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<b>let </b><font color="Maroon">U0</font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><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="E1:23"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">U0</font> is    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U1</font> &amp; <font color="Maroon">U1</font> is    <a href="unialg_2.html#M3" target="_self">SubAlgebra</a> of <font color="Maroon">U2</font> )
 ;<br/>

<a NAME="E2:23"/><b>then </b><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:23"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:23"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D8" target="_self">UNIALG_2:def 8</a><br/>

<b>reconsider </b><font color="Maroon">B1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U1</font> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font> <b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">U2</font>;<br/>

<b>assume </b><a NAME="E7:23"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 ;<br/>

<a NAME="E8:23"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
;<br/>
<a NAME="E9:23"/><i><font color="Green">E51</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">B1</font> &amp; <font color="Maroon">B1</font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:23"/><i><font color="Green">E52</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B2</font> &amp; <font color="Maroon">B2</font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a> )
 
<b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">o2</font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font>;<br/><a NAME="E1:23_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">B2</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o2</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:23_1"/><i><font color="Green">E56</font></i>: 
( <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">U2</font> &amp; 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> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/><a NAME="E4:23_1"/><i><font color="Green">E57</font></i>: 
 <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#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B2</font></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">o21</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font> <b>by </b><i>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:23_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">B1</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o21</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:23_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">o21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , </i>;<br/><a NAME="E8:23_1"/><i><font color="Green">E60</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#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">B1</font></span>)</span>
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">o20</font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U0</font> <b>by </b><i>, , <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E10:23_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">o20</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o21</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B1</font>
 <b>by </b><i>, , <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , <a class="txt" href="unialg_2.html#E2:23"><i><font color="Green">E26</font></i></a>, </i>;<br/><a NAME="E11:23_1"/><i><font color="Green">E79</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o20</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o21</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U2</font> )
 <b>by </b><i/>;<br/><a NAME="E12:23_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">o20</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o21</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span> &amp; <font color="Maroon">o21</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span> )
 <b>by </b><i>, , , , </i>;<br/><a NAME="E13:23_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U1</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E14:23_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font>
 <b>by </b><i/>;<br/><a NAME="E15:23_1"/><b>then </b><i><font color="Green">E81</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T130">FINSEQ_2:130</a></i>;<br/><a NAME="E16:23_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">o2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><a NAME="E17:23_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font>
 <b>by </b><i>, <a class="txt" href="unialg_2.html#E7:23"><i><font color="Green">E29</font></i></a>, </i>;<br/><a NAME="E18:23_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T130">FINSEQ_2:130</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">B</font>;<br/><b>reconsider </b><font color="Maroon">s0</font> = <font color="Maroon">s</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E3:23_1_1"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font> )
 <b>by </b><i>, <a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E4:23_1_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:23_1_1"/><b>then </b>
<font color="Maroon">s</font> is    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B2</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><b>assume </b><a NAME="E7:23_1_1"/><i><font color="Green">E85</font></i>: 
 <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">o2</font>
 ;<br/><a NAME="E8:23_1_1"/><b>then </b>
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B1</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E9:23_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><a NAME="E10:23_1_1"/>
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">B2</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font> </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:23_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">s2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><i><font color="Green">E89</font></i>: <font color="Maroon">o20</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> = 
<span class="p1">(<span class="default"><font color="Maroon">o21</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">o21</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/><a NAME="E13:23_1_1"/>
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font> <a href="finseq_2.html#K3">*</a>  :  <a href="finseq_1.html#K3">len</a> <font color="Olive">w</font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="unialg_2.html#E8:23"><i><font color="Green">E30</font></i></a>, , </i>;<br/><a NAME="E14:23_1_1"/><b>then </b>
<font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o20</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">U0</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#D4">FINSEQ_2:def 4</a></i>;<br/><a NAME="E15:23_1_1"/><b>then </b>
( <font color="Maroon">o20</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">o20</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">o20</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E16:23_1_1"/>
<font color="Maroon">o2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E20:23_1"/>
<font color="Maroon">B</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<i><font color="Green">E90</font></i>:  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U1</font>,<font color="Maroon">B1</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U1</font>
<b>by </b><i/>
;<br/>
<i><font color="Green">E91</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="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B2</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U2</font>
<b>by </b><i/>
;<br/>
<a NAME="E14:23"/><i><font color="Green">E92</font></i>: 
 <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#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_4"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">n</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" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">o2</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U2</font> <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">o21</font> = 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> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">U1</font> <b>by </b><i><a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , , </i>;<br/><a NAME="E4:23_4"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">B</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o2</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:23"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E5:23_4"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">B2</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:23_4"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">B1</font> <a href="unialg_2.html#R2" target="_self">is_closed_on</a> <font color="Maroon">o21</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> = 
<font color="Maroon">o2</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B2</font></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">o2</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B2</font></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E2:23"><i><font color="Green">E26</font></i></a>, </i>
<br/>.= 
<font color="Maroon">o21</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o2</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , , </i>
<br/>.= 
<font color="Maroon">o21</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3">arity</a> <font color="Maroon">o21</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <font color="Maroon">B1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="unialg_2.html#T1" target="_self">Th1</a>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , , </i>
<br/>.= 
<font color="Maroon">o21</font> <a href="unialg_2.html#K2" target="_self">/.</a> <font color="Maroon">B1</font>
<b>by </b><i>, </i>
<br/>.= 
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:23"/>
the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">U0</font> <a href="hidden.html#R1">=</a>  <a href="unialg_2.html#K3" target="_self">Opers</a> <font color="Maroon">U2</font>,<font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="unialg_2.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>

<b>thus </b><a NAME="E17:23"/>
<font color="Maroon">B</font> is <a href="unialg_2.html#V1" target="_self">opers_closed</a>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:23"><i><font color="Green">E20</font></i></a>, </i>;<br/>


</div>
