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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="osalg_1.html#NM3">OrderSortedSign</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="osalg_1.html#NM5">OSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><a NAME="E1:23"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:23"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="osalg_3.html#T16" target="_self">Th16</a></i>;<br/>
<a NAME="E4:23"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <font color="Olive">b<sub>1</sub></font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<a NAME="E1:23_1"/>
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/><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</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="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of 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>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="osalg_3.html#E1:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E4:23_1_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E5:23_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font> <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>7</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:23_1_1"/><i><font color="Green">E20</font></i>: 
( the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<a NAME="E7:23_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:23"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_3.html#D14">MSUALG_3:def 14</a></i>;<br/>
<b>then </b>the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</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>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="osalg_3.html#E1:23_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D25">PBOOLE:def 25</a></i>
<br/>.= 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="osalg_3.html#E5:23_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="relat_1.html#T146">RELAT_1:146</a></i>
;<br/>
<a NAME="E9:23_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E6:23_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>hence </b><a NAME="E10:23_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> is   <a href="funct_2.html#NM1">Function</a> of the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1"/>
<font color="Maroon">c<sub>6</sub></font> is   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="msualg_3.html#K2">id</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>thus </b><a NAME="E6:23"/>
<font color="Maroon">c<sub>5</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>, <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:23"><i><font color="Green">E17</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:23"/>
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R3">is_monomorphism</a>  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="msualg_3.html#T22">MSUALG_3:22</a></i>;<br/>

<b>thus </b><a NAME="E8:23"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a> )
 <b>by </b><i><a class="txt" href="osalg_3.html#E3:23"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msualg_3.html#T4">MSUALG_3:4</a></i>;<br/>

<a NAME="E9:23"/>
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font> is <a href="osalg_1.html#V12">order-sorted</a>
 
<b>by </b><i><a class="txt" href="osalg_3.html#E1:23"><i><font color="Green">E16</font></i></a>, <a class="ref" href="osalg_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E10:23"/><b>then </b>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is   <a href="osalg_1.html#NM4">OrderSortedSet</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="osalg_1.html#T17">OSALG_1:17</a></i>;<br/>
<b>hence </b><a NAME="E11:23"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="osalg_3.html#V1" target="_self">order-sorted</a>
 <b>by </b><i><a class="ref" href="osalg_3.html#T5" target="_self">Th5</a></i>;<br/>


</div>
