<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c1">V</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">Gen</font> = the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> as    <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of  <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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <b>by </b><i><a class="ref" href="msafree2.html#T9" title="MSAFREE2:th.9">MSAFREE2:9</a></i>;<br/>
<a NAME="E2:8_1"/><i><font color="Green" title="E6">A4</font></i>: 
for <font color="Olive" title="b1">C</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">G</font> being   <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</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>() <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Olive" title="b1">C</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] holds <br/> ex <font color="Olive" title="b3">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="Olive" title="b1">C</font> st <br/>( <font color="Olive" title="b3">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="Olive" title="b1">C</font> &amp; <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> &amp; ( for <font color="Olive" title="b4">K</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="Olive" title="b1">C</font>  st <font color="Olive" title="b4">K</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="Olive" title="b1">C</font> &amp; <font color="Olive" title="b4">K</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b4">K</font> ) )
 
<b>by </b><i><a class="txt" href="birkhoff.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a></i>;<br/>
<a NAME="E3:8_1"/><i><font color="Green" title="E7">A5</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" title="E4">A2</font></i></a></i>;<br/>
<a NAME="E4:8_1"/><i><font color="Green" title="E8">A6</font></i>: 
for <font color="Olive" title="b1">A</font> being <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">B</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">strict</a> <a href="msualg_1.html#V5" title="MSUALG_1:attr.5">non-empty</a>  <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of <font color="Olive" title="b1">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">B</font>]
 
<b>by </b><i><a class="txt" href="birkhoff.html#E3:8"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<a NAME="E5:8_1"/><i><font color="Green" title="E9">A7</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="msafree.html#M1" title="MSAFREE:mode.1">GeneratorSet</a> of <font color="Maroon">F<sub>2</sub></font>()
 
<b>from </b><i><a class="ref" href="birkhoff.html#S5" target="_self" title="BIRKHOFF:sch.5">BIRKHOFF:sch 5</a>(<a class="txt" href="birkhoff.html#E2:8_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="birkhoff.html#E3:8_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="birkhoff.html#E4:8_1"><i><font color="Green" title="E8">A6</font></i></a>);<br/></i>
<a NAME="E6:8_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">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" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pzfmisc1.html#D3" title="PZFMISC1:def.3">PZFMISC1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() or  not the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green" title="E10">A8</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<b>assume </b><a NAME="E2:8_1_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>hence </b><a NAME="E3:8_1_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="birkhoff.html#E1:8_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:8_1"/><b>then </b>
 <a href="mssubfam.html#K1" title="MSSUBFAM:func.1">doms</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&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="mssubfam.html#T17" title="MSSUBFAM:th.17">MSSUBFAM:17</a></i>;<br/>
<a NAME="E8:8_1"/><b>then </b><i><font color="Green" title="E10">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeMSA</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">rngs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="equation.html#T14" title="EQUATION:th.14">EQUATION:14</a></i>;<br/>
<a NAME="E9:8_1"/><i><font color="Green" title="E11">A10</font></i>: 
 <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">rngs</a> <font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">rngs</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="birkhoff.html#T1" target="_self" title="BIRKHOFF:th.1">Th1</a></i>;<br/>
<a NAME="E10:8_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K2" title="MSSUBFAM:func.2">rngs</a> <font color="Maroon">F<sub>3</sub></font>()
 
<b>by </b><i><a class="ref" href="equation.html#T13" title="EQUATION:th.13">EQUATION:13</a></i>;<br/>
<a NAME="E11:8_1"/><b>then </b><i><font color="Green" title="E12">A11</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</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#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a> </span>)</span> <a href="pboole.html#K12" title="PBOOLE:func.12">.:.:</a> <font color="Maroon" title="c2">Gen</font>
 
<b>by </b><i><a class="txt" href="birkhoff.html#E8:8_1"><i><font color="Green" title="E10">A9</font></i></a>, <a class="txt" href="birkhoff.html#E9:8_1"><i><font color="Green" title="E11">A10</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>;<br/>
<a NAME="E12:8_1"/>
<font color="Maroon">F<sub>3</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#K2" title="PRE_CIRC:func.2">--&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/>
<b>hence </b><a NAME="E13:8_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="birkhoff.html#K1" title="BIRKHOFF:func.1">-hash</a>  <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</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#K2" title="PRE_CIRC:func.2">--&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="txt" href="birkhoff.html#E5:8_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="birkhoff.html#E11:8_1"><i><font color="Green" title="E12">A11</font></i></a>, <a class="ref" href="equation.html#T25" title="EQUATION:th.25">EQUATION:25</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
