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

<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> 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="b2">B</font>, <font color="Olive" title="b3">C</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <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="b4">F</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/>  for <font color="Olive" title="b5">G</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b3">C</font><br/>  for <font color="Olive" title="b6">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b2">B</font>  st  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Olive" title="b4">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b6">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b6">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b4">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <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="Maroon" title="c2">A</font>,<font color="Olive" title="b1">B</font><br/>  for <font color="Olive" title="b4">G</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">C</font><br/>  for <font color="Olive" title="b5">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Olive" title="b1">B</font>  st  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Olive" title="b3">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b5">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b5">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b3">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>() <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> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/>  for <font color="Olive" title="b2">G</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font><br/>  for <font color="Olive" title="b3">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c3">B</font>  st  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Olive" title="b1">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b3">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Olive" title="b1">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<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">  for <font color="Olive" title="b1">G</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font><br/>  for <font color="Olive" title="b2">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c3">B</font>  st  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">X</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b2">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">G</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</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">X</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <font color="Maroon" title="c3">B</font>  st  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">X</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Olive" title="b1">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>()  <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedSubset</a> of <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"> (  <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> )</span><br/>













<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c7">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span><br/>

<a NAME="E2:12"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> <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="E3:12"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c5">F</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">X</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">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="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K19</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K20</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="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>)) )</span><br/>

<span class="kw">assume </span><a NAME="E1:12_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c8">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"> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K19</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K20</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="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>))</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<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="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
<a NAME="E3:12_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<a NAME="E4:12_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c9">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="funct_6.html#T22" onmouseover="rs('funct_6/T22')" onmouseout="rh()">FUNCT_6:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:12_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">K19</a>(<a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">K20</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="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>))
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</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>
<div><span class="lab"><font color="Green" title="E6">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_2"><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="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</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"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></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="c8">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="c8">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="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/><span class="kw">assume </span><a NAME="E1:12_2"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c8">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="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<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="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/><a NAME="E3:12_2"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">rngs</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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> <font color="Maroon" title="c9">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</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="E4:12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/><a NAME="E5:12_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f9</font> = <font color="Maroon" title="c9">f</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">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E4:12_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/><a NAME="E8:12_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">f9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">gx</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="pboole.html#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<font color="Maroon" title="c12">gx</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="msafree.html#D1" onmouseover="rs('msafree/D1')" onmouseout="rh()">MSAFREE:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c11">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="ref" href="msuhom_1.html#T1" onmouseover="rs('msuhom_1/T1')" onmouseout="rh()">MSUHOM_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="msualg_3.html#T2" onmouseover="rs('msualg_3/T2')" onmouseout="rh()">MSUALG_3:2</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/></div><span class="kw">end;</span></div>
<a NAME="E5:12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> is   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="extens_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE:3</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>
