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<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="finset_1.html#V2" title="FINSET_1:attr.2">V33</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st (  for <font color="Olive" title="b2">A</font>, <font color="Olive" title="b3">B</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <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">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp;  not <font color="Olive" title="b2">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b3">B</font> holds <br/><font color="Olive" title="b3">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">A</font> ) holds <br/> ex <font color="Olive" title="b2">m</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b2">m</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">M</font> &amp; (  for <font color="Olive" title="b3">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Olive" title="b1">M</font> holds <br/><font color="Olive" title="b3">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">m</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</a>() <a href="finset_1.html#V2" title="FINSET_1:attr.2">V33</a>() <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">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>  st <font color="Olive" title="b1">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="b2">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp;  not <font color="Olive" title="b1">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b2">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">A</font> ) implies  ex <font color="Olive" title="b1">m</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; (  for <font color="Olive" title="b2">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b2">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">m</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 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>  st <font color="Olive" title="b1">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="b2">B</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp;  not <font color="Olive" title="b1">A</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b2">B</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">m</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> st <br/>( <font color="Olive" title="b1">m</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; (  for <font color="Olive" title="b2">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b2">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Olive" title="b1">m</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( $2 <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> $1 &amp; (  for <font color="Olive" title="b1">D9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D9</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> $1 holds <br/><font color="Olive" title="b1">D9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2 ) );<br/>
<a NAME="E2:46"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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/> ex <font color="Olive" title="b2">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">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="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies  ex <font color="Olive" title="b1">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">j</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c3">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">  ex <font color="Olive" title="b1">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<a NAME="E2:46_1"/>
<font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c4">c9</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E2:46_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c4">c9</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T134" title="PBOOLE:th.134">PBOOLE:134</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">b9</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E4:46_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c5">b9</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T134" title="PBOOLE:th.134">PBOOLE:134</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">B9</font>, <font color="Maroon" title="c7">C9</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="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</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="c6">B9</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="c3">i</font> or  not <font color="Maroon" title="c7">C9</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="c3">i</font> or <font color="Maroon" title="c6">B9</font>,<font color="Maroon" title="c7">C9</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:46_1_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c6">B9</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="c3">i</font>
 <span class="kw">and </span><br/><a NAME="E6:46_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c7">C9</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="c3">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="c6">B9</font>,<font color="Maroon" title="c7">C9</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/>

<a NAME="E7:46_1_1"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">qc</font> = <font color="Maroon" title="c4">c9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</font></span>)</span>;<br/>
<a NAME="E8:46_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</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="#E1:46_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">qc</font> = <font color="Maroon" title="c4">c9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</font></span>)</span> as   <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="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E10:46_1_1"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E11:46_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E15">A10</font></span>: <font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c7">C9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
;<br/>
<a NAME="E13:46_1_1"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c9">qc</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">j</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="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_2"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
<font color="Maroon" title="c10">j</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="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_1_1_2_1_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:46_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:46_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E12:46_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:46_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font></span><br/><div class="add"><a NAME="E2:46_1_1_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c7">C9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:46_1_1"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:46_1_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:46_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:46_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:46_1_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:46_1_1_2"/>
<font color="Maroon" title="c9">qc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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="c10">j</font>
 ; <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">set </span><font color="Maroon" title="c10">qb</font> = <font color="Maroon" title="c5">b9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</font></span>)</span>;<br/>
<a NAME="E14:46_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</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="#E1:46_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="pzfmisc1.html#T1" title="PZFMISC1:th.1">PZFMISC1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">qb</font> = <font color="Maroon" title="c5">b9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</font></span>)</span> as   <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="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<span class="kw">assume </span><a NAME="E16:46_1_1"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
 not <font color="Maroon" title="c6">B9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">C9</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</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="c7">C9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B9</font></span><br/>

<a NAME="E17:46_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A14</font></span>: <font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c6">B9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
;<br/>
<a NAME="E19:46_1_1"/>
<font color="Maroon" title="c11">qb</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">j</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="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1_1_4"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c12">j</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="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_1_1_4_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> or <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_1"><span class="kw">case </span></a><a NAME="E1:46_1_1_4_1_1_1"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_1_4_1_1_1"/>
<font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:46_1_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E18:46_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_4_1_1_2"><span class="kw">case </span></a><a NAME="E1:46_1_1_4_1_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font></span><br/><div class="add"><a NAME="E2:46_1_1_4_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c6">B9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:46_1_1"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:46_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>;<br/><span class="kw">hence </span><a NAME="E4:46_1_1_4_1_1_2"/>
<font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E1:46_1_1_4"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:46_1_1_4"/>
<font color="Maroon" title="c11">qb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c12">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:46_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">qb</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c9">qc</font> or <font color="Maroon" title="c9">qc</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c11">qb</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E13:46_1_1"><span class="lab"><font color="Green" title="E16">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:46_1_1"/>
<font color="Maroon" title="c7">C9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E16:46_1_1"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E18:46_1_1"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="txt" href="#E12:46_1_1"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</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">then </span><span class="kw">consider </span><font color="Maroon" title="c4">m9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:46_1"/><span class="lab"><font color="Green" title="E9">A16</font></span>: 
( <font color="Maroon" title="c4">m9</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="c3">i</font> &amp; (  for <font color="Olive" title="b1">D9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">D9</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="c3">i</font> holds <br/><font color="Olive" title="b1">D9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">m9</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="mssubfam.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="finset_1.html#T12" title="FINSET_1:th.12">FINSET_1:12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">m9</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">m9</font>]</span><br/>

<span class="kw">thus </span><a NAME="E5:46_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">m9</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1"><span class="lab"><font color="Green" title="E9">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">m</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E4:46"/><span class="lab"><font color="Green" title="E8">A17</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E7">A2</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">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">m</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> &amp; (  for <font color="Olive" title="b1">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b1">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">m</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:46"/>
<font color="Maroon" title="c3">m</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">  for <font color="Olive" title="b1">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b1">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">m</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">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">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c4">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_2"/>
<font color="Maroon" title="c4">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">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c4">i</font></span><br/>

<span class="kw">hence </span><a NAME="E2:46_2"/>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46"><span class="lab"><font color="Green" title="E8">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E6:46"/>
 for <font color="Olive" title="b1">K</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> holds <br/><font color="Olive" title="b1">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">m</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">K</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"> ( <font color="Maroon" title="c4">K</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">M</font> implies <font color="Maroon" title="c4">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_3"/><span class="lab"><font color="Green" title="E9">A18</font></span>: 
<font color="Maroon" title="c4">K</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">K</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">m</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#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">m</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="E2:46_3"/><span class="lab"><font color="Green" title="E10">A19</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="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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="E3:46_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">K</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="#E1:46_3"><span class="lab"><font color="Green" title="E9">A18</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="E4:46_3"/>
<font color="Maroon" title="c4">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">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="#E4:46"><span class="lab"><font color="Green" title="E8">A17</font></span></a>, <a class="txt" href="#E2:46_3"><span class="lab"><font color="Green" title="E10">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
