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

<span class="kw">let </span><font color="Maroon" title="c1">U0</font> be   <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">U1</font>, <font color="Olive" title="b2">U2</font> being    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>  st  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">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="Olive" title="b2">U2</font> holds <br/><font color="Olive" title="b1">U1</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Olive" title="b2">U2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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> implies <font color="Maroon" title="c2">U1</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font> )</span><br/>



<a NAME="E1:21"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T7" target="_self" title="UNIALG_2:th.7">Th10</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:21"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">U1</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font></span><br/>

<span class="kw">hence </span><a NAME="E4:21"/>
 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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">UNIALG_2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   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>  st <font color="Olive" title="b1">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> holds <br/>(  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="Olive" title="b1">B</font>) &amp; <font color="Olive" title="b1">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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> implies (  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="c5">B</font>) &amp; <font color="Maroon" title="c5">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E5:21"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c5">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  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="c5">B</font>) &amp; <font color="Maroon" title="c5">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>  )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<a NAME="E7:21"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
 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="c6">B2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<a NAME="E8:21"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c6">B2</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<a NAME="E9:21"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
 <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="c5">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def7</a></span>;<br/>
<a NAME="E10:21"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T7" target="_self" title="UNIALG_2:th.7">Th10</a></span>;<br/>
<a NAME="E11:21"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
<font color="Maroon" title="c4">B1</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<a NAME="E12:21"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<font color="Maroon" title="c5">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="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">o2</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c3">U2</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">UNIALG_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c7">o2</font></span><br/><span class="kw">let </span><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="c5">B</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D3" target="_self" title="UNIALG_2:def.3">UNIALG_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o2</font> implies <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="c5">B</font> )</span><br/>



<span class="kw">consider </span><font color="Maroon" title="c9">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:21_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">and </span><br/><a NAME="E3:21_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
 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="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">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="c9">n</font> as   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E5:21_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
 <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="unialg_2.html#T6" target="_self" title="UNIALG_2:th.6">Th9</a></span>;<br/>
<a NAME="E6:21_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> <font color="Maroon" title="c5">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E7:21_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
 <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">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:21_1"/><span class="kw">then </span>
<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="c6">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">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="c6">B2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:21_1"/><span class="lab"><font color="Green" title="E25">A14</font></span>: 
 <a href="margrel1.html#K19" title="MARGREL1:func.19">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c5">B</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">tb2</font> = <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B2</font>;<br/>
<a NAME="E12:21_1"/><span class="lab"><font color="Green" title="E26">A15</font></span>: 
<font color="Maroon" title="c6">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c10">o0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def5</a></span>;<br/>
<span class="lab"><font color="Green" title="E27">A16</font></span>: <font color="Maroon" title="c7">o2</font> = 
<font color="Maroon" title="c10">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c6">B2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E2:21_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="txt" href="#E3:21_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c10">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="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21_1"><span class="lab"><font color="Green" title="E26">A15</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def6</a></span>
;<br/>
<a NAME="E14:21_1"/><span class="lab"><font color="Green" title="E28">A17</font></span>: 
<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="c10">o0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def5</a></span>;<br/>
<a NAME="E15:21_1"/>
<span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">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="c6">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="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#D4" title="FINSEQ_2:def.4">FINSEQ_2:def 4</a></span>;<br/>
<a NAME="E16:21_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c10">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E5:21_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a></span>;<br/>
<a NAME="E17:21_1"/><span class="kw">then </span>
<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="c10">o0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:21_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="c5">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E11:21_1"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="txt" href="#E14:21_1"><span class="lab"><font color="Green" title="E28">A17</font></span></a>, <a class="txt" href="#E5:21_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="unialg_2.html#D3" target="_self" title="UNIALG_2:def.3">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:21"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
 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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">B</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:21_2"/><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c5">B</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><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#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">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#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c3">U2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:21_2"/><span class="lab"><font color="Green" title="E23">A20</font></span>: 
( <font color="Maroon" title="c9">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="c6">B2</font> &amp;  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c9">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K19" title="MARGREL1:func.19">arity</a> <font color="Maroon" title="c8">o0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def7</a>, <a class="ref" href="unialg_2.html#T6" target="_self" title="UNIALG_2:th.6">Th9</a></span>;<br/><a NAME="E5:21_2"/><span class="lab"><font color="Green" title="E24">A21</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c9">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def5</a></span>;<br/><a NAME="E6:21_2"/><span class="lab"><font color="Green" title="E25">A22</font></span>: 
<font color="Maroon" title="c6">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def5</a></span>;<br/><a NAME="E7:21_2"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:21"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def5</a></span>;<br/><span class="kw">thus </span> 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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def7</a></span>
<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="margrel1.html#K19" title="MARGREL1:func.19">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_2"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def6</a></span>
<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="margrel1.html#K19" title="MARGREL1:func.19">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">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="margrel1.html#K19" title="MARGREL1:func.19">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="margrel1.html#T21" title="MARGREL1:th.21">MARGREL1:21</a></span>
<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="margrel1.html#K19" title="MARGREL1:func.19">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">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="margrel1.html#K19" title="MARGREL1:func.19">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">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<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="c6">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="margrel1.html#K19" title="MARGREL1:func.19">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">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_2"><span class="lab"><font color="Green" title="E25">A22</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">o2</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c5">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_2"><span class="lab"><font color="Green" title="E24">A21</font></span></a>, <a class="txt" href="#E4:21_2"><span class="lab"><font color="Green" title="E23">A20</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def6</a></span>
<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="c5">B</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="txt" href="#E1:21_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:21"/>
 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="c5">B</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E10:21"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E9:21"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> </span><br/>

<span class="kw">thus </span><a NAME="E16:21"/>
<font color="Maroon" title="c5">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:21"><span class="lab"><font color="Green" title="E20">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
