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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span>( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp;  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp; (  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="c1">U0</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/><font color="Olive" title="b1">B</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>  ) );<br/>
<span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="xfamily.html#S1" onmouseover="rs('xfamily/S1')" onmouseout="rh()">XFAMILY:sch 1</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c5">P</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">X</font>;<br/>
<a NAME="E3:32_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font> &amp; (  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="c1">U0</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/><font color="Olive" title="b1">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="ref" href="unialg_2.html#T4" target="_self" onmouseover="rs('unialg_2/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="zfmisc_1.html#D1" onmouseover="rs('zfmisc_1/D1')" onmouseout="rh()">ZFMISC_1:def 1</a></span>;<br/>
<a NAME="E4:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">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> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E5:32_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font></span><br/><a NAME="E2:32_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E7:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="setfam_1.html#T5" onmouseover="rs('setfam_1/T5')" onmouseout="rh()">SETFAM_1:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">P</font> =  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c4">X</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="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E5:32_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c7">E</font> =  <a href="unialg_2.html#K4" title="UNIALG_2:func.4">UniAlgSetClosed</a> <font color="Maroon" title="c6">P</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</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="c7">E</font> &amp; (  for <font color="Olive" title="b1">U1</font> being    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Maroon" title="c2">A</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="b1">U1</font> holds <br/><font color="Maroon" title="c7">E</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Olive" title="b1">U1</font> ) )</span><br/>

<a NAME="E9:32_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c6">P</font> is  <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">o</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U0</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D4" target="_self" onmouseover="rs('unialg_2/D4')" onmouseout="rh()">UNIALG_2:def 4</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">P</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c6">P</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D3" target="_self" onmouseover="rs('unialg_2/D3')" onmouseout="rh()">UNIALG_2:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c8">o</font> implies <font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:32_1_1_2"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c8">o</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">a</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">a</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">a</font> )</span><br/><a NAME="E1:32_1_1_2_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/><span class="kw">assume </span><a NAME="E2:32_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c10">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">a</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">a0</font> = <font color="Maroon" title="c10">a</font> as   <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="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/><a NAME="E4:32_1_1_2_1"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">a0</font> &amp;  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">a0</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E2:32_1_1_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">a0</font> = <font color="Maroon" title="c11">a0</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="txt" href="#E1:32"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>;<br/><a NAME="E6:32_1_1_2_1"/>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">a0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a></span>;<br/><a NAME="E7:32_1_1_2_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">a0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_2_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">s0</font> = <font color="Maroon" title="c9">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c12">a0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/><a NAME="E9:32_1_1_2_1"/>
<font color="Maroon" title="c12">a0</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="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E2:32_1_1_2_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>;<br/><a NAME="E10:32_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">a0</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c8">o</font>
 ;<br/><a NAME="E11:32_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">a0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1_2"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E12:32_1_1_2_1"/>
<font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:32_1_1_2"/>
<font color="Maroon" title="c8">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" onmouseover="rs('setfam_1/D1')" onmouseout="rh()">SETFAM_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E10:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c7">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c6">P</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> (<font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c6">P</font>)</span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D8" target="_self" onmouseover="rs('unialg_2/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E11:32_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:32_1_1"/>
<font color="Maroon" title="c2">A</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="c7">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> being    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Maroon" title="c2">A</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="b1">U1</font> holds <br/><font color="Maroon" title="c7">E</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Olive" title="b1">U1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">U1</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</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="c8">U1</font> implies <font color="Maroon" title="c7">E</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c8">U1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E13:32_1_1"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<font color="Maroon" title="c2">A</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="c8">U1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">E</font> is    <a href="unialg_2.html#M1" title="UNIALG_2:mode.1">SubAlgebra</a> of <font color="Maroon" title="c8">U1</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c9">u1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font>;<br/>
<a NAME="E14:32_1_1"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c10">B</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">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" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:32_1_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font><span class="kw"> such that </span><br/><a NAME="E4:32_1_1_3"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E5:32_1_1_3"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 ex <font color="Olive" title="b1">o</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U0</font> st <br/>(  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">o</font> )
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">o0</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c1">U0</font><span class="kw"> such that </span><br/><a NAME="E7:32_1_1_3"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
 <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:32_1_1_3"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c13">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_3"><span class="lab"><font color="Green" title="E26">A14</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:32_1_1_3"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c13">o0</font>
 <span class="kw">and </span><br/><a NAME="E11:32_1_1_3"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_3"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c13">o0</font> = 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="txt" href="#E7:32_1_1_3"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="margrel1.html#T22" onmouseover="rs('margrel1/T22')" onmouseout="rh()">MARGREL1:22</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T94" onmouseover="rs('finseq_2/T94')" onmouseout="rh()">FINSEQ_2:94</a></span>
;<br/>
<a NAME="E13:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c10">B</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1_3"><span class="lab"><font color="Green" title="E29">A17</font></span></a></span>;<br/>
<a NAME="E14:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T94" onmouseover="rs('finseq_2/T94')" onmouseout="rh()">FINSEQ_2:94</a></span>;<br/>
