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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  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> ;<br/><b>let </b><font color="Maroon" title="c3">C</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">A</font>;<br/><b>let </b><font color="Maroon" title="c4">InpFs</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c5">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B</font>;<br/>







<a NAME="E1:23"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">InpFs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K5" title="FUNCT_5:func.5">curry'</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_6.html#D12" title="FUNCT_6:def.12">FUNCT_6:def 12</a></i>;<br/>
<a NAME="E2:23"/><i><font color="Green" title="E6">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">InpFs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:23"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span> <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" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:23_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</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"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c6">i</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"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:23_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c7">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="c4">InpFs</font>
 <b>and </b><br/><a NAME="E5:23_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">InpFs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>and </b><br/><a NAME="E6:23_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c9">y</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="c8">g</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D4" title="FUNCT_5:def.4">FUNCT_5:def 4</a></i>;<br/>
<a NAME="E7:23_1_1"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</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="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E8:23_1_1"/>
<font color="Maroon" title="c8">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="pre_circ.html#E5:23_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E9:23_1_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E7:23_1_1"><i><font color="Green" title="E11">A7</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="E10:23_1_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<b>hence </b><a NAME="E11:23_1_1"/>
<font color="Maroon" title="c6">i</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" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E3:23_1_1"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="pre_circ.html#E6:23_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">i1</font>, <font color="Maroon" title="c7">i2</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="relat_1.html#D3" title="RELAT_1:def.3">RELAT_1:def 3</a><br/>

<b>assume </b><a NAME="E2:23_1"/><i><font color="Green" title="E7">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<a NAME="E3:23_1"/><b>then </b><i><font color="Green" title="E8">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E4:23_1"/><i><font color="Green" title="E9">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">InpFs</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E2:23_1"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c4">InpFs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E6:23_1"/><i><font color="Green" title="E10">A11</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span> <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="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E7:23_1"/>
<font color="Maroon" title="c8">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span>,<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E8:23_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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> </span>)</span>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E6:23_1"><i><font color="Green" title="E10">A11</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="E9:23_1"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E10:23_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">g</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23_1"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E11:23_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i1</font>,<font color="Maroon" title="c7">i2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span>
 <b>by </b><i><a class="txt" href="pre_circ.html#E3:23_1"><i><font color="Green" title="E8">A9</font></i></a>, <a class="txt" href="pre_circ.html#E4:23_1"><i><font color="Green" title="E9">A10</font></i></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:23"/><i><font color="Green" title="E7">A12</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K5" title="FUNCT_5:func.5">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 <b>and </b><br/><a NAME="E6:23"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span>
 <b>and </b><br/><a NAME="E7:23"/><i><font color="Green" title="E8">A13</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">A</font> holds <br/><font color="Maroon" title="c6">g</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">b</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T39" title="FUNCT_5:th.39">FUNCT_5:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="pre_circ.html#E1:23"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="pre_circ.html#E5:23"><i><font color="Green" title="E7">A12</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">c</font>
;<br/>

<b>thus </b><a NAME="E9:23"/>
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">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#D4" title="PBOOLE:def.4">PBOOLE:def 4</a><br/>

<b>assume </b><a NAME="E10:23"/><i><font color="Green" title="E9">A14</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c4">InpFs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E12:23"/><i><font color="Green" title="E10">A15</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="txt" href="pre_circ.html#E10:23"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/>
<a NAME="E13:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E10:23"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:23"/><b>then </b><i><font color="Green" title="E11">A16</font></i>: 
<font color="Maroon" title="c9">h</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E10:23"><i><font color="Green" title="E9">A14</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/>
<a NAME="E15:23"/><b>then </b><i><font color="Green" title="E12">A17</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="pre_circ.html#E12:23"><i><font color="Green" title="E10">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry</a> <font color="Maroon" title="c4">InpFs</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="pre_circ.html#E1:23"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="pre_circ.html#E5:23"><i><font color="Green" title="E7">A12</font></i></a>, <a class="txt" href="pre_circ.html#E7:23"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="pre_circ.html#E10:23"><i><font color="Green" title="E9">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="txt" href="pre_circ.html#E2:23"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pre_circ.html#E10:23"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="pre_circ.html#E15:23"><i><font color="Green" title="E12">A17</font></i></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></i>
;<br/>
<b>hence </b><a NAME="E17:23"/>
<font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="pre_circ.html#E12:23"><i><font color="Green" title="E10">A15</font></i></a>, <a class="txt" href="pre_circ.html#E14:23"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>


</div>
