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

<b>let </b><font color="Maroon" title="c2">A</font> be  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a1</font>, <font color="Olive" title="b2">b1</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 <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">b2</font> being <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b3">a2</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="Olive" title="b1">a1</font> &amp; <font color="Olive" title="b4">b2</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="Olive" title="b2">b1</font> holds <br/> <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a2</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Olive" title="b4">b2</font></span><br/><b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">b1</font> be  <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 <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a2</font>, <font color="Olive" title="b2">b2</font> being <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> <font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">a2</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="c3">a1</font> &amp; <font color="Olive" title="b2">b2</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="c4">b1</font> holds <br/> <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a2</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Olive" title="b2">b2</font></span><br/><b>let </b><font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">b2</font> be  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a2</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="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</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="c4">b1</font> implies  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c6">b2</font> )</span><br/>





<b>assume </b><a NAME="E1:30"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c5">a2</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="c3">a1</font> &amp; <font color="Maroon" title="c6">b2</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="c4">b1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c6">b2</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">MSA</font> =  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> as    <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> <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/>
<a NAME="E3:30"/>
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> <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="c2">A</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="c8">MSA</font> = <font color="Maroon" title="c7">MSA</font> as  <a href="msualg_1.html#V3" title="MSUALG_1:attr.3">strict</a> <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> <font color="Maroon" title="c2">A</font> ;<br/>
<a NAME="E5:30"/>
for <font color="Olive" title="b1">B</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">A</font></span>)</span>  st <font color="Olive" title="b1">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font> holds <br/><font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Olive" title="b1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">B</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <font color="Maroon" title="c2">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font> implies <font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c9">B</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c9">B</font></span><br/>

