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

<b>let </b><font color="Maroon" title="c1">U0</font> be  <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font> be    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>;<br/>



<b>thus </b><a NAME="E1:12"/>
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c2">U1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">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" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E7">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font>
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">o</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E8">A2</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  ex <font color="Olive" title="b1">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">o1</font> being   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E6:12_1"/><i><font color="Green" title="E9">A3</font></i>: 
(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">o1</font> )
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<a NAME="E7:12_1"/>
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T18" title="UNIALG_2:th.18">UNIALG_2:18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E9:12_1"/><i><font color="Green" title="E10">A4</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E1:12_1"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green" title="E8">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:12_1"/><i><font color="Green" title="E11">A5</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> &amp; <font color="Maroon" title="c5">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green" title="E11">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E14:12_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="unialg_2.html#T10" title="UNIALG_2:th.10">UNIALG_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">o</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:12_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c9">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E17:12_1"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o1</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E11:12_1"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E18:12_1"/>
<font color="Maroon" title="c9">A</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E19:12_1"/><b>then </b><i><font color="Green" title="E13">A7</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c5">o1</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/>
<i><font color="Green" title="E14">A8</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">o1</font> = 
0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c9">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

<br/>.= 
0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c9">A</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>
;<br/>
<font color="Maroon" title="c10">o</font> = 
<font color="Maroon" title="c5">o1</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c9">A</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E17:12_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c5">o1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c9">A</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="unialg_3.html#E19:12_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="ref" href="unialg_2.html#D6" title="UNIALG_2:def.6">UNIALG_2:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c5">o1</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E20:12_1"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>
;<br/>
<a NAME="E22:12_1"/><b>then </b>
<font color="Maroon" title="c6">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">U1</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">o</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="unialg_3.html#E9:12_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E23:12_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c2">U1</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="unialg_3.html#E9:12_1"><i><font color="Green" title="E10">A4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:12"/>
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c2">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</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" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c2">U1</font>
 ;<br/>

<a NAME="E2:12_2"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">U1</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">o</font> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon" title="c4">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E4:12_2"/><i><font color="Green" title="E7">A9</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  ex <font color="Olive" title="b1">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">o</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c5">o</font> being   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font><b> such that </b><br/><a NAME="E6:12_2"/><i><font color="Green" title="E8">A10</font></i>: 
(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">o</font> )
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_2"><i><font color="Green" title="E7">A9</font></i></a></i>;<br/>
<a NAME="E7:12_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E8:12_2"/><b>then </b>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">u1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E10:12_2"/><i><font color="Green" title="E9">A11</font></i>: 
<font color="Maroon" title="c6">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:12_2"/><i><font color="Green" title="E10">A12</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> &amp; <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green" title="E10">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E15:12_2"/><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="unialg_2.html#T10" title="UNIALG_2:th.10">UNIALG_2:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">o1</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E17:12_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c9">A</font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E15:12_2"><i><font color="Green" title="E11">A13</font></i></a>, <a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E18:12_2"/><b>then </b><i><font color="Green" title="E12">A14</font></i>: 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c9">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">o1</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c9">A</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7" title="UNIALG_2:def.7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E19:12_2"/>
<font color="Maroon" title="c9">A</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E20:12_2"/><b>then </b><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c10">o1</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E21:12_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c2">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>;<br/>
<a NAME="E22:12_2"/><b>then </b><i><font color="Green" title="E14">A16</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c2">U1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E23:12_2"/>
<font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c1">U0</font> <a href="unialg_2.html#R1" title="UNIALG_2:pred.1">are_similar</a> 
 
<b>by </b><i><a class="ref" href="unialg_2.html#T16" title="UNIALG_2:th.16">UNIALG_2:16</a></i>;<br/>
<a NAME="E24:12_2"/><b>then </b>
 <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D2" title="UNIALG_2:def.2">UNIALG_2:def 2</a></i>;<br/>
<b>then </b><i><font color="Green" title="E15">A17</font></i>:  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c10">o1</font> = 
<span class="p1">(<span class="default"><a href="unialg_1.html#K4" title="UNIALG_1:func.4">signature</a> <font color="Maroon" title="c2">U1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E22:12_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">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" title="E8">A10</font></i></a>, <a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="unialg_3.html#E22:12_2"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="unialg_1.html#D11" title="UNIALG_1:def.11">UNIALG_1:def 11</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E16">A18</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">o1</font> = 
0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c9">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

<br/>.= 
0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c9">A</font>
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>
;<br/>
<font color="Maroon" title="c5">o</font> = 
<font color="Maroon" title="c10">o1</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c9">A</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E12:12_2"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="unialg_3.html#E18:12_2"><i><font color="Green" title="E12">A14</font></i></a>, <a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c10">o1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c9">A</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_3.html#E20:12_2"><i><font color="Green" title="E13">A15</font></i></a>, <a class="txt" href="unialg_3.html#E25:12_2"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="unialg_2.html#D6" title="UNIALG_2:def.6">UNIALG_2:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c10">o1</font>
<b>by </b><i><a class="txt" href="unialg_3.html#E26:12_2"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="relat_1.html#T98" title="RELAT_1:th.98">RELAT_1:98</a></i>
;<br/>
<a NAME="E28:12_2"/><b>then </b>
<font color="Maroon" title="c6">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">a1</font> where <font color="Olive" title="b1">a1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font> :  ex <font color="Olive" title="b2">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Olive" title="b1">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">o</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="unialg_3.html#E6:12_2"><i><font color="Green" title="E8">A10</font></i></a>, <a class="txt" href="unialg_3.html#E10:12_2"><i><font color="Green" title="E9">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E29:12_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font>
 <b>by </b><i><a class="txt" href="unialg_3.html#E4:12_2"><i><font color="Green" title="E7">A9</font></i></a>, <a class="txt" href="unialg_3.html#E10:12_2"><i><font color="Green" title="E9">A11</font></i></a></i>;<br/>


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


</div>
