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<span class="kw">let </span><font color="Maroon" title="c1">U0</font>, <font color="Maroon" title="c2">U1</font>, <font color="Maroon" title="c3">U2</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">U0</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c2">U1</font> &amp; <font color="Maroon" title="c2">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font> implies <font color="Maroon" title="c1">U0</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c1">U0</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c2">U1</font>
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c2">U1</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">U0</font> is    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c3">U2</font></span><br/>

<a NAME="E3:24"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">B2</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="c3">U2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<a NAME="E5:24"/><span class="lab"><font color="Green" title="E16">A4</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="c3">U2</font>,<font color="Maroon" title="c4">B2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<a NAME="E6:24"/>
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="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:24"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">U2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">UNIALG_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">U0</font> holds <br/>( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</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="Red">::  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="c1">U0</font> implies ( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</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="E8:24"/><span class="lab"><font color="Green" title="E17">A5</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="c1">U0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">U0</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">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</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="c2">U1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<a NAME="E10:24"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</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="c2">U1</font>,<font color="Maroon" title="c6">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<a NAME="E11:24"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
<font color="Maroon" title="c4">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="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<span class="lab"><font color="Green" title="E20">A8</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="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="c4">B2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>
<br/>.= 
 <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#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>
;<br/>
<a NAME="E13:24"/><span class="lab"><font color="Green" title="E21">A9</font></span>: 
<font color="Maroon" title="c6">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="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E22">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">consider </span><font color="Maroon" title="c8">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="E2:24_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c8">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="E3:24_2"/><span class="lab"><font color="Green" title="E24">A12</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="c8">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#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/><a NAME="E4:24_2"/><span class="lab"><font color="Green" title="E25">A13</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> <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="c4">B2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">o21</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:24_2"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">o21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c9">o21</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="c2">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T2" target="_self" title="UNIALG_2:th.2">Th2</a></span>;<br/><a NAME="E7:24_2"/><span class="lab"><font color="Green" title="E27">A15</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> <span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c2">U1</font>,<font color="Maroon" title="c6">B1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">o20</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:24_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">o20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c10">o20</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T2" target="_self" title="UNIALG_2:th.2">Th2</a></span>;<br/><a NAME="E10:24_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c7">o2</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="c3">U2</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">o2</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="c7">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">o2</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c7">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T2" target="_self" title="UNIALG_2:th.2">Th2</a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E11:24_2"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c4">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c7">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></span>;<br/><a NAME="E12:24_2"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c9">o21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c4">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E3:24_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>;<br/><a NAME="E13:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">o21</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">o2</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="c7">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>;<br/><a NAME="E14:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <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="c9">o21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E10:24_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="unialg_2.html#T1" target="_self" title="UNIALG_2:th.1">Th1</a>, <a class="ref" href="finseq_2.html#T130" title="FINSEQ_2:th.130">FINSEQ_2:130</a></span>;<br/><a NAME="E15:24_2"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<font color="Maroon" title="c6">B1</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c9">o21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></span>;<br/><a NAME="E16:24_2"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c10">o20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o21</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c6">B1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E5:24"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E2:24_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E4:24_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E7:24_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>;<br/><a NAME="E17:24_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">o20</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">o21</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="c9">o21</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:24_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>;<br/><a NAME="E18:24_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c10">o20</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c9">o21</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="c2">U1</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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="c9">o21</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E6:24_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/><a NAME="E19:24_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A23</font></span>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c10">o20</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="c9">o21</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#T1" target="_self" title="UNIALG_2:th.1">Th1</a>, <a class="ref" href="finseq_2.html#T130" title="FINSEQ_2:th.130">FINSEQ_2:130</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c5">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</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="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="c11">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">reconsider </span><font color="Maroon" title="c12">s1</font> = <font color="Maroon" title="c11">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c6">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="#E8:24"><span class="lab"><font color="Green" title="E17">A5</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">reconsider </span><font color="Maroon" title="c13">s0</font> = <font color="Maroon" title="c11">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E3:24_2_1"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">o20</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="txt" href="#E8:24"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E4:24_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">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="E5:24_2_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:24_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">s</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">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="c14">s2</font> = <font color="Maroon" title="c11">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c4">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">assume </span><a NAME="E8:24_2_1"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">s</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="c7">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">B</font></span><br/><a NAME="E9:24_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c4">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="c7">o2</font> </span>}</span> 