<a NAME="E15:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c13">o0</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="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c10">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_1_3"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E10:32_1_1_3"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E16:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E18:32_1_1_3"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="tarski.html#R2" title="TARSKI: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>
 <span class="kw">and </span><br/><a NAME="E19:32_1_1_3"/><span class="lab"><font color="Green" title="E33">A21</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="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" onmouseover="rs('finseq_2/T10')" onmouseout="rh()">FINSEQ_2:10</a></span>;<br/>
<a NAME="E20:32_1_1_3"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c15">n</font> <a href="tarski.html#R2" title="TARSKI: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="c8">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:32_1_1_3"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="unialg_2.html#T7" target="_self" onmouseover="rs('unialg_2/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">o1</font> =  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c8">U1</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c15">n</font> as   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c8">U1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E22:32_1_1_3"/>
 the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c8">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="c10">B</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E23:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c16">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c10">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:32_1_1_3"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E20:32_1_1_3"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E24:32_1_1_3"/>
<font color="Maroon" title="c10">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="ref" href="unialg_2.html#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E25:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o0</font>
 
;<br/>
<a NAME="E26:32_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c10">B</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_1_1_3"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<a NAME="E27:32_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<font color="Maroon" title="c16">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32_1_1_3"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E28:32_1_1_3"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">o1</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="c8">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<font color="Maroon" title="c16">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c10">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32_1_1_3"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E25:32_1_1_3"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c12">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_1_3"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E16:32_1_1_3"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E30:32_1_1_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E27:32_1_1_3"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="txt" href="#E28:32_1_1_3"><span class="lab"><font color="Green" title="E38">A26</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E15:32_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> &amp; (  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="c1">U0</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="c8">U1</font> holds <br/><font color="Olive" title="b1">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="ref" href="unialg_2.html#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E16:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A27</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">U1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E13:32_1_1"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E14:32_1_1"><span class="lab"><font color="Green" title="E24">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:32_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c8">U1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="unialg_2.html#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">UNIALG_2:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c8">U1</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="c7">E</font> holds <br/>(  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">E</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="c8">U1</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="c10">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="c8">U1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">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="c7">E</font> implies (  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">E</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="c8">U1</font>,<font color="Maroon" title="c10">B</font>) &amp; <font color="Maroon" title="c10">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="E18:32_1_1"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
<font color="Maroon" title="c10">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="c7">E</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c7">E</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="c8">U1</font>,<font color="Maroon" title="c10">B</font>) &amp; <font color="Maroon" title="c10">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="c11">u11</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">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" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E20:32_1_1"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
 the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c8">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="c11">u11</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D7" target="_self" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<a NAME="E21:32_1_1"/><span class="lab"><font color="Green" title="E28">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="c1">U0</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="c1">U0</font>,<font color="Maroon" title="c11">u11</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E22:32_1_1"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c11">u11</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" onmouseover="rs('unialg_2/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E30">A32</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">o1</font> being   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c8">U1</font> holds  <font color="Maroon" title="c10">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Olive" title="b1">o1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">o1</font> be   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c8">U1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c12">o1</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c13">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E2:32_1_1_4"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c13">n</font> <a href="tarski.html#R2" title="TARSKI: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="c8">U1</font>
 <span class="kw">and </span><br/><a NAME="E3:32_1_1_4"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<font color="Maroon" title="c12">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c8">U1</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c13">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" onmouseover="rs('finseq_2/T10')" onmouseout="rh()">FINSEQ_2:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">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="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c13">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="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E2:32_1_1_4"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:32_1_1_4"/><span class="lab"><font color="Green" title="E33">A35</font></span>: 
 <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c12">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1_4"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E3:32_1_1_4"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="unialg_2.html#T6" target="_self" onmouseover="rs('unialg_2/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E6:32_1_1_4"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
<font color="Maroon" title="c11">u11</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c14">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:32_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_4_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c10">B</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c12">o1</font> holds <br/><font color="Maroon" title="c12">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c10">B</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c12">o1</font> implies <font color="Maroon" title="c12">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font> )</span><br/><a NAME="E1:32_1_1_4_1"/><span class="lab"><font color="Green" title="E35">A37</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c14">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">sE</font> = <font color="Maroon" title="c15">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c6">P</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="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/><a NAME="E3:32_1_1_4_1"/>
