2"/><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c21">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">m</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c21">A1</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c7">b1</font></span>)</span> = 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_2.html#K11" title="UNIALG_2:func.11">UniAlg_meet</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c8">a2</font>,<font color="Maroon" title="c9">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
<br/>.= 
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">a3</font> <a href="unialg_2.html#K5" title="UNIALG_2:func.5">/\</a> <font color="Maroon" title="c11">b3</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E21:37_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="unialg_2.html#D17" title="UNIALG_2:def.17">UNIALG_2:def 17</a></i>
<br/>.= 
<font color="Maroon" title="c19">g1</font> <a href="msualg_2.html#K11" title="MSUALG_2:func.11">/\</a> <font color="Maroon" title="c20">g2</font>
<b>by </b><i><a class="ref" href="mssublat.html#T31" target="_self" title="MSSUBLAT:th.31">Th31</a></i>
<br/>.= 
the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E7:37_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="mssublat.html#E9:37_2"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="msualg_2.html#D22" title="MSUALG_2:def.22">MSUALG_2:def 22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D2" title="LATTICES:def.2">LATTICES:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E23:37_2"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a1</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="mssublat.html#E14:37_2"><i><font color="Green" title="E32">A7</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as    <a href="lattice4.html#M1" title="LATTICE4:mode.1">Homomorphism</a> of  <a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #), <a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c6">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D5" title="LATTICE4:def.5">LATTICE4:def 5</a><br/>

<a NAME="E9:37"/><i><font color="Green" title="E29">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E10:37"/>
<font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D2" title="LATTICE4:def.2">LATTICE4:def 2</a>,<a class="ref" href="lattice4.html#D4" title="LATTICE4:def.4">LATTICE4:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:37_3_1"/><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <b>and </b><br/><a NAME="E2:37_3_1"/><i><font color="Green" title="E31">A11</font></i>: 
<font color="Maroon" title="c8">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 <b>and </b><br/><a NAME="E3:37_3_1"/><i><font color="Green" title="E32">A12</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x2</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">y1</font> = <font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c10">y2</font> = <font color="Maroon" title="c8">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="txt" href="mssublat.html#E1:37_3_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="txt" href="mssublat.html#E2:37_3_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">A1</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E6:37_3_1"/><i><font color="Green" title="E33">A13</font></i>: 
( <font color="Maroon" title="c11">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c11">A1</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c12">A2</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E8:37_3_1"/><i><font color="Green" title="E34">A14</font></i>: 
( <font color="Maroon" title="c12">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c12">A2</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/><i><font color="Green" title="E35">A15</font></i>:  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c11">A1</font> = 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>
<br/>.= 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c12">A2</font>
<b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>
;<br/><b>thus </b><font color="Maroon" title="c7">x1</font> = 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c11">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E6:37_3_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>
<br/>.= 
<font color="Maroon" title="c8">x2</font>
<b>by </b><i><a class="txt" href="mssublat.html#E3:37_3_1"><i><font color="Green" title="E32">A12</font></i></a>, <a class="txt" href="mssublat.html#E6:37_3_1"><i><font color="Green" title="E33">A13</font></i></a>, <a class="txt" href="mssublat.html#E8:37_3_1"><i><font color="Green" title="E34">A14</font></i></a>, <a class="txt" href="mssublat.html#E9:37_3_1"><i><font color="Green" title="E35">A15</font></i></a>, <a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:37_3"/>
<font color="Maroon" title="c6">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


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

<b>thus </b><a NAME="E11:37"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="lattice4.html#D3" title="LATTICE4:def.3">LATTICE4:def 3</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:37_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:37_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K17" title="MSUALG_2:func.17">MSSubAlLattice</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">x</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="msualg_2.html#D20" title="MSUALG_2:def.20">MSUALG_2:def 20</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">C</font> =  <a href="msualg_2.html#K2" title="MSUALG_2:func.2">Constants</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c8">y</font> <b>by </b><i><a class="ref" href="msualg_2.html#T11" title="MSUALG_2:th.11">MSUALG_2:11</a></i>;<br/>
<a NAME="E5:37_4"/>
( <font color="Maroon" title="c9">C</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c9">C</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> )
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E6:37_4"/><b>then </b>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon" title="c8">y</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 
<b>by </b><i><a class="ref" href="pboole.html#T143" title="PBOOLE:th.143">PBOOLE:143</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>;<br/>
<a NAME="E8:37_4"/>
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c10">y</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)</span>)</span>
 
<b>by </b><i><a class="ref" href="mssublat.html#T20" target="_self" title="MSSUBLAT:th.20">Th20</a></i>;<br/>
<a NAME="E9:37_4"/><b>then </b>
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c10">y</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">y1</font> =  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="unialg_2.html#K9" title="UNIALG_2:func.9">Sub</a> <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #) <b>by </b><i><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></i>;<br/>
<a NAME="E11:37_4"/><i><font color="Green" title="E30">A16</font></i>: 
<font color="Maroon" title="c11">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E9:37"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">A1</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Z</font>,the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">Z</font> #)<b> such that </b><br/><a NAME="E13:37_4"/><i><font color="Green" title="E31">A17</font></i>: 
( <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c12">A1</font> )
 <b>by </b><i><a class="txt" href="mssublat.html#E5:37"><i><font color="Green" title="E28">A3</font></i></a></i>;<br/>
<a NAME="E14:37_4"/>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="mssublat.html#E2:37"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="mssublat.html#E13:37_4"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="mssublat.html#T26" target="_self" title="MSSUBLAT:th.26">Th26</a></i>;<br/>
<b>hence </b><a NAME="E15:37_4"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="mssublat.html#E11:37_4"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E12:37"/>
verum
 ;<br/>


</div>
