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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="unialg_2.html#V2">with_const_op</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="unialg_2.html#M3">SubAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:12"/>
 <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>2</sub></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="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:12_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:12_1"/><i><font color="Green">E9</font></i>: 
(  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E7:12_1"/>
 <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T18">UNIALG_2:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:12_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:12_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green">E8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:12_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>5</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>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E14:12_1"/>
<font color="Maroon">c<sub>8</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>2</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#T10">UNIALG_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</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>8</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="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:12_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E17:12_1"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E18:12_1"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E19:12_1"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E20:12_1"/>
 <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="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="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E21:12_1"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <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>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E22:12_1"/>
<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> 
 
<b>by </b><i><a class="ref" href="unialg_2.html#T16">UNIALG_2:16</a></i>;<br/>
<a NAME="E23:12_1"/><b>then </b>
 <a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E15</font></i>:  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>10</sub></font> = 
<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>8</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E21:12_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="unialg_3.html#E21:12_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
;<br/>
<i><font color="Green">E16</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
0 <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/>
<font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E17:12_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="unialg_3.html#E19:12_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E25:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
;<br/>
<a NAME="E27:12_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="unialg_3.html#E9:12_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="unialg_3.html#E24:12_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E28:12_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="unialg_3.html#E9:12_1"><i><font color="Green">E10</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:12"/>
 <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:12_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:12_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:12_2"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:12_2"/><i><font color="Green">E8</font></i>: 
(  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<a NAME="E7:12_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E8:12_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:12_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:12_2"/><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>7</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>2</sub></font> &amp; <font color="Maroon">c<sub>5</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>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E15:12_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>8</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="txt" href="unialg_3.html#E12:12_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="unialg_2.html#T10">UNIALG_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</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>8</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="E17:12_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E15:12_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E18:12_2"/><b>then </b><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E19:12_2"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="unialg_2.html#V1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E20:12_2"/><b>then </b><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="unialg_2.html#R2">is_closed_on</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E21:12_2"/>
 <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> <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>2</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E22:12_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <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>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:12_2"/>
<font color="Maroon">c<sub>2</sub></font>,<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#T16">UNIALG_2:16</a></i>;<br/>
<a NAME="E24:12_2"/><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> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E15</font></i>:  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E22:12_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="unialg_3.html#E22:12_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>
;<br/>
<b>then </b><i><font color="Green">E16</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> = 
0 <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>

<br/>.= 
0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/>
<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>10</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="unialg_3.html#E18:12_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_3.html#E20:12_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="unialg_3.html#E25:12_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="unialg_2.html#D6">UNIALG_2:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="unialg_3.html#E26:12_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T98">RELAT_1:98</a></i>
;<br/>
<a NAME="E28:12_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="unialg_1.html#K2">arity</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="unialg_3.html#E10:12_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="unialg_3.html#E25:12_2"><i><font color="Green">E15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E29:12_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="unialg_2.html#K6">Constants</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_2"><i><font color="Green">E7</font></i></a>, <a class="txt" href="unialg_3.html#E10:12_2"><i><font color="Green">E9</font></i></a></i>;<br/>


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


</div>
