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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K2" title="FUNCT_6:func.2">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#K8" title="PBOOLE:func.8">**</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K2" title="FUNCT_6:func.2">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#K8" title="PBOOLE:func.8">**</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="funct_6.html#K2" title="FUNCT_6:func.2">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#K8" title="PBOOLE:func.8">**</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#K2" title="FUNCT_6:func.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="PBOOLE:func.8">**</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="relset_1.html#K1" title="RELSET_1:func.1">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="relset_1.html#K1" title="RELSET_1:func.1">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</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#NK1" title="RELAT_1:NK.1">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#K8" title="PBOOLE:func.8">**</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#D19" onmouseover="rs('pboole/D19')" onmouseout="rh()">PBOOLE:def 19</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E5">A4</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">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#K8" title="PBOOLE:func.8">**</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="Olive" title="b1">i</font></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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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#K8" title="PBOOLE:func.8">**</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K8" title="PBOOLE:func.8">**</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#K5" title="RELAT_1:func.5">|</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" onmouseover="rs('catalg_1/D1')" onmouseout="rh()">Def1</a></span>;<br/><a NAME="E3:3_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E4:3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">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="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/><a NAME="E5:3_1"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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></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#K5" title="RELAT_1:func.5">|</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#K5" title="RELAT_1:func.5">|</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#K6" title="RELAT_1:func.6">|`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</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#K5" title="RELAT_1:func.5">|</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#T83" onmouseover="rs('relat_1/T83')" onmouseout="rh()">RELAT_1:83</a>, <a class="ref" href="relat_1.html#T94" onmouseover="rs('relat_1/T94')" onmouseout="rh()">RELAT_1:94</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#K5" title="RELAT_1:func.5">|</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#K5" title="RELAT_1:func.5">|</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#K5" title="RELAT_1:func.5">|</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#T1" onmouseover="rs('monoid_1/T1')" onmouseout="rh()">MONOID_1:1</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#K5" title="RELAT_1:func.5">|</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#K5" title="RELAT_1:func.5">|</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" onmouseover="rs('catalg_1/D1')" onmouseout="rh()">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#K8" title="PBOOLE:func.8">**</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#D19" onmouseover="rs('pboole/D19')" onmouseout="rh()">PBOOLE:def 19</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</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#K8" title="PBOOLE:func.8">**</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#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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