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<div class="add">

<a NAME="E1:23"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="unialg_1.html#NM2">Universal_Algebra</a>  st  <a href="msualg_1.html#K6">signature</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Olive">B</font> holds <br/> <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">B</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:23_1"/>
 <a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="unialg_2.html#R1">are_similar</a> 
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:23_1"/>
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="msuhom_1.html#T10">MSUHOM_1:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:23"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="unialg_1.html#NM2">Universal_Algebra</a>  st  <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">B</font> holds <br/> <a href="msualg_1.html#K6">signature</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Olive">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>, <font color="Maroon">B</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green">E33</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">B</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">ff1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">gg1</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span>,<span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="msualg_1.html#T7">MSUALG_1:7</a></i>;<br/>
<a NAME="E4:23_2"/><i><font color="Green">E34</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span>,<font color="Maroon">ff1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<a NAME="E5:23_2"/><i><font color="Green">E49</font></i>: 
 <a href="msualg_1.html#K7">MSSign</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#G1">ManySortedSign</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span></span>)</span>,<font color="Maroon">gg1</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> #)
 
<b>by </b><i><a class="ref" href="msualg_1.html#T16">MSUALG_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span>
 ;<br/><a NAME="E2:23_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</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="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>;<br/><b>then </b><i><font color="Green">E52</font></i>: <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="alg_1.html#T1">ALG_1:1</a></i>
;<br/><b>reconsider </b><font color="Maroon">m</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , , , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><b>reconsider </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/><a NAME="E6:23_2_1"/>
<span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">m</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><a NAME="E8:23_2_1"/>
<span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span> <a href="finseq_2.html#K3">*</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="pboole.html#K8">*--&gt;</a> 0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="finseq_1.html#NM1">FinSequence</a> ;<br/><b>thus </b><span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:23_2"/>
 <a href="msualg_1.html#K6">signature</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="unialg_1.html#NM2">Universal_Algebra</a> holds <br/> (  <a href="msualg_1.html#K6">signature</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K6">signature</a> <font color="Olive">B</font> iff  <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K7">MSSign</a> <font color="Olive">B</font> )
 <b>by </b><i/>;<br/>


</div>
