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

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E2:10_1"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E3:10_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c2">f</font> = <font color="Maroon" title="c1">f</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() holds <br/><font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</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="Red">::  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">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() implies <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E5:10_1"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span><br/>

<a NAME="E6:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:10_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:10_1"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="txt" href="#E6:10_1"><span class="lab"><font color="Green" title="E7">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
