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<div class="add">

<b>set </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> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default">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> </span>)</span></span>)</span> as    <a href="msafree.html#M1">GeneratorSet</a> of  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default">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> </span>)</span> <b>by </b><i><a class="ref" href="msafree2.html#T9">MSAFREE2:9</a></i>;<br/>
<a NAME="E2:8_1"/><i><font color="Green">E6</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>3</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>4</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 <font color="Olive">b<sub>4</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>4</sub></font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:8"><i><font color="Green">E3</font></i></a></i>;<br/>
<a NAME="E3:8_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>2</sub></font>()]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E2:8"><i><font color="Green">E4</font></i></a></i>;<br/>
<a NAME="E4:8_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="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_2.html#M1">MSSubAlgebra</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/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:8"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E5:8_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K13">.:.:</a> <span class="p1">(<span class="default">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> </span>)</span> is  <a href="relat_1.html#V2">V5</a>  <a href="msafree.html#M1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>from </b><i><a class="ref" href="birkhoff.html#S5" target="_self">BIRKHOFF:sch 5</a>(<a class="txt" href="birkhoff.html#E2:8_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="birkhoff.html#E3:8_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="birkhoff.html#E4:8_1"><i><font color="Green">E8</font></i></a>);<br/></i>
<a NAME="E6:8_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default">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> </span>)</span></span>)</span> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pzfmisc1.html#D3">PZFMISC1:def 3</a><br/>

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>assume </b><a NAME="E2:8_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>hence </b><a NAME="E3:8_1_1"/>
the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default">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> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:8_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pboole.html#D16">PBOOLE:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8_1"/><b>then </b>
 <a href="mssubfam.html#K1">doms</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </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="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default">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> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="mssubfam.html#T17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E8:8_1"/><b>then </b><i><font color="Green">E10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="pboole.html#K13">.:.:</a> the <a href="msualg_1.html#U4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11">FreeMSA</a> <span class="p2">(<span class="default">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> </span>)</span></span>)</span> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K2">rngs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T14">EQUATION:14</a></i>;<br/>
<a NAME="E9:8_1"/><i><font color="Green">E11</font></i>: 
 <a href="mssubfam.html#K2">rngs</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K2">rngs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="birkhoff.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:8_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K13">.:.:</a> <span class="p1">(<span class="default">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> </span>)</span> <a href="pboole.html#R2">c=</a>  <a href="mssubfam.html#K2">rngs</a> <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="equation.html#T13">EQUATION:13</a></i>;<br/>
<a NAME="E11:8_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K13">.:.:</a> <span class="p1">(<span class="default">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> </span>)</span> <a href="pboole.html#R2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a> </span>)</span> <a href="pboole.html#K13">.:.:</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E8:8_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="birkhoff.html#E9:8_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pboole.html#T15">PBOOLE:15</a></i>;<br/>
<a NAME="E12:8_1"/>
<font color="Maroon">F<sub>3</sub></font>() <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> <span class="p1">(<span class="default">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> </span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E13:8_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" target="_self">-hash</a>  <a href="msualg_3.html#R2">is_epimorphism</a>  <a href="msafree.html#K11">FreeMSA</a> <span class="p1">(<span class="default">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> </span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="txt" href="birkhoff.html#E5:8_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="birkhoff.html#E11:8_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="equation.html#T25">EQUATION:25</a></i>;<br/>


</div>
