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

<b>let </b><font color="Maroon">U1</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">U2</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">U3</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">U2</font>,<font color="Maroon">U3</font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<b>let </b><font color="Maroon">h1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U1</font>,<font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">h2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">U2</font>,<font color="Maroon">U3</font>;<br/>



<a NAME="E3:33"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U3</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i>, , <a class="ref" href="unialg_2.html#T4">UNIALG_2:4</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E51</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:33"/><i><font color="Green">E52</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">U3</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E53</font></i>: 
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:33"/>
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green">E55</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>

;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:33_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/><i><font color="Green">E61</font></i>: <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E6:33_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
 <b>by </b><i>, , , <a class="ref" href="msuhom_1.html#T1" target="_self">Th1</a>, <a class="ref" href="pboole.html#D24">PBOOLE:def 24</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:33"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>;<br/>
<a NAME="E12:33"/><i><font color="Green">E94</font></i>: 
 <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span> is    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:33"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:33_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/><b>hence </b><a NAME="E4:33_3"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:33"/><i><font color="Green">E99</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">g</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_4"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E2:33_4"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E3:33_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> = 
<font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i><a class="ref" href="msuhom_1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
<b>by </b><i>, , , , <a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="ref" href="msuhom_1.html#T2" target="_self">Th2</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E17:33"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h1</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E3:33_5"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default">0 <a href="cqc_lang.html#K3">.--&gt;</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h2</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#T6">CQC_LANG:6</a></i>
;<br/>
<a NAME="E5:33_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> is   <a href="funct_1.html#NM1">Function</a>
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:33_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p2">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i>, , , <a class="txt" href="msuhom_1.html#E1:33"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:33"/>
<span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h2</font></span>)</span> <a href="pboole.html#K12">**</a> <span class="p1">(<span class="default"><a href="msuhom_1.html#K3" target="_self">MSAlg</a> <font color="Maroon">h1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msuhom_1.html#K3" target="_self">MSAlg</a> <span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
