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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#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:34"/><i><font color="Green">E29</font></i>: 
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E2:34"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon">c<sub>3</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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:34_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:34"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</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> holds <br/> <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 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="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:34_2"/>
<font color="Maroon">c<sub>3</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="E2:34_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</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>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E3:34_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 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>3</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:34"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="msualg_3.html#V2" target="_self">"onto"</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:34"/>
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E2:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="msualg_3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E6:34"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:34"><i><font color="Green">E29</font></i></a>, <a class="ref" href="msualg_3.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E7:34"/><i><font color="Green">E33</font></i>: 
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:34"><i><font color="Green">E29</font></i></a>, <a class="txt" href="msualg_3.html#E4:34"><i><font color="Green">E31</font></i></a>, <a class="ref" href="msualg_3.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E8:34"/>
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E6:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self">Def12</a></i>;<br/>

<b>take </b>
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D13" target="_self">MSUALG_3:def 13</a><br/>

<b>thus </b><a NAME="E9:34"/>
 <a href="msualg_3.html#K2" target="_self">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E6:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="msualg_3.html#E7:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="msualg_3.html#D12" target="_self">Def12</a></i>;<br/>


</div>
