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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">Y</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>





<b>assume </b><a NAME="E1:144"/><i><font color="Green" title="E77">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font> iff <font color="Olive" title="b1">x</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font> )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E2:144"/><i><font color="Green" title="E78">A2</font></i>: 
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">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="pboole.html#D16" target="_self" title="PBOOLE:def.16">Def16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E4:144"/><i><font color="Green" title="E79">A3</font></i>: 
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" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S1" target="_self" title="PBOOLE:sch.1">PBOOLE:sch 1</a>(<a class="txt" href="pboole.html#E2:144"><i><font color="Green" title="E78">A2</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:144_1"/><i><font color="Green" title="E80">A4</font></i>: 
<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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c5">i</font>,$1,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/><b>consider </b><font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:144_1_1"/><i><font color="Green" title="E81">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E3:144_1_1"/><i><font color="Green" title="E82">A6</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c7">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">u</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">g</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="pboole.html#E2:144_1_1"><i><font color="Green" title="E81">A5</font></i></a>, <a class="ref" href="pboole.html#D3" target="_self" title="PBOOLE:def.3">Def3</a></i>;<br/><i><font color="Green" title="E83">A7</font></i>: <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="pboole.html#E1:144_1"><i><font color="Green" title="E80">A4</font></i></a>, <a class="txt" href="pboole.html#E3:144_1_1"><i><font color="Green" title="E82">A6</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">e</font>
<b>by </b><i><a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:144_1_1_2"/><i><font color="Green" title="E84">A8</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/><a NAME="E2:144_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:144_1_1_2_1"/><i><font color="Green" title="E85">A9</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:144_1_1_2_1_1"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:144_1_1_2_1_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:144_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="pboole.html#E5:144_1_1"><i><font color="Green" title="E83">A7</font></i></a>, <a class="txt" href="pboole.html#E1:144_1_1_2"><i><font color="Green" title="E84">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:144_1_1_2_1_1_2"/><i><font color="Green" title="E86">A10</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<b>by </b><i><a class="txt" href="pboole.html#E3:144_1_1"><i><font color="Green" title="E82">A6</font></i></a>, <a class="txt" href="pboole.html#E1:144_1_1_2_1"><i><font color="Green" title="E85">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
<b>by </b><i><a class="txt" href="pboole.html#E1:144_1_1_2_1_1_2"><i><font color="Green" title="E86">A10</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E3:144_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="pboole.html#E4:144"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="pboole.html#E1:144_1_1_2_1"><i><font color="Green" title="E85">A9</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:144_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:144_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144_1"><i><font color="Green" title="E80">A4</font></i></a>, <a class="txt" href="pboole.html#E5:144_1_1"><i><font color="Green" title="E83">A7</font></i></a>, <a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">Def4</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E7:144_1_1"/><i><font color="Green" title="E84">A11</font></i>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/><a NAME="E8:144_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="144_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E1:144_1_1_3"/><i><font color="Green" title="E85">A12</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:144_1_1_3_1"/>
( <font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:144_1_1_3_1_1"/>
<font color="Maroon" title="c9">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:144_1_1_3_1_1"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="pboole.html#E5:144_1_1"><i><font color="Green" title="E83">A7</font></i></a>, <a class="txt" href="pboole.html#E7:144_1_1"><i><font color="Green" title="E84">A11</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="144_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:144_1_1_3_1_2"/><i><font color="Green" title="E86">A13</font></i>: 
<font color="Maroon" title="c9">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font>
 ;<br/><div class="add"><a NAME="E2:144_1_1_3_1_2"/>
<font color="Maroon" title="c4">f</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="pboole.html#E4:144"><i><font color="Green" title="E79">A3</font></i></a>, <a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">Def4</a></i>;<br/><a NAME="E3:144_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E87">A14</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> = 
 <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">IFEQ</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c9">u</font>,<font color="Maroon" title="c6">e</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font></span>)</span>
<b>by </b><i><a class="txt" href="pboole.html#E3:144_1_1"><i><font color="Green" title="E82">A6</font></i></a>, <a class="txt" href="pboole.html#E1:144_1_1_3"><i><font color="Green" title="E85">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
<b>by </b><i><a class="txt" href="pboole.html#E1:144_1_1_3_1_2"><i><font color="Green" title="E86">A13</font></i></a>, <a class="ref" href="cqc_lang.html#D1" title="CQC_LANG:def.1">CQC_LANG:def 1</a></i>
;<br/><b>hence </b><a NAME="E5:144_1_1_3_1_2"/>
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">u</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144_1_1_3"><i><font color="Green" title="E85">A12</font></i></a>, <a class="txt" href="pboole.html#E3:144_1_1_3_1_2"><i><font color="Green" title="E87">A14</font></i></a>, <a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E9:144_1_1"/><b>then </b>
<font color="Maroon" title="c8">g</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a> <font color="Maroon" title="c3">X</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144"><i><font color="Green" title="E77">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E10:144_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="pboole.html#E1:144_1"><i><font color="Green" title="E80">A4</font></i></a>, <a class="txt" href="pboole.html#E5:144_1_1"><i><font color="Green" title="E83">A7</font></i></a>, <a class="ref" href="pboole.html#D4" target="_self" title="PBOOLE:def.4">Def4</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:144_1"/>
<font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:144"/>
<font color="Maroon" title="c3">X</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" target="_self" title="PBOOLE:th.3">Th3</a></i>;<br/>


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