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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font> being   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font> holds   <a href="closure1.html#K6" title="CLOSURE1:func.6">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(#  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Olive" title="b1">D</font>, the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Olive" title="b1">D</font> #)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">D</font> be   <a href="closure1.html#NM2" title="CLOSURE1:NM.2">MSClosureSystem</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="closure1.html#K6" title="CLOSURE1:func.6">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(#  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>, the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> #)</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">M</font> =  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F</font> =  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">I</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X1</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c6">X1</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T134" title="PBOOLE:th.134">PBOOLE:134</a></span>;<br/>
<a NAME="E3:82"/>
 the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:82_1"/>
 the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D10" title="PBOOLE:def.10">PBOOLE:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:82_1_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">i</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="c1">S</font> or  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:82_1_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c7">i</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="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Fi</font> =  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:82_1_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<a NAME="E6:82_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c6">X1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">x</font></span>)</span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E8:82_1_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E9:82_1_1"/>
 the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E10:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/>
<a NAME="E11:82_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<a NAME="E12:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:82_1_1"/>
<font color="Maroon" title="c7">i</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"><font color="Maroon" title="c7">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A8</font></span>: <font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
;<br/>
<a NAME="E15:82_1_1"/>
<font color="Maroon" title="c11">X</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D14" title="PBOOLE:def.14">PBOOLE:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">s</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="c1">S</font> or <font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_1_1_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c12">s</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="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1_1_2_1"/>
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_2_1_1"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_1_2_1_1"/>
<font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E14:82_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E10:82_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_1_2_1_2"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font></span><br/><div class="add"><a NAME="E2:82_1_1_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:82_1_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:82_1_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_1_2_1_2"/>
<font color="Maroon" title="c11">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:82_1_1_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">X</font> = <font color="Maroon" title="c11">X</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">SF</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E18:82_1_1"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
 for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c13">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c12">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T31" target="_self" title="CLOSURE1:th.31">Th31</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">Q</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <span class="p1">(<span class="default"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E20:82_1_1"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E21:82_1_1"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c13">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="mssubfam.html#D1" title="MSSUBFAM:def.1">MSSUBFAM:def 1</a></span>;<br/>
<a NAME="E22:82_1_1"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c13">SF</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Fi</font> : <font color="Maroon" title="c12">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">z</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E18:82_1_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="closure1.html#T32" target="_self" title="CLOSURE1:th.32">Th32</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">Z1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q</font> implies <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Z1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:82_1_1_3"/>
<font color="Maroon" title="c15">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Z1</font></span><br/><a NAME="E2:82_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Fi</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Z1</font> &amp; <font color="Maroon" title="c12">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:82_1_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E22:82_1_1"><span class="lab"><font color="Green" title="E44">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:82_1_1_3"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E20:82_1_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="mssubfam.html#T5" title="MSSUBFAM:th.5">MSSUBFAM:5</a></span>;<br/>
<a NAME="E25:82_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">Fi</font> : <font color="Maroon" title="c10">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<a NAME="E26:82_1_1"/><span class="kw">then </span>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:82_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E20:82_1_1"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="txt" href="#E22:82_1_1"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="mssubfam.html#T2" title="MSSUBFAM:th.2">MSSUBFAM:2</a></span>;<br/>
<a NAME="E27:82_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c14">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:82_1_1"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E28:82_1_1"/>
<span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c12">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c13">SF</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:82_1_1"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="closure1.html#D14" target="_self" title="CLOSURE1:def.14">Def15</a></span>;<br/>
