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<b>let </b><font color="Maroon">MS</font> be  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> non <a href="msualg_1.html#V2">void</a> <a href="msualg_1.html#V6">segmental</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">A</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">MS</font>;<br/><b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font>;<br/>





<a NAME="E1:18"/><i><font color="Green">E32</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font> &amp; ( for <font color="Olive">T</font> being  <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">A</font>  st <font color="Olive">T</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> holds <br/>( <font color="Olive">T</font> is <a href="msualg_2.html#V3">opers_closed</a> &amp; the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">B</font> <a href="pboole.html#R6">=</a>  <a href="msualg_2.html#K4">Opers</a> <font color="Maroon">A</font>,<font color="Olive">T</font> ) ) )
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E2:18"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:18"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E49</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">A</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E5:18"/>
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1">(<span class="default"><a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">B</font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">B</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">B</font></span>)</span> as   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <b>by </b><i><a class="ref" href="msualg_1.html#D17">MSUALG_1:def 17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">A</font></span>)</span> as   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="msualg_1.html#D17">MSUALG_1:def 17</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E50</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="pboole.html#R2">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E9:18"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E10:18"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon">cr</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:18"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">cr</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">cr</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:18"/>
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E14:18"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>then consider </b><font color="Maroon">dr</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E16:18"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">dr</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">MS</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">dr</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E17:18"/>
<font color="Maroon">cr</font> <a href="hidden.html#R1">=</a> <font color="Maroon">dr</font>
 
<b>by </b><i>, , <a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E18:18"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">B</font></span>)</span> is   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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