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

<a NAME="E1:14_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</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>()  st <font color="Olive">A</font>,<font color="Olive">B</font> <a href="msualg_3.html#R6">are_isomorphic</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E2:14_1"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">A</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</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">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E3:14_1"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">A</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">R</font> being  <a href="msualg_4.html#NM5">MSCongruence</a> of <font color="Olive">A</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="msualg_4.html#K14">QuotMSAlg</a> <font color="Olive">A</font>,<font color="Olive">R</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E4:14_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">I</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">I</font>,<font color="Maroon">F<sub>1</sub></font>()  st ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">I</font> holds <br/> ex <font color="Olive">A</font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] ) ) holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Olive">F</font>]
 
<b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">s</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font> : for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> </span>}</span>  );<br/>
<b>set </b><font color="Maroon">I</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1"/>
<font color="Maroon">w</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>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:14_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font>
 ;<br/><b>take </b><font color="Maroon">d</font> = <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> : for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> </span>}</span> ;<br/><b>thus </b><a NAME="E4:14_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">w</font>,<font color="Maroon">d</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">E</font> being    <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>()<b> such that </b><br/><a NAME="E7:14_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</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>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<a NAME="E8:14_1"/>
<font color="Maroon">E</font> is    <a href="pboole.html#M4">ManySortedSubset</a> of  <a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a>,<a class="ref" href="pboole.html#D23">PBOOLE:def 23</a><br/>

<b>assume </b><a NAME="E1:14_1_2"/>
<font color="Maroon">j</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>then consider </b><font color="Maroon">s</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:14_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:14_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">e</font> where <font color="Olive">e</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> : for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> </span>}</span> 
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:14_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E7:14_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E8:14_1_2"/>
for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Maroon">z</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E9:14_1_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">E</font> as   <a href="equation.html#NM1">EqualSet</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>take </b>
<font color="Maroon">E</font>
;<br/>

<b>let </b><font color="Maroon">A</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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/><b>thus </b><a NAME="E2:14_1_3"/>
<font color="Maroon">A</font> <a href="equation.html#R2">|=</a> <font color="Maroon">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s1</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6">EQUATION:def 6</a><br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>;<br/>



<b>assume </b><a NAME="E1:14_1_3_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
 ;<br/>

<a NAME="E2:14_1_3_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s1</font>,<font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>]
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font><b> such that </b><br/><a NAME="E4:14_1_3_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:14_1_3_1"/><i><font color="Green">E84</font></i>: 
for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">h</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">A</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="E6:14_1_3_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">h</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">A</font>
 ;<br/>

<a NAME="E7:14_1_3_1"/>
<font color="Maroon">A</font> <a href="equation.html#R1">|=</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:14_1_3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">s1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i>, , <a class="ref" href="equation.html#D5">EQUATION:def 5</a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E11:14_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">A</font> <a href="equation.html#R2">|=</a> <font color="Maroon">E</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>()] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="equation.html#R2">|=</a> <font color="Maroon">E</font>;<br/>
<a NAME="E12:14_1"/><i><font color="Green">E88</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</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>()  st <font color="Olive">A</font>,<font color="Olive">B</font> <a href="msualg_3.html#R6">are_isomorphic</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T36">EQUATION:36</a></i>;<br/>
<a NAME="E13:14_1"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">A</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</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">A</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T34">EQUATION:34</a></i>;<br/>
<a NAME="E14:14_1"/><i><font color="Green">E91</font></i>: 
for <font color="Olive">I</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">F</font> being   <a href="pralg_2.html#M2">MSAlgebra-Family</a> of <font color="Olive">I</font>,<font color="Maroon">F<sub>1</sub></font>()  st ( for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Olive">I</font> holds <br/> ex <font color="Olive">A</font> being   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>] ) ) holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="pralg_2.html#K15">product</a> <font color="Olive">F</font>]
 
<b>by </b><i><a class="ref" href="equation.html#T40">EQUATION:40</a></i>;<br/>
<b>set </b><font color="Maroon">XX</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>consider </b><font color="Maroon">K</font> being  <a href="msualg_1.html#V4">strict</a> <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>(), <font color="Maroon">F</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="Maroon">K</font><b> such that </b><br/><a NAME="E16:14_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">K</font>]
 <b>and </b><br/><a NAME="E17:14_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">B</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">G</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</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/> ex <font color="Olive">H</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Olive">B</font> st <br/>( <font color="Olive">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Olive">B</font> &amp; <font color="Olive">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Olive">G</font> &amp; ( for <font color="Olive">W</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Olive">B</font>  st <font color="Olive">W</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Olive">B</font> &amp; <font color="Olive">W</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Olive">G</font> holds <br/><font color="Olive">H</font> <a href="pboole.html#R6">=</a> <font color="Olive">W</font> ) )
 <b>from </b><i><a class="ref" href="birkhoff.html#S2" target="_self">BIRKHOFF:sch 2</a>(, , );<br/></i>
<a NAME="E18:14_1"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">D</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>() holds <br/> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">D</font>] iff for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>  st ( for <font color="Olive">M</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">M</font>] holds <br/><font color="Olive">M</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> ) holds <br/><font color="Olive">D</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</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>thus </b><a NAME="E1:14_1_4"/>
( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">D</font>] implies for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>  st ( for <font color="Olive">B</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Olive">B</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> ) holds <br/><font color="Maroon">D</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_4_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">D</font>]
 ;<br/>

