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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">U1</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">U2</font> be  <a href="msualg_1.html#V4">strict</a> <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">F</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">F</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<b>set </b><font color="Maroon">FF</font> = <font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font>;<br/>
<b>thus </b><a NAME="E2:39"/>
( <font color="Maroon">F</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> implies  <a href="msualg_3.html#K7" target="_self">Image</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 ;<br/>

<a NAME="E2:39_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">F</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E34</font></i>: 
<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>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,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> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E3:39_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 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>
 <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E5:39_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>;<br/><a NAME="E6:39_1_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 <font color="Maroon">U1</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>  )
 <b>by </b><i>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><a NAME="E7:39_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E8:39_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 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>
 <b>by </b><i><a class="txt" href="msualg_3.html#E2:39_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="msualg_3.html#E1:39_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:39_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>
<a NAME="E5:39_1"/>
<font color="Maroon">U2</font> is  <a href="msualg_1.html#V4">strict</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T6">MSUALG_2:6</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1"/>
 <a href="msualg_3.html#K7" target="_self">Image</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E1:39"><i><font color="Green">E30</font></i></a>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:39"/>
 <a href="msualg_3.html#K7" target="_self">Image</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">U2</font>
 ;<br/>

<a NAME="E4:39"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">F</font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:39"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E5:39"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:39_2"/>
<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>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="msualg_3.html#K1" target="_self">.</a> <font color="Maroon">i</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,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> ;<br/>
<a NAME="E4:39_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 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>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>;<br/>
<a NAME="E5:39_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:39_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 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>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:39"/><b>then </b>
<font color="Maroon">F</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:39"/>
<font color="Maroon">F</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E1:39"><i><font color="Green">E30</font></i></a>, </i>;<br/>


</div>
