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<b>let </b><font color="Maroon">A1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:11_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> iff  ex <font color="Olive">M</font>, <font color="Olive">N</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ex <font color="Olive">f</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Olive">M</font>,<font color="Olive">N</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">M</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">N</font> &amp; <font color="Olive">f</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">M</font>,<font color="Olive">N</font> ) )
 ;<br/>

<b>assume </b><a NAME="E2:11_1_2"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> iff  ex <font color="Olive">M</font>, <font color="Olive">N</font> being <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font> ex <font color="Olive">f</font> being  <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Olive">M</font>,<font color="Olive">N</font> st <br/>( <font color="Olive">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Olive">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">M</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Olive">N</font> &amp; <font color="Olive">f</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Olive">M</font>,<font color="Olive">N</font> ) )
 ;<br/>

<b>thus </b><a NAME="E3:11_1_2"/>
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:11_1_2_1"/>
<font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:11_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">M</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">N</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">f</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:11_1_2_1_1"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">M</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">f</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">M</font>,<font color="Maroon">N</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:11_1_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:11_1_2_1"/>
<font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:11_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">M</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">N</font> being  <a href="msualg_1.html#V4">strict</a> <a href="msualg_6.html#V1">feasible</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font>, <font color="Maroon">f</font> being   <a href="msualg_3.html#NM1">ManySortedFunction</a> of <font color="Maroon">M</font>,<font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:11_1_2_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">M</font> <a href="pzfmisc1.html#R1">is_transformable_to</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">N</font> &amp; <font color="Maroon">f</font> <a href="msualg_3.html#R1">is_homomorphism</a> <font color="Maroon">M</font>,<font color="Maroon">N</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:11_1_2_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 <b>by </b><i>, </i>;<br/>


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


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


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