<b>set </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c8">MSA</font>;<br/>
<a NAME="E2:30_1"/>
<font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/>
<a NAME="E3:30_1"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D14" title="MSUALG_1:def.14">MSUALG_1:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8" title="PBOOLE:func.8">*--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="E5:30_1"/><i><font color="Green" title="E28">A4</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">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> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c1">z</font></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/>
<a NAME="E6:30_1"/><i><font color="Green" title="E29">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E7:30_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">b1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">K</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">b1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E6:30_1"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">M1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c8">MSA</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</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="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">x</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<a NAME="E11:30_1"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c9">B</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c8">MSA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c13">M1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">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_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span> or <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c15">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_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<b>then </b><i><font color="Green" title="E32">A8</font></i>: <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="mssublat.html#T28" target="_self" title="MSSUBLAT:th.28">Th28</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">b1</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<i><font color="Green" title="E33">A9</font></i>: <font color="Maroon" title="c13">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E3:30_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E1:30_1_1_1"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/>
<a NAME="E4:30_1_1_1"/>
<font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c12">K</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D14" title="UNIALG_2:def.14">UNIALG_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1_1"/>
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">a2</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">M1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="#E2:30_1_1_1"><i><font color="Green" title="E32">A8</font></i></a>, <a class="txt" href="#E3:30_1_1_1"><i><font color="Green" title="E33">A9</font></i></a>, <a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_1_1"/>
<font color="Maroon" title="c9">B</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c8">MSA</font>
 <b>by </b><i><a class="txt" href="#E1:30_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:30_1"/>
for <font color="Olive" title="b1">U1</font> being   <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> <font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c9">B</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Olive" title="b1">U1</font> holds <br/><font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">U1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">U1</font> be    <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> <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c15">U1</font> implies <font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1_2"/><i><font color="Green" title="E31">A10</font></i>: 
<font color="Maroon" title="c9">B</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c15">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_1"/>
( <font color="Maroon" title="c15">U1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> or  not <font color="Maroon" title="c15">U1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_1_1"/>
<font color="Maroon" title="c15">U1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">U11</font> = <font color="Maroon" title="c15">U1</font> as  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">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> <font color="Maroon" title="c2">A</font> ;<br/><a NAME="E3:30_1_2_1_1"/><i><font color="Green" title="E32">A11</font></i>: 
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</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> <font color="Maroon" title="c2">A</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="E4:30_1_2_1_1"/><b>then </b>
 <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>;<br/><a NAME="E5:30_1_2_1_1"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
 <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</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="c16">U11</font></span>)</span>
 <b>by </b><i><a class="ref" href="mssublat.html#T7" target="_self" title="MSSUBLAT:th.7">Th7</a></i>;<br/><a NAME="E6:30_1_2_1_1"/>
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">U11</font>
 <b>by </b><i><a class="txt" href="#E1:30_1_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:30_1_2_1_1"/><b>then </b><i><font color="Green" title="E34">A13</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">U11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><a NAME="E8:30_1_2_1_1"/><i><font color="Green" title="E35">A14</font></i>: 
 <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">U11</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c16">U11</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</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="c16">U11</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="mssublat.html#T26" target="_self" title="MSSUBLAT:th.26">Th26</a></i>;<br/><a NAME="E9:30_1_2_1_1"/>
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</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="c16">U11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/><a NAME="E10:30_1_2_1_1"/><b>then </b><i><font color="Green" title="E36">A15</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">U11</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E8:30_1_2_1_1"><i><font color="Green" title="E35">A14</font></i></a>, <a class="ref" href="msualg_1.html#D14" title="MSUALG_1:def.14">MSUALG_1:def 14</a></i>;<br/><i><font color="Green" title="E37">A16</font></i>: <font color="Maroon" title="c9">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="#E1:30_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="mssublat.html#T28" target="_self" title="MSSUBLAT:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c12">K</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c17">A1</font> =  <a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</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 <font color="Maroon" title="c2">A</font> <b>by </b><i><a class="txt" href="#E3:30_1_2_1_1"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="msualg_1.html#T15" title="MSUALG_1:th.15">MSUALG_1:15</a></i>;<br/><a NAME="E13:30_1_2_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">b1</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 <font color="Maroon" title="c17">A1</font>
 <b>by </b><i><a class="txt" href="#E7:30_1_2_1_1"><i><font color="Green" title="E34">A13</font></i></a>, <a class="txt" href="#E10:30_1_2_1_1"><i><font color="Green" title="E36">A15</font></i></a>, <a class="txt" href="#E11:30_1_2_1_1"><i><font color="Green" title="E37">A16</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>;<br/><a NAME="E14:30_1_2_1_1"/><b>then </b>
 <a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <font color="Maroon" title="c12">K</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> <font color="Maroon" title="c16">U11</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>;<br/><a NAME="E15:30_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</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> <font color="Maroon" title="c16">U11</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D14" title="UNIALG_2:def.14">UNIALG_2:def 14</a></i>;<br/><a NAME="E16:30_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">MSA</font> is    <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> <span class="p1">(<span class="default"><a href="msualg_1.html#K13" title="MSUALG_1:func.13">1-Alg</a> <font color="Maroon" title="c16">U11</font></span>)</span>
 <b>by </b><i><a class="ref" href="mssublat.html#T12" target="_self" title="MSSUBLAT:th.12">Th12</a></i>;<br/><a NAME="E17:30_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c16">U11</font>,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c16">U11</font> #)
 <b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E5:30_1_2_1_1"><i><font color="Green" title="E33">A12</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="E18:30_1_2_1_1"/>
<font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font>
 <b>by </b><i><a class="ref" href="mssublat.html#T21" target="_self" title="MSSUBLAT:th.21">Th21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_1_2"/>
not <font color="Maroon" title="c15">U1</font> is <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font></span><br/><div class="add"><a NAME="E2:30_1_2_1_2"/><b>then </b>
not the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c15">U1</font> is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>
 <b>by </b><i><a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:30_1_2_1_2"/><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c16">i</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_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>and </b><br/><a NAME="E5:30_1_2_1_2"/><i><font color="Green" title="E33">A18</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c15">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E6:30_1_2_1_2"/>
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c15">U1</font>
 <b>by </b><i><a class="txt" href="#E1:30_1_2"><i><font color="Green" title="E31">A10</font></i></a>, <a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/><a NAME="E7:30_1_2_1_2"/><b>then </b><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c15">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c17">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">a1</font>;<br/><a NAME="E9:30_1_2_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">0a1</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">a1</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</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="msualg_1.html#K7" title="MSUALG_1:func.7">MSSign</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E11:30_1_2_1_2"/>
<font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K8" title="MSUALG_1:func.8">MSSorts</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K9" title="MSUALG_1:func.9">MSCharact</a> <font color="Maroon" title="c3">a1</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="msualg_1.html#D16" title="MSUALG_1:def.16">MSUALG_1:def 16</a></i>;<br/><a NAME="E12:30_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">B</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">0a1</font> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font>
 <b>by </b><i><a class="txt" href="#E1:30_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="msualg_1.html#D14" title="MSUALG_1:def.14">MSUALG_1:def 14</a></i>;<br/><a NAME="E13:30_1_2_1_2"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">0a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D7" title="PBOOLE:def.7">PBOOLE:def 7</a></i>;<br/><a NAME="E14:30_1_2_1_2"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E15:30_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c18">0a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</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 <font color="Maroon" title="c3">a1</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E16:30_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <b>by </b><i><a class="txt" href="#E13:30_1_2_1_2"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E17:30_1_2_1_2"/>
<font color="Maroon" title="c8">MSA</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c15">U1</font>
 <b>by </b><i><a class="txt" href="#E5:30_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="#E4:30_1_2_1_2"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="#E5:30_1_2_1_2"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="#E7:30_1_2_1_2"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E13:30_1"/>
<font color="Maroon" title="c8">MSA</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_2.html#K12" title="MSUALG_2:func.12">GenMSAlg</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="#E11:30_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="msualg_2.html#D18" title="MSUALG_2:def.18">MSUALG_2:def 18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
 <a href="msualg_1.html#K10" title="MSUALG_1:func.10">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a1</font> <a href="unialg_2.html#K8" title="UNIALG_2:func.8">"\/"</a> <font color="Maroon" title="c4">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a2</font> <a href="msualg_2.html#K13" title="MSUALG_2:func.13">"\/"</a> <font color="Maroon" title="c6">b2</font>
 <b>by </b><i><a class="ref" href="msualg_2.html#D19" title="MSUALG_2:def.19">MSUALG_2:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
