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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="msafree.html#V2">free</a> <a href="msualg_6.html#V1">feasible</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="msafree.html#V1">free</a>  <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>4</sub></font> be   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E19</font></i>: 
 <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> is <a href="msualg_6.html#V1">feasible</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="msualg_2.html#NM1">MSSubset</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D18">MSUALG_2:def 18</a></i>;<br/>
<a NAME="E4:35"/>
<font color="Maroon">c<sub>5</sub></font> is    <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="pboole.html#R6">=</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="extens_1.html#T22">EXTENS_1:22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D4">MSAFREE:def 4</a><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font> ;<br/>
<a NAME="E6:35"/>
<font color="Maroon">c<sub>4</sub></font> is    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R2">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E1:35"><i><font color="Green">E18</font></i></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>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D6">MSAFREE:def 6</a><br/>

<a NAME="E8:35"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="msafree.html#V1">free</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree.html#D5">MSAFREE:def 5</a><br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>6</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E2:35_3"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="msafree.html#K1">||</a> <font color="Maroon">c<sub>7</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="equation.html#T7" target="_self">Th7</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:35_3"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:35_3"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="msafree.html#K1">||</a> <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="msafree.html#D5">MSAFREE:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> as   <a href="msualg_2.html#NM1">MSSubset</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="msualg_2.html#D10">MSUALG_2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="msafree.html#K1">||</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E8:35_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msualg_2.html#K12">GenMSAlg</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="equation.html#E2:35"><i><font color="Green">E19</font></i></a>, <a class="txt" href="equation.html#E4:35_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="equation.html#T24" target="_self">Th24</a></i>;<br/>

<b>thus </b><font color="Maroon">c<sub>9</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="msafree.html#K1">||</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="equation.html#E2:35_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="equation.html#E5:35_3"><i><font color="Green">E22</font></i></a>, <a class="ref" href="equation.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font> <a href="msafree.html#K1">||</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="equation.html#T6" target="_self">Th6</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:35"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="msafree.html#V1">free</a>
 ;<br/>


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