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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> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="pboole.html#K10">|]</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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="E1:41_1"/>
<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/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><i><font color="Green">E16</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">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>5</sub></font></span>)</span>,<span class="p2">(<span class="default">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>5</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="txt" href="msualg_9.html#E1:41"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#D21">PBOOLE:def 21</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1">nabla</a> <span class="p1">(<span class="default">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>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="eqrel_1.html#D1">EQREL_1:def 1</a></i>
;<br/><b>thus </b>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msualg_4.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_4.html#D8">MSUALG_4:def 8</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">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>5</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="msualg_9.html#E3:41_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_9.html#T5" target="_self">Th5</a></i>
<br/>.= 
<span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="pzfmisc1.html#K1">}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="pzfmisc1.html#D1">PZFMISC1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:41"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1"><a href="pzfmisc1.html#K1">{</a><span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="pzfmisc1.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


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