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>



<b>assume </b><a NAME="E2:14_1_4_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">B</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Olive">B</font> <a href="equation.html#R1">|=</a> <font color="Maroon">e</font>
 ;<br/>

<a NAME="E3:14_1_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>]
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E5:14_1_4_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> : for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Olive">f</font> </span>}</span>  )
 ;<br/>
<a NAME="E6:14_1_4_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:14_1_4_1"/>
<font color="Maroon">D</font> <a href="equation.html#R1">|=</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="equation.html#D6">EQUATION:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:14_1_4"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">s</font> being  <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>  st ( for <font color="Olive">B</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Olive">B</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font> ) holds <br/><font color="Maroon">D</font> <a href="equation.html#R1">|=</a> <font color="Olive">e</font>
 ;<br/>

<b>let </b><font color="Maroon">s</font> be   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(); <i><font color="Red">:: according to </font></i><a class="ref" href="equation.html#D6">EQUATION:def 6</a><br/><b>let </b><font color="Maroon">e</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>;<br/>



<b>assume </b><a NAME="E3:14_1_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<a NAME="E4:14_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>]
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="msualg_1.html#NM1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:14_1_4"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> &amp; <font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">f</font> where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> : for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Olive">f</font> </span>}</span>  )
 ;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="equation.html#K4">Equations</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E8:14_1_4"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; ( for <font color="Olive">A</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>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Olive">A</font> <a href="equation.html#R1">|=</a> <font color="Maroon">f</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E9:14_1_4"/>
<font color="Maroon">D</font> <a href="equation.html#R1">|=</a> <font color="Maroon">e</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:14_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">K</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E20:14_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">K</font>]
 
<b>from </b><i><a class="ref" href="birkhoff.html#S11" target="_self">BIRKHOFF:sch 11</a>(, , , , , );<br/></i>
<a NAME="E21:14_1"/><i><font color="Green">E109</font></i>: 
for <font color="Olive">M</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">G</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">M</font>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">M</font>] holds <br/> ex <font color="Olive">H</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Olive">M</font> st <br/>( <font color="Olive">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Olive">M</font> &amp; <font color="Olive">G</font> <a href="pboole.html#R6">=</a> <font color="Olive">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">M</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>let </b><font color="Maroon">G</font> be    <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">M</font>;<br/>



<b>assume </b><a NAME="E1:14_1_5"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">M</font>]
 ;<br/>

<a NAME="E2:14_1_5"/><b>then </b>
 ex <font color="Olive">H</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Maroon">M</font> st <br/>( <font color="Olive">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> &amp; <font color="Olive">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> &amp; ( for <font color="Olive">W</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Maroon">M</font>  st <font color="Olive">W</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> &amp; <font color="Olive">W</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> holds <br/><font color="Olive">H</font> <a href="pboole.html#R6">=</a> <font color="Olive">W</font> ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:14_1_5"/>
 ex <font color="Olive">H</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Maroon">M</font> st <br/>( <font color="Olive">H</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">K</font>,<font color="Maroon">M</font> &amp; <font color="Olive">H</font> <a href="msualg_3.html#K3">**</a> <font color="Maroon">F</font> <a href="pboole.html#R6">=</a> <font color="Maroon">G</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E22:14_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">B</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be  <a href="msualg_1.html#V4">strict</a> <a href="msualg_1.html#V5">non-empty</a> <a href="msafree2.html#V3">finitely-generated</a>  <a href="msualg_2.html#M1">MSSubAlgebra</a> of <font color="Maroon">A</font>;<br/>

<a NAME="E1:14_1_6"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">B</font>]
 
<b>by </b><i>, <a class="ref" href="equation.html#T34">EQUATION:34</a></i>;<br/>
<b>thus </b><a NAME="E2:14_1_6"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">B</font>]
 <b>from </b><i><a class="ref" href="birkhoff.html#S7" target="_self">BIRKHOFF:sch 7</a>(, , , , );</i><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E23:14_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>from </b><i><a class="ref" href="birkhoff.html#S9" target="_self">BIRKHOFF:sch 9</a>(, , , , );</i><br/>


</div>
