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<span class="kw">let </span><font color="Maroon" title="c1">I</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">  for <font color="Olive" title="b1">M</font>, <font color="Olive" title="b2">A</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b3">F</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">M</font>,<font color="Olive" title="b1">M</font> holds <br/> ( ( <font color="Olive" title="b2">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp; <font color="Olive" title="b3">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Olive" title="b2">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">A</font> ) iff <font color="Olive" title="b2">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Olive" title="b3">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">A</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <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">F</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</font> holds <br/> ( ( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b1">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> ) iff <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Olive" title="b1">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c2">M</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="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> ) iff <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Maroon" title="c4">F</font> )</span><br/>







<span class="kw">thus </span><a NAME="E1:67"/>
( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> implies <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Maroon" title="c4">F</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="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Maroon" title="c4">F</font> implies ( <font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">and </span><br/><a NAME="E2:67_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</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="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Maroon" title="c4">F</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">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#D1" title="PBOOLE:def.1">PBOOLE:def 1</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="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:67_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" title="PBOOLE:def.15">PBOOLE:def 15</a></span>;<br/>
<a NAME="E5:67_1"/>
<font color="Maroon" title="c5">i</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="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E6:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A4</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E7:67_1"/>
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T17" title="MSSUBFAM:th.17">MSSUBFAM:17</a></span>;<br/>
<a NAME="E8:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="mssubfam.html#T14" title="MSSUBFAM:th.14">MSSUBFAM:14</a></span>;<br/>
<a NAME="E9:67_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:67_1"/>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="txt" href="#E6:67_1"><span class="lab"><font color="Green" title="E27">A4</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">assume </span><a NAME="E2:67"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="closure1.html#K5" title="CLOSURE1:func.5">MSFixPoints</a> <font color="Maroon" title="c4">F</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="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:67"/>
<font color="Maroon" title="c3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</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="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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#D1" title="PBOOLE:def.1">PBOOLE:def 1</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="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<a NAME="E1:67_2"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T17" title="MSSUBFAM:th.17">MSSUBFAM:17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:67_2"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">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="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<a NAME="E3:67_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>;<br/>
<a NAME="E4:67_2"/><span class="kw">then </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> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="ref" href="closure1.html#D12" target="_self" title="CLOSURE1:def.12">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:67_2"/>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67_2"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E1:67_2"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="mssubfam.html#T14" title="MSSUBFAM:th.14">MSSUBFAM:14</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="hs2(this)" href="javascript:()" title="67_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</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="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:67_3"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</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"><font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/><a NAME="E2:67_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <font color="Maroon" title="c4">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>;<br/><a NAME="E3:67_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A9</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> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_3"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="closure1.html#D12" target="_self" title="CLOSURE1:def.12">Def13</a></span>;<br/><a NAME="E4:67_3"/>
<font color="Maroon" title="c5">i</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="c4">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_3"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:67_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67_3"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</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="E5:67"/>
<font color="Maroon" title="c4">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c3">A</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
