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<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>assume </b><a NAME="E1:11"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">U1</font>,<font color="Maroon">U2</font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<b>let </b><font color="Maroon">o1</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font>;<br/><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font>;<br/><b>let </b><font color="Maroon">o</font> be   <a href="unialg_2.html#NM5">operation</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U2</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>







<b>assume </b><a NAME="E2:11"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U2</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> )
 ;<br/>

<a NAME="E3:11"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U2</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font></span>)</span> #)
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:11"/><i><font color="Green">E39</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:11"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">U2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:11"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E7:11"/><i><font color="Green">E43</font></i>: 
 <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font>
 
<b>by </b><i>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<a NAME="E8:11"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E9:11"/><b>then </b><i><font color="Green">E44</font></i>: 
( <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K4">signature</a> <font color="Maroon">U2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> )
 
<b>by </b><i>, , , <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E10:11"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:11"/>
for <font color="Olive">x</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U2</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font> holds <br/> <a href="finseq_1.html#K3">len</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">U1</font>,<font color="Maroon">U2</font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o</font>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s</font> where <font color="Olive">s</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="pralg_1.html#T1" target="_self">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E4:11_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> )
 ;<br/>
<b>thus </b><a NAME="E5:11_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:11"/>
(  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o1</font> &amp;  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o2</font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>


</div>
