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<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:25"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:25"/><i><font color="Green">E21</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:25"/><i><font color="Green">E22</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E4:25"/><i><font color="Green">E23</font></i>: 
 <a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_1.html#K12">the_charact_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#D19">MSUALG_1:def 19</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E24</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:25_1_1"/><i><font color="Green">E26</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> st the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pboole.html#T151">PBOOLE:151</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E2:25_1_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:25_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> st the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="pboole.html#NM3">Component</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="mssublat.html#E1:25_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pboole.html#T151">PBOOLE:151</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E2:25_1_1_2"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div><b>then </b>the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="msualg_1.html#K11">the_sort_of</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E2:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mssublat.html#E3:25"><i><font color="Green">E22</font></i></a>, <a class="txt" href="mssublat.html#E4:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#D17">MSUALG_1:def 17</a></i>
<br/>.= 
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E2:25_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="msualg_1.html#D17">MSUALG_1:def 17</a></i>
;<br/><b>hence </b><a NAME="E5:25_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:25_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div><b>end;</b></div>
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> = 
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="msualg_1.html#K13">1-Alg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="mssublat.html#E3:25"><i><font color="Green">E22</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
<br/>.= 
the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="mssublat.html#E2:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="mssublat.html#E4:25"><i><font color="Green">E23</font></i></a>, <a class="ref" href="msualg_1.html#D18">MSUALG_1:def 18</a></i>
;<br/>
<b>hence </b><a NAME="E7:25"/>
 <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>2</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G3">MSAlgebra</a>(# the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U5">Charact</a> of <font color="Maroon">c<sub>3</sub></font> #)
 <b>by </b><i><a class="txt" href="mssublat.html#E5:25"><i><font color="Green">E24</font></i></a></i>;<br/>


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