<font color="Maroon" title="c15">s</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c11">u11</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T24" onmouseover="rs('finseq_2/T24')" onmouseout="rh()">FINSEQ_2:24</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">s1</font> = <font color="Maroon" title="c15">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c11">u11</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" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/><span class="lab"><font color="Green" title="E36">A38</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c14">o0</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="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font></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="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T22" onmouseover="rs('margrel1/T22')" onmouseout="rh()">MARGREL1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T21" onmouseover="rs('margrel1/T21')" onmouseout="rh()">MARGREL1:21</a></span>
;<br/><span class="kw">assume </span><a NAME="E6:32_1_1_4_1"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c12">o1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font></span><br/><a NAME="E7:32_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c11">u11</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">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_4"><span class="lab"><font color="Green" title="E33">A35</font></span></a></span>;<br/><a NAME="E8:32_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A40</font></span>: 
<font color="Maroon" title="c17">s1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_4_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="finseq_2.html#D4" onmouseover="rs('finseq_2/D4')" onmouseout="rh()">FINSEQ_2:def 4</a></span>;<br/><font color="Maroon" title="c12">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c11">u11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E2:32_1_1_4"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E3:32_1_1_4"><span class="lab"><font color="Green" title="E32">A34</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_4"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c14">o0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">sE</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1_4_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
;<br/><span class="kw">hence </span><a NAME="E10:32_1_1_4_1"/>
<font color="Maroon" title="c12">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E5:32_1_1_4"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="txt" href="#E6:32_1_1_4_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E1:32_1_1_4_1"><span class="lab"><font color="Green" title="E35">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:32_1_1_4"/>
<font color="Maroon" title="c10">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c12">o1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E24:32_1_1"/><span class="lab"><font color="Green" title="E31">A41</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="c8">U1</font>,<font color="Maroon" title="c10">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="c8">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E25:32_1_1"/><span class="lab"><font color="Green" title="E32">A42</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="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="c1">U0</font>,<font color="Maroon" title="c6">P</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>;<br/>
<a NAME="E26:32_1_1"/><span class="lab"><font color="Green" title="E33">A43</font></span>: 
<font color="Maroon" title="c6">P</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="c8">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:32_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI: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> holds <br/> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">E</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Olive" title="b1">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="c8">U1</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI: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="c7">E</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">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="c8">U1</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">n</font> )</span><br/><span class="kw">assume </span><a NAME="E1:32_1_1_5"/><span class="lab"><font color="Green" title="E34">A44</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c7">E</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">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="c8">U1</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">n</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">o0</font> =  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">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" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">o1</font> =  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c8">U1</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">n</font> as   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c8">U1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><a NAME="E4:32_1_1_5"/><span class="lab"><font color="Green" title="E35">A45</font></span>: 
<font color="Maroon" title="c11">u11</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:32_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a></span>;<br/><a NAME="E5:32_1_1_5"/><span class="lab"><font color="Green" title="E36">A46</font></span>: 
<font color="Maroon" title="c6">P</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></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="c7">E</font> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">n</font> = 
<font color="Maroon" title="c13">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c6">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E25:32_1_1"><span class="lab"><font color="Green" title="E32">A42</font></span></a>, <a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c13">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_5"><span class="lab"><font color="Green" title="E36">A46</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c13">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">P</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:32_1_1"><span class="lab"><font color="Green" title="E33">A43</font></span></a>, <a class="ref" href="margrel1.html#T21" onmouseover="rs('margrel1/T21')" onmouseout="rh()">MARGREL1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">u11</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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" onmouseover="rs('relat_1/T71')" onmouseout="rh()">RELAT_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c11">u11</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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_1_1_5"><span class="lab"><font color="Green" title="E35">A45</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<font color="Maroon" title="c14">o1</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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c13">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c6">P</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c14">o1</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#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c14">o1</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c10">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="unialg_2.html#T6" target="_self" onmouseover="rs('unialg_2/T6')" onmouseout="rh()">Th6</a></span>
<br/>.= 
<font color="Maroon" title="c14">o1</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c10">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:32_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="unialg_2.html#D5" target="_self" onmouseover="rs('unialg_2/D5')" onmouseout="rh()">Def5</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="c8">U1</font>,<font color="Maroon" title="c10">B</font>)</span>)</span> <a href="partfun1.html#K8" title="PARTFUN1:func.8">.</a> <font color="Maroon" title="c12">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E24:32_1_1"><span class="lab"><font color="Green" title="E31">A41</font></span></a>, <a class="txt" href="#E1:32_1_1_5"><span class="lab"><font color="Green" title="E34">A44</font></span></a>, <a class="ref" href="unialg_2.html#D6" target="_self" onmouseover="rs('unialg_2/D6')" onmouseout="rh()">Def6</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E28:32_1_1"/>
(  the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">E</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="c8">U1</font>,<font color="Maroon" title="c10">B</font>) &amp; <font color="Maroon" title="c10">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="#E10:32_1_1"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E20:32_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E25:32_1_1"><span class="lab"><font color="Green" title="E32">A42</font></span></a>, <a class="txt" href="#E21:32_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E23:32_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E24:32_1_1"><span class="lab"><font color="Green" title="E31">A41</font></span></a>, <a class="ref" href="finseq_1.html#T13" onmouseover="rs('finseq_1/T13')" onmouseout="rh()">FINSEQ_1:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
