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<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>  =  <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">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="c6">X</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="E2:29_1_1"/><i><font color="Green" title="E11">A1</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="c6">X</font> <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> <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#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<a NAME="E3:29_1_1"/>
<font color="Maroon" title="c6">X</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</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#D18" title="PBOOLE:def.18">PBOOLE:def 18</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<a NAME="E2:29_1_1_1"/><b>then </b><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E3:29_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:29_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="msualg_9.html#E1:29_1_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Bi</font> = <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="msualg_9.html#E1:29_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Xi</font> = <font color="Maroon" title="c6">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="msualg_9.html#E3:29_1_1_1"><i><font color="Green" title="E14">A4</font></i></a></i>;<br/>
<a NAME="E6:29_1_1_1"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:29_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E7:29_1_1_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <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="c7">i</font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E2:29_1_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:29_1_1_1"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">Xi</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="c7">i</font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>;<br/>
<a NAME="E9:29_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Bi</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">q</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:29_1_1_1_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">Xi</font>
 ;<br/>

<b>assume </b><a NAME="E2:29_1_1_1_1"/><i><font color="Green" title="E18">A8</font></i>: 
not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Bi</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:29_1_1_1_1"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">Xi</font>
 <b>and </b><br/><a NAME="E5:29_1_1_1_1"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c9">Xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E1:29_1_1_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:29_1_1_1_1"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="msualg_9.html#E4:29_1_1_1_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>;<br/>
<a NAME="E7:29_1_1_1_1"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c9">Xi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="msualg_9.html#E3:29_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>;<br/>
<a NAME="E8:29_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="pboole.html#K10" title="PBOOLE:func.10">[|</a><span class="default"><font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span><a href="pboole.html#K10" title="PBOOLE:func.10">|]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E6:29_1_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E9:29_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A13</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p2">(<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="c7">i</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E1:29_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="pboole.html#D21" title="PBOOLE:def.21">PBOOLE:def 21</a></i>;<br/>
<a NAME="E10:29_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="msualg_9.html#E6:29_1_1_1_1"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E11:29_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:29_1_1_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="msualg_9.html#E5:29_1_1_1_1"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="msualg_9.html#E7:29_1_1_1_1"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="msualg_9.html#E9:29_1_1_1_1"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:29_1_1_1"/>
<font color="Maroon" title="c6">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="msualg_9.html#E8:29_1_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c7">X</font>
;<br/>

<b>thus </b><a NAME="E5:29_1_1"/>
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="c7">X</font> <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>  <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="msualg_9.html#E2:29_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/>


</div>
