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<span class="kw">consider </span><font color="Maroon" title="c1">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>()<span class="kw"> such that </span><br/><a NAME="E2:5_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c1">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>()
 <span class="kw">and </span><br/><a NAME="E3:5_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>4</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c1">H</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">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#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">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/>

<span class="kw">thus </span><a NAME="E4:5_1"/>
<font color="Maroon" title="c1">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>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">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/>

<a NAME="E5:5_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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<font color="Maroon">F<sub>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="birkhoff.html#D1" target="_self" onmouseover="rs('birkhoff/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E6:5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c1">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span>,<font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:5_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="msualg_3.html#T10" onmouseover="rs('msualg_3/T10')" onmouseout="rh()">MSUALG_3:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> as  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">h1</font> = <font color="Maroon">F<sub>5</sub></font>() as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">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="E9:5_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
<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="c2">SN</font>,<font color="Maroon">F<sub>3</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="birkhoff.html#D1" target="_self" onmouseover="rs('birkhoff/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<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="c2">SN</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>() <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p1">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> <font color="Maroon" title="c2">SN</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="birkhoff.html#D1" target="_self" onmouseover="rs('birkhoff/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c1">H</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><a href="msafree.html#K15" title="MSAFREE:func.15">Reverse</a> <font color="Maroon" title="c2">SN</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="pboole.html#T140" onmouseover="rs('pboole/T140')" onmouseout="rh()">PBOOLE:140</a></span>
<br/>.= 
<font color="Maroon" title="c1">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="birkhoff.html#D1" target="_self" onmouseover="rs('birkhoff/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="extens_1.html#T4" onmouseover="rs('extens_1/T4')" onmouseout="rh()">EXTENS_1:4</a></span>
;<br/>
<span class="kw">hence </span><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#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:5_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E6:5_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="extens_1.html#T14" onmouseover="rs('extens_1/T14')" onmouseout="rh()">EXTENS_1:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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