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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<a NAME="E2:12"/><b>then </b>
<span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pralg_1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:12"/><b>then </b>
 <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="pralg_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then </b><i><font color="Green">E11</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
;<br/>
<a NAME="E5:12"/><i><font color="Green">E12</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></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#K3">signature</a> <font color="Maroon">c<sub>1</sub></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#K3">signature</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></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#K3">signature</a> <span class="p3"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>2</sub></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">c<sub>2</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></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> the <a href="unialg_1.html#U1">charact</a> of <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</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#K3">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:12_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E3:12_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:12_2"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E1:12_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E6:12_2"/>
<font color="Maroon">c<sub>3</sub></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 <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="pralg_1.html#E4:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E1:12_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = 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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:12_2"/><i><font color="Green">E16</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E4:12"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E1:12_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pralg_1.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E1:12_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E11:12_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>hence </b><a NAME="E12:12_2"/>
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <span class="p2"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pralg_1.html#E1:12"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pralg_1.html#E5:12"><i><font color="Green">E12</font></i></a>, <a class="txt" href="pralg_1.html#E2:12_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="pralg_1.html#E5:12_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="pralg_1.html#E8:12_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pralg_1.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:12"/><b>then </b>
 <a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">signature</a> <span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E4:12"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>
<b>hence </b><a NAME="E8:12"/>
<span class="p1"><a href="pralg_1.html#K5" target="_self">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="pralg_1.html#K5" target="_self">:]</a></span>,<font color="Maroon">c<sub>1</sub></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/>


</div>
