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

<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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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>, <font color="Olive" title="b4">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">A</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b2">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:3"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 <a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E2:3"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
<font color="Maroon" title="c1">I</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
;<br/>
<a NAME="E3:3"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E4:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E5">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
<font color="Maroon" title="c6">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E2:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="catalg_1.html#D1" target="_self" title="CATALG_1:def.1">Def1</a></span>;<br/><a NAME="E3:3_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E4:3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_6.html#K3" title="FUNCT_6:func.3">rngs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="funct_6.html#T31" title="FUNCT_6:th.31">FUNCT_6:31</a></span>;<br/><a NAME="E5:3_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></span>;<br/><a NAME="E6:3_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="relat_1.html#K8" title="RELAT_1:func.8">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T112" title="RELAT_1:th.112">RELAT_1:112</a>, <a class="ref" href="relat_1.html#T125" title="RELAT_1:th.125">RELAT_1:125</a></span>;<br/><a NAME="E7:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_1.html#T2" title="MONOID_1:th.2">MONOID_1:2</a></span>;<br/><a NAME="E8:3_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="catalg_1.html#D1" target="_self" title="CATALG_1:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E8">A7</font></span></a>, <a class="ref" href="pboole.html#D24" title="PBOOLE:def.24">PBOOLE:def 24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">B</font></span>)</span> <a href="pboole.html#K10" title="PBOOLE:func.10">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="catalg_1.html#K1" title="CATALG_1:func.1">-MSF</a> <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E5">A4</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
