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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#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="relat_1.html#V2">V2</a>  <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</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><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>6</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>
 ;<br/>

<b>reconsider </b><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>6</sub></font></span>)</span> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E4:30"/>
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>6</sub></font></span>)</span> is    <a href="pboole.html#M4">ManySortedSubset</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>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_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>6</sub></font></span>)</span> <a href="pboole.html#R2">c=</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>6</sub></font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>8</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>6</sub></font></span>)</span> as  <a href="relat_1.html#V2">V2</a> <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> ;<br/>
<a NAME="E6:30"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is    <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="msafree2.html#T9">MSAFREE2:9</a></i>;<br/>
<a NAME="E7:30"/>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T22">EXTENS_1:22</a></i>;<br/>
<a NAME="E8:30"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E6:30"><i><font color="Green">E17</font></i></a>, <a class="ref" href="msafree.html#T3">MSAFREE:3</a></i>;<br/>
<a NAME="E9:30"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</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>6</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>6</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="ref" href="extens_1.html#T6">EXTENS_1:6</a></i>;<br/>
<a NAME="E2:30_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>6</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="equation.html#E1:30"><i><font color="Green">E15</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<b>hence </b><a NAME="E3:30_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R2">c=</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>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="equation.html#E2:30"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_3.html#D14">MSUALG_3:def 14</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>


</div><b>end;</b></div>
<a NAME="E10:30"/><b>then </b>
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T21">EXTENS_1:21</a></i>;<br/>
<a NAME="E11:30"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is    <a href="msualg_2.html#M1">MSSubAlgebra</a> of  <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="equation.html#E8:30"><i><font color="Green">E18</font></i></a>, <a class="ref" href="msafree.html#T3">MSAFREE:3</a></i>;<br/>
<a NAME="E12:30"/><b>then </b>
 <a href="msualg_3.html#K7">Image</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#T8">MSUALG_2:8</a></i>;<br/>
<b>hence </b><a NAME="E13:30"/>
<font color="Maroon">c<sub>6</sub></font> <a href="msualg_3.html#R2">is_epimorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:30"><i><font color="Green">E16</font></i></a>, <a class="ref" href="msualg_3.html#T19">MSUALG_3:19</a></i>;<br/>


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