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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">U1</font>, <font color="Olive" title="b3">U2</font> being   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">U0</font>  st the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b2">U1</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="Olive" title="b3">U2</font> holds <br/><font color="Olive" title="b2">U1</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b3">U2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U0</font> be    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being   <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font>  st the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">U1</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="Olive" title="b2">U2</font> holds <br/><font color="Olive" title="b1">U1</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b2">U2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c4">U2</font> be    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c2">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</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="c4">U2</font> implies <font color="Maroon" title="c3">U1</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c4">U2</font> )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">B1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font>, <font color="Maroon" title="c6">B2</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c4">U2</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c2">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:28"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</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="c4">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">U1</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Maroon" title="c4">U2</font></span><br/>

<span class="kw">hence </span><a NAME="E3:28"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c3">U1</font> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c4">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">MSUALG_2:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font> being  <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c4">U2</font>  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="c3">U1</font> holds <br/>( <font color="Olive" title="b1">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Olive" title="b1">B</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">B</font> be   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <font color="Maroon" title="c4">U2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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="c3">U1</font> implies ( <font color="Maroon" title="c7">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font> ) )</span><br/>

<a NAME="E4:28"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c5">B1</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">O</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">P</font> =  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font>;<br/>
<a NAME="E5:28"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E6:28"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c6">B2</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<a NAME="E7:28"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c4">U2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c6">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D10" target="_self" title="MSUALG_2:def.10">Def10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E8:28"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<font color="Maroon" title="c7">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="c3">U1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>  &amp; the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font> )</span><br/>

<a NAME="E9:28"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c7">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Olive" title="b1">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">o</font></span><br/>

<a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">Def7</a></span>;<br/>
<a NAME="E2:28_1"/><span class="lab"><font color="Green" title="E25">A9</font></span>: 
<font color="Maroon" title="c6">B2</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">Def7</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A10</font></span>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c4">U2</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<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="c10">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c10">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c6">B2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" target="_self" title="MSUALG_2:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="msualg_2.html#D8" target="_self" title="MSUALG_2:def.8">Def8</a></span>
;<br/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c3">U1</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<font color="Maroon" title="c5">B1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c10">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c5">B1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" target="_self" title="MSUALG_2:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="msualg_2.html#D8" target="_self" title="MSUALG_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="msualg_2.html#T3" target="_self" title="MSUALG_2:th.3">Th3</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c4">U2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></span>
;<br/>
<a NAME="E5:28_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c4">U2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c3">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E6:28_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<font color="Maroon" title="c4">U2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:28_1"/>
<font color="Maroon" title="c7">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c10">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="msualg_2.html#D6" target="_self" title="MSUALG_2:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:28"/>
<font color="Maroon" title="c7">B</font> is  <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font></span><br/>

<a NAME="E11:28"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> holds <br/>the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> implies the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:28_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">o</font> = <font color="Maroon" title="c10">x</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:28_2"/><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c5">B1</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c11">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">Def7</a></span>;<br/>
<a NAME="E4:28_2"/><span class="lab"><font color="Green" title="E25">A12</font></span>: 
<font color="Maroon" title="c6">B2</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c11">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="msualg_2.html#D7" target="_self" title="MSUALG_2:def.7">Def7</a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A13</font></span>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U2</font> = 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c2">U0</font>,<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="c11">o</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c11">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c6">B2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" target="_self" title="MSUALG_2:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a>, <a class="ref" href="msualg_2.html#D8" target="_self" title="MSUALG_2:def.8">Def8</a></span>
;<br/>
<span class="kw">thus </span>the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<font color="Maroon" title="c11">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c5">B1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="msualg_2.html#D9" target="_self" title="MSUALG_2:def.9">Def9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="msualg_2.html#D8" target="_self" title="MSUALG_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c2">U0</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">B2</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="msualg_2.html#T3" target="_self" title="MSUALG_2:th.3">Th3</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c11">o</font>,<font color="Maroon" title="c4">U2</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">B1</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_2"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></span>
<br/>.= 
<font color="Maroon" title="c11">o</font> <a href="msualg_2.html#K3" title="MSUALG_2:func.3">/.</a> <font color="Maroon" title="c7">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="#E9:28"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="msualg_2.html#D8" target="_self" title="MSUALG_2:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" target="_self" title="MSUALG_2:def.9">Def9</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:28"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c3">U1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_2.html#K4" title="MSUALG_2:func.4">Opers</a> <font color="Maroon" title="c4">U2</font>,<font color="Maroon" title="c7">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
