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<b>reconsider </b><font color="Maroon">c<sub>1</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a>  as  <a href="relat_1.html#V2">V5</a>  <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() ;<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">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E2:6_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="equation.html#K3">TermAlg</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span>,<font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D5">EQUATION:def 5</a><br/>

<b>assume </b><a NAME="E3:6_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="equation.html#K3">TermAlg</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> as    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> ,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">F<sub>3</sub></font>() as    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> ,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() ;<br/>
<a NAME="E7:6_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:6_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E8:6_1"/><i><font color="Green">E8</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="numbers.html#K5">NAT</a> ,the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>3</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:6"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E10:6_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E11:6_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="birkhoff.html#K1" target="_self">-hash</a>  <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span>
 <b>from </b><i><a class="ref" href="birkhoff.html#S3" target="_self">BIRKHOFF:sch 3</a>(<a class="txt" href="birkhoff.html#E7:6_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="birkhoff.html#E8:6_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<a NAME="E12:6_1"/><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="birkhoff.html#K1" target="_self">-hash</a>  <a href="msualg_3.html#R1">is_homomorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E13:6_1"/>
 <a href="mssubfam.html#K2">rngs</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="extens_1.html#T14">EXTENS_1:14</a></i>;<br/>
<a NAME="E14:6_1"/><b>then </b><i><font color="Green">E11</font></i>: 
(  <a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_3.html#V1">"1-1"</a> &amp;  <a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font> is <a href="msualg_3.html#V2">"onto"</a> )
 
<b>by </b><i><a class="ref" href="extens_1.html#T13">EXTENS_1:13</a>, <a class="ref" href="msualg_9.html#T12">MSUALG_9:12</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p4">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="autalg_1.html#T13">AUTALG_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2">id</a> <span class="p2">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E14:6_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="msualg_3.html#T5">MSUALG_3:5</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_3.html#T3">MSUALG_3:3</a></i>
;<br/>
<a NAME="E16:6_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="birkhoff.html#K1" target="_self">-hash</a>  <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:6_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="birkhoff.html#E12:6_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="extens_1.html#T18">EXTENS_1:18</a></i>;<br/>
<i><font color="Green">E13</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p5">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>5</sub></font>() = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>5</sub></font>()
<b>by </b><i><a class="txt" href="birkhoff.html#E11:6_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>6</sub></font>()</span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E1:6"><i><font color="Green">E3</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>6</sub></font>()
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="msafree.html#K1">||</a> <span class="p5">(<span class="default"><a href="msafree.html#K13">FreeGen</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="msualg_3.html#K3">**</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="msafree.html#K15">Reverse</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="msualg_3.html#K4">""</a> </span>)</span></span>)</span> <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">F<sub>6</sub></font>()
<b>by </b><i><a class="txt" href="birkhoff.html#E11:6_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="msualg_3.html#T2">MSUALG_3:2</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>6</sub></font>()</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">F<sub>5</sub></font>()
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">F<sub>5</sub></font>(),<font color="Maroon">F<sub>6</sub></font>()</span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> </span>)</span>
<b>by </b><i><a class="txt" href="birkhoff.html#E16:6_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="birkhoff.html#E17:6_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>
;<br/>


</div>
