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<b>let </b><font color="Maroon" title="c1">U0</font> be   <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>, <font color="Maroon" title="c3">U2</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>assume </b><a NAME="E1:25"/><i><font color="Green" title="E15">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font>
 ;<br/>

<b>hence </b><a NAME="E2:25"/>
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="c3">U2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">UNIALG_2:def 8</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">B1</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" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">B2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</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" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green" title="E16">A2</font></i>: 
( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c4">B1</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c5">B2</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" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">B</font> be  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="c3">U2</font>;<br/>

<b>assume </b><a NAME="E6:25"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font>
 ;<br/>

<a NAME="E7:25"/><i><font color="Green" title="E18">A4</font></i>: 
(  <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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;  <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="c3">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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;  <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="c4">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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;  <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="c5">B2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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;  <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="c3">U2</font>,<font color="Maroon" title="c6">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">U2</font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a>, <a class="ref" href="unialg_2.html#T10" target="_self" title="UNIALG_2:th.10">Th10</a></i>;<br/>
<a NAME="E8:25"/><i><font color="Green" title="E19">A5</font></i>: 
<font color="Maroon" title="c6">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">o2</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c3">U2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">UNIALG_2:def 5</a><br/><b>let </b><font color="Maroon" title="c8">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c6">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">UNIALG_2:def 4</a><br/>



<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E20">A6</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font>
 ;<br/>

<a NAME="E2:25_1"/>
( <font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:25_1"/><b>then </b><i><font color="Green" title="E21">A7</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c5">B2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:25_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">B2</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">s2</font> = <font color="Maroon" title="c8">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">B2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c8">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">B1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="unialg_2.html#E6:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E7:25_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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_2.html#E3:25_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:25_1"/><b>then </b>
<font color="Maroon" title="c8">s</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 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="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">s0</font> = <font color="Maroon" title="c8">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E11:25_1"/><i><font color="Green" title="E22">A8</font></i>: 
( <font color="Maroon" title="c12">n</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="c3">U2</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">o0</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="c12">n</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="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E11:25_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">tb2</font> = <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font>;<br/>
<b>set </b><font color="Maroon" title="c15">tb0</font> = <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <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>;<br/>
<a NAME="E13:25_1"/><i><font color="Green" title="E23">A9</font></i>: 
( <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c5">B2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">v</font> where <font color="Olive" title="b1">v</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">B1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font> </span>}</span>  &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></i>;<br/>
<a NAME="E14:25_1"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c5">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o0</font> &amp; <font color="Maroon" title="c4">B1</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o0</font> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E5:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/>
<a NAME="E15:25_1"/><i><font color="Green" title="E25">A11</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E11:25_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="unialg_2.html#T9" target="_self" title="UNIALG_2:th.9">Th9</a></i>;<br/>
<i><font color="Green" title="E26">A12</font></i>: <font color="Maroon" title="c7">o2</font> = 
<font color="Maroon" title="c13">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c5">B2</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E5:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="unialg_2.html#E11:25_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c13">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E14:25_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>
;<br/>
<a NAME="E17:25_1"/>
( <font color="Maroon" title="c11">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <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> &amp; <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font> &amp; <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">s0</font> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E1:25_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="unialg_2.html#E13:25_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="unialg_2.html#E15:25_1"><i><font color="Green" title="E25">A11</font></i></a></i>;<br/>
<a NAME="E18:25_1"/><b>then </b>
<font color="Maroon" title="c7">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s1</font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E16:25_1"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E19:25_1"/>
<font color="Maroon" title="c7">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E6:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="unialg_2.html#E14:25_1"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="unialg_2.html#E15:25_1"><i><font color="Green" title="E25">A11</font></i></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:25_2"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c7">n</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>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">o0</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="c7">n</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/><b>reconsider </b><font color="Maroon" title="c9">o1</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="c7">n</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="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <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="c10">o2</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c3">U2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c3">U2</font> <b>by </b><i><a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:25_2"/><i><font color="Green" title="E21">A14</font></i>: 
( <font color="Maroon" title="c9">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c4">B1</font> &amp; <font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c5">B2</font> )
 <b>by </b><i><a class="txt" href="unialg_2.html#E5:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></i>;<br/><a NAME="E6:25_2"/><i><font color="Green" title="E22">A15</font></i>: 
( <font color="Maroon" title="c4">B1</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o0</font> &amp; <font color="Maroon" title="c5">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o0</font> &amp; <font color="Maroon" title="c6">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c10">o2</font> )
 <b>by </b><i><a class="txt" href="unialg_2.html#E5:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="unialg_2.html#E8:25"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/><a NAME="E7:25_2"/><i><font color="Green" title="E23">A16</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c10">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="unialg_2.html#T9" target="_self" title="UNIALG_2:th.9">Th9</a></i>;<br/><b>thus </b>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">n</font> = 
<font color="Maroon" title="c8">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c4">B1</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E5:25"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></i>
<br/>.= 
<font color="Maroon" title="c8">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B1</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E6:25_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c8">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E1:25"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="unialg_2.html#T1" target="_self" title="UNIALG_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c5">B2</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E6:25"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c5">B2</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c8">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B</font></span>)</span>
<b>by </b><i><a class="txt" href="unialg_2.html#E6:25_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c10">o2</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c6">B</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E5:25_2"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="unialg_2.html#E6:25_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="unialg_2.html#E7:25_2"><i><font color="Green" title="E23">A16</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c6">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="unialg_2.html#E1:25_2"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:25"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c3">U2</font>,<font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E7:25"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>

<b>thus </b><a NAME="E11:25"/>
<font color="Maroon" title="c6">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 <b>by </b><i><a class="txt" href="unialg_2.html#E8:25"><i><font color="Green" title="E19">A5</font></i></a></i>;<br/>


</div>