 ;<br/><a NAME="E10:24_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c14">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="c7">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B2</font>
 <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="E11:24_2_1"/>
<font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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 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">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="c10">o20</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:24_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E8:24_2_1"><span class="lab"><font color="Green" title="E37">A25</font></span></a></span>;<br/><a NAME="E12:24_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c10">o20</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>
 <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="E13:24_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
<font color="Maroon" title="c10">o20</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">o20</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E14:24_2_1"/>
<font color="Maroon" title="c12">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">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">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="c9">o21</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E8:24_2_1"><span class="lab"><font color="Green" title="E37">A25</font></span></a></span>;<br/><a NAME="E15:24_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A28</font></span>: 
<font color="Maroon" title="c12">s1</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="c9">o21</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font>
 <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/><font color="Maroon" title="c10">o20</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">o21</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="c9">o21</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font></span>)</span></span>)</span> <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="#E15:24_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E16:24_2"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">o21</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="#E15:24_2_1"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">o2</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="c7">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">B2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E12:24_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c7">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_2_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/><span class="kw">hence </span><a NAME="E17:24_2_1"/>
<font color="Maroon" title="c7">o2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="#E13:24_2_1"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="txt" href="#E3:24_2_1"><span class="lab"><font color="Green" title="E36">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E21:24_2"/>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E23">A29</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="c1">U0</font> = 
 <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="c2">U1</font>,<font color="Maroon" title="c6">B1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></span>
<br/>.= 
 <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>
;<br/>
<a NAME="E16:24"/><span class="lab"><font color="Green" title="E24">A30</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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>;<br/>
<a NAME="E17:24"/><span class="lab"><font color="Green" title="E25">A31</font></span>: 
<font color="Maroon" title="c5">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E17">A5</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_4"><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="Red">::  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> <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> implies <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_4"/><span class="lab"><font color="Green" title="E26">A32</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> <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">o21</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E4:24_4"/><span class="lab"><font color="Green" title="E27">A33</font></span>: 
<font color="Maroon" title="c4">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></span>;<br/><a NAME="E5:24_4"/><span class="lab"><font color="Green" title="E28">A34</font></span>: 
<font color="Maroon" title="c6">B1</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c9">o21</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E21">A9</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></span>;<br/><span class="kw">thus </span><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> = 
<font color="Maroon" title="c8">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="#E1:24_4"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c8">o2</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">o2</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="#E14:24"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c8">o2</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">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">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">o2</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>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E25">A31</font></span></a>, <a class="ref" href="unialg_2.html#T1" target="_self" title="UNIALG_2:th.1">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">o2</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">o2</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c4">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">o2</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="ref" href="relat_1.html#T100" title="RELAT_1:th.100">RELAT_1:100</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">o2</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c4">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">o2</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="#E4:24_4"><span class="lab"><font color="Green" title="E27">A33</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c9">o21</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">o2</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="#E5:24"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>
<br/>.= 
<font color="Maroon" title="c9">o21</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="c9">o21</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">B1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E8:24"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="unialg_2.html#T9" target="_self" title="UNIALG_2:th.9">Th9</a></span>
<br/>.= 
<font color="Maroon" title="c9">o21</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c6">B1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_4"><span class="lab"><font color="Green" title="E28">A34</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></span>
<br/>.= 
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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="txt" href="#E1:24_4"><span class="lab"><font color="Green" title="E26">A32</font></span></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:24"/>
the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</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="#E15:24"><span class="lab"><font color="Green" title="E23">A29</font></span></a>, <a class="txt" href="#E12:24"><span class="lab"><font color="Green" title="E20">A8</font></span></a>, <a class="txt" href="#E16:24"><span class="lab"><font color="Green" title="E24">A30</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E20:24"/>
<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="#E14:24"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
