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<b>reconsider </b><font color="Maroon" title="c1">SN</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>consider </b><font color="Maroon" title="c2">H</font> being   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()<b> such that </b><br/><a NAME="E3:5_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()
 <b>and </b><br/><a NAME="E4:5_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>()
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="#E2:5"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">h1</font> = <font color="Maroon">F<sub>5</sub></font>() as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c1">SN</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon">F<sub>3</sub></font>() ;<br/>
<a NAME="E6:5_1"/><i><font color="Green" title="E7">A5</font></i>: 
<font color="Maroon" title="c3">h1</font> <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c1">SN</font>,<font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self" title="BIRKHOFF:def.1">Def1</a></i>;<br/>
<a NAME="E7:5_1"/>
<font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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" title="BIRKHOFF:def.1">Def1</a></i>;<br/>
<a NAME="E8:5_1"/><b>then </b><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="txt" href="#E3:5_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="msualg_3.html#T10" title="MSUALG_3:th.10">MSUALG_3:10</a></i>;<br/>
<i><font color="Green" title="E9">A7</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">h1</font> <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c1">SN</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>() <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> <font color="Maroon" title="c1">SN</font></span>)</span>
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self" title="BIRKHOFF:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c2">H</font> <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K11" title="PBOOLE:func.11">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> <font color="Maroon" title="c1">SN</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:5_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="pboole.html#T154" title="PBOOLE:th.154">PBOOLE:154</a></i>
<br/>.= 
<font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <span class="p3">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="birkhoff.html#D1" target="_self" title="BIRKHOFF:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span></span>)</span> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <span class="p2">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="extens_1.html#T8" title="EXTENS_1:th.8">EXTENS_1:8</a></i>
;<br/>
<b>take </b>
<font color="Maroon" title="c2">H</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">F<sub>5</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> )</span><br/>

<b>thus </b><a NAME="E10:5_1"/>
<font color="Maroon" title="c2">H</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="txt" href="#E3:5_1"><i><font color="Green" title="E5">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>5</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span></span><br/>

<b>thus </b><a NAME="E11:5_1"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E6:5_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="#E8:5_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="#E9:5_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="ref" href="extens_1.html#T18" title="EXTENS_1:th.18">EXTENS_1:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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