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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>, <font color="Maroon">U2</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">H</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">H1</font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">U2</font>,<font color="Maroon">U1</font>;<br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">H1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">H</font> <a href="msualg_3.html#K4" target="_self">""</a>  )
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">H</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/>;<br/>
<a NAME="E3:31"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">H</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font> &amp; <font color="Maroon">H</font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:31"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">H</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H1</font> &amp; <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Olive">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><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>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E51</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>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">H</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="E5:31_1"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, <a class="txt" href="msualg_3.html#E2:31"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E6:31_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#T62">FUNCT_1:62</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1"/>
<font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31"/>
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">H1</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">U1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_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="E1:31_2"/><i><font color="Green">E55</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/>

<a NAME="E2:31_2"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">H</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="E4:31_2"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, , </i>;<br/>
<a NAME="E5:31_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T55">FUNCT_1:55</a></i>;<br/>
<a NAME="E6:31_2"/>
( 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="E7:31_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> </span>)</span> <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="txt" href="msualg_3.html#E2:31"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E8:31_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">H1</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">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="msualg_3.html#D1" target="_self">Def1</a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:31"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">H1</font> is <a href="msualg_3.html#V2" target="_self">"onto"</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:31"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">H1</font> is <a href="msualg_3.html#V1" target="_self">"1-1"</a>
 
<b>by </b><i><a class="txt" href="msualg_3.html#E1:31"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E9:31"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">H1</font> <a href="msualg_3.html#R1" target="_self">is_homomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:31"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">H1</font> <a href="msualg_3.html#R2" target="_self">is_epimorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:31"/>
<font color="Maroon">H1</font> <a href="msualg_3.html#R3" target="_self">is_monomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="txt" href="msualg_3.html#E2:31"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:31"/>
<font color="Maroon">H1</font> <a href="msualg_3.html#R4" target="_self">is_isomorphism</a> <font color="Maroon">U2</font>,<font color="Maroon">U1</font>
 <b>by </b><i><a class="txt" href="msualg_3.html#E3:31"><i><font color="Green">E34</font></i></a>, </i>;<br/>


</div>