<a NAME="E29:82_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E14:82_1_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E21:82_1_1"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E27:82_1_1"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E1:82_1_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E30:82_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E12:82_1_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="closure1.html#D12" target="_self" title="CLOSURE1:def.12">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">i</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="c1">S</font> or <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:82_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
<font color="Maroon" title="c7">i</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="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:82_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<a NAME="E4:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A18</font></span>: 
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="closure1.html#D12" target="_self" title="CLOSURE1:def.12">Def13</a></span>;<br/>
<a NAME="E5:82_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">X1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span> <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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c6">X1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">x</font></span>)</span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E7:82_1"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E8:82_1"/>
<font color="Maroon" title="c7">i</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"><font color="Maroon" title="c7">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E38">A20</font></span>: <font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
;<br/>
<a NAME="E10:82_1"/>
 <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E11:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<span class="p1">(<span class="default"><a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/>
<a NAME="E12:82_1"/>
<font color="Maroon" title="c9">X</font> is    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D14" title="PBOOLE:def.14">PBOOLE:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">s</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="c1">S</font> or <font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:82_1_3"/><span class="lab"><font color="Green" title="E40">A22</font></span>: 
<font color="Maroon" title="c10">s</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="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:82_1_3_1"/>
( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> or <font color="Maroon" title="c10">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:82_1_3_1_1"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:82_1_3_1_1"/>
<font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:82_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E9:82_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E11:82_1"><span class="lab"><font color="Green" title="E39">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="82_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:82_1_3_1_2"/>
<font color="Maroon" title="c10">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span><br/><div class="add"><a NAME="E2:82_1_3_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c8">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:82_1"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:82_1_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">X1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:82_1_3_1_2"/>
<font color="Maroon" title="c9">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E1:82_1_3"><span class="lab"><font color="Green" title="E40">A22</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">X</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">Element</a> of  <a href="mssubfam.html#K5" title="MSSUBFAM:func.5">bool</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font> ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>] <span class="kw">means </span><font color="Maroon" title="c10">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">SF</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="mssubfam.html#NM1" title="MSSUBFAM:NM.1">MSSubsetFamily</a> of  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font><span class="kw"> such that </span><br/><a NAME="E15:82_1"/><span class="lab"><font color="Green" title="E40">A23</font></span>: 
 for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c11">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c10">X</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">Y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#T31" target="_self" title="CLOSURE1:th.31">Th31</a></span>;<br/>
<a NAME="E16:82_1"/><span class="lab"><font color="Green" title="E41">A24</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E17:82_1"/>
( (  for <font color="Olive" title="b1">Y</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c11">SF</font> iff ( <font color="Olive" title="b1">Y</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Y</font>] ) ) ) implies <font color="Maroon" title="c11">SF</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="closure1.html#S1" target="_self" title="CLOSURE1:sch.1">CLOSURE1:sch 1</a>();<br/></span>
<a NAME="E18:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A25</font></span>: 
 <a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c11">SF</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="mssubfam.html#D5" title="MSSUBFAM:def.5">MSSUBFAM:def 5</a></span>;<br/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K4" title="MSSUBFAM:func.4">meet</a> <font color="Maroon" title="c11">SF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="closure1.html#K2" title="CLOSURE1:func.2">..</a> <font color="Maroon" title="c10">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:82_1"><span class="lab"><font color="Green" title="E40">A23</font></span></a>, <a class="ref" href="closure1.html#D14" target="_self" title="CLOSURE1:def.14">Def15</a></span>
<br/>.= 
<font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E4:82_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E9:82_1"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="txt" href="#E16:82_1"><span class="lab"><font color="Green" title="E41">A24</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:82_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82_1"><span class="lab"><font color="Green" title="E34">A16</font></span></a>, <a class="txt" href="#E18:82_1"><span class="lab"><font color="Green" title="E42">A25</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:82"/>
 <a href="closure1.html#K6" title="CLOSURE1:func.6">ClOp-&gt;ClSys</a> <span class="p1">(<span class="default"><a href="closure1.html#K7" title="CLOSURE1:func.7">ClSys-&gt;ClOp</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="closure1.html#G1" title="CLOSURE1:aggr.1">MSClosureStr</a>(#  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">D</font>, the <a href="closure1.html#U1" title="CLOSURE1:sel.1">Family</a> of <font color="Maroon" title="c2">D</font> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="closure1.html#D13" target="_self" title="CLOSURE1:def.13">Def14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
