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<b>let </b><font color="Maroon">S</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">U0</font> be  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4" target="_self">Equations</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/><b>let </b><font color="Maroon">U2</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">U0</font>;<br/>









<b>assume </b><a NAME="E1:47"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">U0</font> <a href="equation.html#R1" target="_self">|=</a> <font color="Maroon">e</font>
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">U2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5" target="_self">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E2:47"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">h</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>,<font color="Maroon">U2</font>
 ;<br/>

<a NAME="E3:47"/><i><font color="Green">E39</font></i>: 
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 ;<br/>

<b>hence </b><a NAME="E2:47_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> is   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">U0</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">h</font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">U0</font> <b>by </b><i>, <a class="ref" href="extens_1.html#T9">EXTENS_1:9</a></i>;<br/>
<a NAME="E6:47"/>
<font color="Maroon">f1</font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3" target="_self">TermAlg</a> <font color="Maroon">S</font>,<font color="Maroon">U0</font>
 
<b>by </b><i>, <a class="ref" href="msualg_9.html#T16">MSUALG_9:16</a></i>;<br/>
<b>hence </b><a NAME="E7:47"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i>, </i>;<br/>


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